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Table of ContentsClick on a topic to go to that section
Area of Rectangles
Area of Irregular Figures Area of Shaded Regions
Area of ParallelogramsArea of Right Triangles
Area of TrapezoidsMixed Review
3-Dimensional Solids
Surface AreaVolume
More Polygons in the Coordinate PlaneGlossary & Standards
Surface Area and Volume Application Problems
Area of Acute and Obtuse Triangles
Nets
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Area is:
10 ft
5 ft
Areathe number of square units (units2) it takes to cover the surface of a figure.
ALWAYS label units2!!!
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How many 1 ft2 tiles does it take to cover the rectangle?
Use the squares to find out!
Look for a faster way than covering the whole figure.
10 ft
5 ft
Area Practice
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A = length(width)A = lw
A = side(side)A = s2
The Area (A) of a rectangle is found by using the formula:
The Area (A) of a square is found by using the formula:
Area
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1 What is the Area (A) of the figure?
13 ft
7 ft
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2 Find the area of the figure below.
8
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3 Michelle needs new carpeting for her bedroom that is 12 feet by 9 feet. Does Michelle need to find the area or perimeter of her bedroom in order to figure out how much carpet to order?
A Area
B Perimeter
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4 Now solve the problem....
Michelle needs new carpeting for her bedroom that is 12 feet by 9 feet. How many square feet of carpet does Michelle need to order?
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5 A rectangle measures 3 in by 4 in. If the lengths of each side double, what is the effect on the area?
A The area doubles
B The area quadruples
C The area is cut in half
D There is no effect
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6 The area of a desktop is 24 sq. units. The length of the desktop is 6 units. What is the width of the desktop?
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7 The 6th grade class at Immersion Middle School is building a giant I for their school. The I will be 10 ft. tall and 2 ft. wide. How large will the I be if measured in square inches?
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8 The lumber that will be used to make the Immersion School I is 6 in by 1 ft. How many pieces of wood are needed to complete the project?
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11 units
How can we find the area of this parallelogram? Cut out your parallelogram and work with your table to come up with a way to determine the area.
10 units
15 units
click
Area of a Parallelogram
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9 What is the area of the parallelogram?
10 units
15 units
click
11 units
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Area of a ParallelogramLet's use the same process as we did for the rectangle. How many 1 ft2 tiles fit across the bottom of the parallelogram?
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Area of a ParallelogramLet's use the same process as we did for the rectangle. If we build the parallelogram with rows of ten 1 ft2 tiles, what happens?
How tall is the parallelogram?How can you tell?
10 ft
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How does this help us find the area of the parallelogram?
10 ft
How do you find the area of a parallelogram?
4 ft
Area of a Parallelogram
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A = base(height)A = bh
The Area (A) of a parallelogram is found by using the formula:
Note: The base & height always form a right angle!
Area of a Parallelogram
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Example.
Find the area of the figure.
6 cm
6 cm
2 cm 2 cm1.7 cm
Parallelogram Area Practice
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Try These.
Find the area of the figures.
10.4 in
8.7 in6.2 in
13 m
15 m
13 m
16 m
Parallelogram Area Practice
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10 Find the area.
10 ft 9 ft
11 ft
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11 Find the area.
15 in
15 in
10 in 11 in11 in
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12 Find the area.
8.4 m
13.1 m
8.4 m
12.2 m
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13 Find the area.
13 cm
12 cm
7 cm
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14 A box with a square opening is squashed into the rhombus shown below. What is the area of the opening?
7 in.
14 in
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Solving for Missing InformationA parallelogram has an area of 137.7 cm2 and a base of 9 cm. Write an equation that relates the area to the base and height, h. Solve the equation to determine the length of the height.
Step 1: Plug in known information.
A = bh
=
Step 2: Use inverse operations to solve
( (
137.7 cm2 9 cm
Abh
information
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15 The height of a parallelogram is 12.6 feet and the area is 88.2 square feet. Write an equation that relates the area to the height and the base, b. Solve the equation to determine the length of the base.
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16 The height of a parallelogram is 54 inches and the area is 972 square inches. Write an equation that relates the area to the height and the base, b. Solve the equation to determine the length of the base.
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Area of a TriangleLet's use the same process as we did for the rectangle & parallelogram. How many 1 ft2 tiles fit across the bottom of the triangle?
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Area of a TriangleIf we continue to build the triangle with rows of thirteen 1 ft2 tiles what happens?
How tall is the triangle? How can you tell?
13 ft
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How does this help us find the area of the triangle?
Find the area of the rectangle, then divide by 232.5 ft2
See that the rectangle we built is twice as large as the triangle. How do you find the area of a triangle?
13 ft
5 ft
Area of a Triangle
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The Area (A) of a triangle is found by using the formula:
Area of a Triangle
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Try this. What is the area of the right triangle below?
4 units
14 units
14.7 units
Area of a Triangle Practice
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17 Calculate the area.
7 cm.
8 cm.10.5 cm
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18 Calculate the area.
9.9 m9 m
4.1 m
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19 Calculate the area.
7 in7 in
9.9 in
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Solving for Missing Information
A triangle has an area of 70.8 cm2 and a base of 6 cm. Write an equation that relates the area to the base and height, h. Solve the equation to determine the length of the height.
Step 1: Plug in known information.
A = bh
=
Step 2: Use inverse operations to solve
( (
70.8 cm2 6 cm
Abh
information
( (
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20 If the area of a triangle is 117 square cm and its base is 20 cm, write an equation that relates the area to the height, h, and the base. Solve the equation to determine the height.
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21 Fran is surveying a plot of land in the shape of a right triangle. The area of the land is 45,000 sq. meters. If the base of the triangular plot is 180 m long, what is the height, in meters, of the triangle? Write and solve an equation.
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What is the difference between these three triangles?
Triangles
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The height of the right triangle is easy to find, it is a side. It does not always need to be a side of the triangle. The height of a triangle is also called the altitude, which is a line segment from a vertex of the triangle and perpendicular to the opposite side.
hh
b
b
b
h
Triangle Altitudes
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Is the formula for the area of a right triangle true for all triangles?Let's see!
Triangle Area
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Example.
Find the area of the figure.
8 cm
11 cm 11 cm
11 cm
Triangle Area
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Try These.
Find the area of the figures.
13 ft
11 ft
10 ft 12 ft 1420
16
16
Triangle Area Practice
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22 Find the area.
8 in
6 in
10 in9 in
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23 Find the area.
14 m
9 m10 m 12 m
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24 Find the area.
10 in.
6 in.14 in.
5 in.
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26 Chauncey is building a storage bench for his son's playroom The storage bench will fit into the corner and then go along the wall to form a triangle. Chancey wants to buy a cover for the bench. If the storage bench is ft. along one wall and ft. along the other wall, how big will the cover have to be to cover the entire bench?
(Problem derived from )
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Use what you know to try and figure out how can we calculate the area of this triangle.
Hint
Triangle Area
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11.5 7+ 10 28.5
49- 28.5 20.5
Triangle Sum
Difference
Trianglea = 1/2bha = 1/2(2)(7)a = 7 u2
Triangle a = 1/2bha = 1/2(3)(7)a = 11.5 u2
Trianglea = 1/2bha = 1/2(5)(4)a = 10 u2
a = lwa = 7(7)a = 49 u2
The shaded triangle is 20.5 u2
Square
Step 1: Calculate the area of the square Step 2: Calculate the area of the triangles.Step 3: Find the sum of the areas of the triangles.
Step 4: Subtract the sum of the triangle areas from the rectangle area.
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27 What is the area of the shaded figure? Students type their answers here
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28 What is the area of the shaded figure? Students type their answers here
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29 What is the area of the shaded figure? Students type their answers here
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This method can be used with any shape, as long as you can find the base and height of the
triangles that form the surrounding rectangle.
Area of Any Shape
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30 What is the area of the shaded figure? Students type their answers here
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31 What is the area of the shaded figure? Students type their answers here
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Area of a Trapezoid· Draw a diagonal line to break the trapezoid into two triangles.· Find the area of each triangle· Add the area of each triangle together
See the diagram below. 10 in
12 in
5 in
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The Area (A) of a trapezoid is also found by using the formula:
Note: The base & height always form a right angle!
10 in
12 in
5 in
Area of a Trapezoid
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Example.
Find the area of the figure by drawing a diagonal and splitting it into two triangles.
12 cm
10 cm 11 cm
9 cm
Trapezoid Area Practice
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Try These.
Find the area of the figures using the formula.
12 ft
9 ft
7 ft 8 ft
13
10
8 ft 7 86
Trapezoid Area Practice
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32 Find the area of the trapezoid by drawing a diagonal.
9 m
11 m
8.5 m
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33 Find the area of the trapezoid using the formula.
20 cm
13 cm
12 cm
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34 The shape of the state of Arkansas resembles a trapezoid. The population density of Arkansas is 54.8 people per square mi. What is the approximate total population of this state?
280 mi
210 mi
235 mi
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35 Each of the four sides of this tent are congruent. How much fabric was used to make all four sides of this tent?
23 in.
36.5 in.
32 in.
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36 Find the area of the figure.
5 cm
4 cm 3 cm 4 cm
11 cm
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37 Find the area of the figure.
8 yd
10.5 yd
10.5 yd 10.5 yd
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38 Find the area of the figure.
4.7 m
7.2 m
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39 Find the area of the figure.
9 in 7 in
15 in
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40 Find the area of the figure by drawing a diagonal and creating triangles.
17 cm
16 cm 15 cm 16 cm
22 cm
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41 Find the area of the figure.
7 in 5.2 in
12.4 in
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42 Find the area of the figure.
11 yd12 yd
13 yd
12 yd
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43 Find the area of the figure.
4.6 m
8.7 m
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44 The Andersons were going on a long sailing trip during the summer. However, one of the sails on their sailboat ripped, and they have to replace it. The sail is pictured below.
If the sailboat sails are on sale for $2 a square foot, how much will the new sail cost?
Derived from
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45 A wall is 56" wide. You want to center a picture frame that is 20" wide on the wall. How much space will there be between the edge of the wall and the frame?
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46 Daniel decided to walk the perimeter of his triangular backyard. He walked 26.2 feet north and 19.5 feet west and back to his starting point. What is the area of Daniel's backyard?
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47 If the area of a parallelogram is sq. km. and the base is km., write an equation that relates the area to the base, b, and the height. Solve the equation to determine the height.
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48 If the area of a right triangle is sq. ft. and the height is ft., write an equation that relates the area to the base, b, and the height. Solve the equation to determine the base.
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49 Below is a drawing of a wall that is to be covered with either wallpaper or paint. It is 8 ft. high and 16 ft. long. The window, mirror and fireplace will not be painted or papered. The window measures 18 in. by 14 ft. The fireplace is 5 ft. wide and 3 ft. high, while the mirror above the fireplace is 4 ft. by 2 ft.
Part A: How many square feet of wallpaper are needed to cover the wall?
Derived from continued
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50 Part B: The wallpaper is sold in rolls that are 18 in. wide and 33 ft. long. Rolls of solid color wallpaper will be used so patterns do not have to match up. What is the area of one roll of wallpaper?
Continued from previous page.
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51 Part C: How many rolls would be needed to cover the wall?
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52 Part D: This week the rolls of wallpaper are on sale for $11.99/ roll. Find the cost of covering the wall with wallpaper.
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53 Part E: A gallon of special textured paint covers 200 ft2 and is on sale for $22.99/ gallon. The wall needs to be painted twice (the wall needs two coats of paint). Find the cost of using paint to cover the wall.
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54 The area of a rectangular patio is square yards, and its length is yards. What is the patio's width in yards?
A
B
C
D
From PARCC PBA sample test non-calculator #3
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55 Joanne buys a rectangular rug with an area of 35/4 square meters. The length of the rug is 7/2 meters. What is the width, in meters, of the rug?
From PARCC EOY sample test non-calculator #1
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Area of Irregular FiguresMethod #1
1. Divide the irregular figure into smaller figures (that you know how to find the area of)
2. Label each small figure and label the new lengths and widths of each shape
3. Find the area of each shape
4. Add the areas
5. Label your answer
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Example:Find the area of the figure.
12 m
8 m
4 m2 m
12 m6 m
4 m2 m #1
#2
2 m
Irregular Figure Area
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Area of Irregular FiguresMethod #2
1. Create one large, closed figure
2. Label the small added figure and label the new lengths and widths of each shape
3. Find the area of the new, large figure
4. Subtract the areas
5. Label your answer
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Example:Find the area of the figure.
12 m
8 m
4 m2 m
8 m Whole Rectangle
Extra Rectangle
12 m
8 m
4 m2 m
Irregular Figure Area
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Try This:Find the area of the figure.
3m5m
3m8m
Irregular Figure Area Practice
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Try This:Find the area of the figure.
18 ft
12 ft
6 ft
10 ft
Irregular Figure Area Practice
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56 Find the area.
4'
3'
1'
2'
10'
8'
5'
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57 Find the area.
12
101320
25
10
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58 Find the area.
8 cm 18 cm
9 cm
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59 Find the area.
4 ft 9 ft
6 ft
7 ft
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60 Find the area.
8 mm
8 mm
8 mm10 mm
14 mm
14 mm
6 mm
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61 Cara wants to put new carpet in both of her bedrooms. How much carpet will she need?
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62 How many rectangular tiles are needed to cover this floor?
2 m1 m
Tiles
(Drag and drop to check.)
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Area of a Shaded Region
1. Find area of whole figure.
2. Find area of unshaded figure(s).
3. Subtract unshaded area from whole figure.
4. Label answer with units2.
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Example
Find the area of the shaded region.
8 ft
10 ft
3 ft3 ft
Area Whole Rectangle
Area Unshaded Square
Area Shaded Region
Shaded Region Area
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Try ThisFind the area of the shaded region.
14 cm
12 cm
Shaded Region Area
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Try This
Find the area of the shaded region.
16 m
12 m6 m
8 m2 m
Shaded Region Area
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63 Find the area of the shaded region.
11'
8'
3'4'
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64 Find the area of the shaded region.
16"
17"
15"7"
5"
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13"
14"
8"
5"
9"
4"
65 Find the area of the shaded region.
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4 yd
4 yd
3 yd
8 yd
4 yd
4 yd
66 Find the area of the shaded region.
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67 A cement path 2 feet wide is poured around a rectangular pool. If the pool is 13 feet by 9 feet, how much cement was needed to create the path?
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68 Logan wants to paint his trapezoid-shaped wall shown below. He of course will not be painting over his window. One gallon of paint will cover 50 sq. feet. How many gallons of paint will he need?
23 ft
18 ft
13 ft5 ft
4 ft
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69 An advertising company is designing a new logo that consists of a shaded triangle inside a parallelogram.
Part A
What is the area, in square units, of parallelogram ABCD?
square units
From PARCC EOY sample test calculator #7
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70 Part B
In the new logo, what fraction of the parallelogram is shaded?
square units
From PARCC EOY sample test calculator #7
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3-Dimensional SolidsCategories & Characteristics of 3-D Solids:
Prisms1. Have 2 congruent, polygon bases which are parallel to one another2. Sides are rectangular (parallelograms)3. Named by the shape of their base
Pyramids1. Have 1 polygon base with a vertex opposite it2. Sides are triangular3. Named by the shape of their base
click to reveal
click to reveal
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Sort the figures. If you are incorrect, the figure will be sent back.3-Dimensional Figures
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3-Dimensional SolidsCategories & Characteristics of 3-D Solids:
Cylinders1. Have 2 congruent, circular bases which are parallel to one another2. Sides are curved
Cones1. Have 1 circular bases with a vertex opposite it2. Sides are curved
click to reveal
click to reveal
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3-Dimensional SolidsVocabulary Words for 3-D Solids:
Face Flat surface of a Polyhedron
Edge Line segment formed where 2 faces meet
Vertex (Vertices) Point where 3 or more faces/edges meet
Face
edge
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A polyhedron is a 3-D figure whose faces are all polygons.
Polyhedron Not Polyhedron
Sort the figures to the appropriate side.
Polyhedron
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71 Name the figure.
A rectangular prism
B triangular prism
C triangular pyramid
D cylinder
E square cone
F square pyramid
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72 Name the figure.
A rectangular prism
B triangular prism
C triangular pyramid
D cylinder E coneF square pyramid
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73 Name the figure.
A rectangular prism
B triangular prism
C triangular pyramid
D pentagonal prism
E cone
F square pyramid
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74 Name the figure.
A rectangular prism
B triangular prism
C triangular pyramid
D pentagonal prism
E cone
F square pyramid
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75 Name the figure.
A rectangular prism
B cylinder
C triangular pyramid
D pentagonal prism
E coneF square pyramid
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NetsNets are two-dimensional drawings that represent the surface area of three-dimensional shapes.
There is more than one way to draw a net for a cube, however not all nets can be folded into a cube...
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NetsExploratory Challenge Lab
Click for Link to LabThere are some six square arranglements on your page. Sort each of the six arrangements into one of two piles, those that are nets of a cube and those that are not.
click to reveal answersDerived from
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Click for a web site with interactive 3-D figures and nets.
Interactive 3-D Figures and Nets
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Nets for prisms will have rectangular faces and two bases for which the shape is named.
Notice the two triangles areopposite from one another (bases).
Prism Nets
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76 Name the figure represented by the net.
A rectangular prism
B cylinder
C triangular prism
D pentagonal prism
E coneF square pyramid
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77 Name the figure represented by the net.
A rectangular prism B cylinder
C triangular prism
D pentagonal prism
E coneF square pyramid
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Use the packaging explorer to view more examples of nets.Interactive Nets
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For each figure, find the number of faces, vertices and edges. What is the relationship between the number of faces, vertices and edges of 3D Figures?
Name Faces Vertices Edges Cube 6 8 12
Rectangular Prism 6 8 12
Triangular Prism 5 6 9
Triangular Pyramid 4 4 6
Square Pyramid 5 5 8
Pentagonal Pyramid 6 6 10
Octagonal Prism 10 16 24
3D Figure Patterns
Mat
h Pr
actic
e
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Euler's Formula
F + V - 2 = E
The number of edges is 2 less than the sum of the faces and vertices.
click to reveal
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78 How many faces does a cube have?
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79 How many vertices does a triangular prism have?
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80 How many edges does a square pyramid have?
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81 Paige has a figure whose faces are all congruent, and it has 4 vertices. Which shape does Paige have?
A triangular pyramid
B triangular prism
C cube
D square
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82 Jonathan has 2 cubes. Henry has a square pyramid. How many edges do they have all together?
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83 Which of these nets can be folded to form a cube?
A
B
C
D
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Surface AreaSurface area is the sum of the areas of all outside faces of a 3-D figure.
To find surface area, you must find the area of each face of the figure then add them together.
6 in
2 in7 in
What type of figure is pictured?
How many surfaces are there?
How do you find the area of each surface?
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Surface Area
6 in
2 in7 in
7 in2 in
2 in6 in
A net is helpful in calculating surface area.
Simply label each section and find the area of each.
#2 #4
6 in
#1
#3
#5
#6
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7 in2 in
2 in6 in
#2 #4
6 in
#1
#3
#5
#6
#1 #2 #3 #4 #5 #6
Example
Surface Area Example
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Try This
Find the surface area of figure using the given net.
#1
#2 #3 #4
#5
12 cm
Surface Area Practice
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84 Find the surface area of the figure given its net.
7 yd
7 yd
7 yd
7 ydSince all of the faces are the same, you can find the area of one face and multiply it by 6 to calculate the surface area of a cube.
What pattern did you notice while finding the surface area of a cube?
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85 Find the surface area of the figure given its net.
9 cm
25 cm
12 cm
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86 The figure below represents a present you want to wrap for your friend's birthday. How many square centimeters of wrapping paper will you need? On the grid on the next slide, the distance between grid lines represents one centimeter. Use the grid to draw the net for the given figure. Then, calculate its surface area.
4 cm
4 cm8 cm
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87 Draw the net for the given figure, and calculate its surface area.
7 ft7 ft
11 ft
4 ft
12 ft
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88 This is a net of a right rectangular prism.
Part A
Which prism can be made using the net?
A
B
C
D
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89 Part B
What is the surface area, in square feet, of the prism?
From PARCC EOY sample test calculator #12
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Volume Activity
Take unit cubes and create a rectangular prism with dimensions of 4 x 2 x 1.
What happens to the volume if you add another layer and make it 4 x 2 x 2?
What happens to the volume is you add another layer and make it 4 x 2 x 3?
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Volume
- Volume is the amount of space occupied by or inside a 3-D Figure.- The number of cubic units needed to fill a 3-D Figure (layering).
Label:Units3 or cubic units
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Volume Formulas
Formula 1
V= lwh, where l = length, w = width, h = height
Multiply the length, width, and height of the rectangular prism.
Formula 2
V=Bh, where B = area of base, h = height
Find the area of the rectangular prism's base and multiply it by the height.
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Find the Volume.
5 m
8 m
2 m
Volume Practice
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Example
Each of the small cubes in the prism shown have a length, width and height of 1/4 inch.
The formula for volume is lwh.
Therefore the volume of one of the small cubes is:
Multiply the numerators together, then multiply the denominators. In other words, multiply across.
Forget how to multiply fractions?
Volume Example
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To calculate the volume of the whole prism, count the number of cubes, and multiply it by the volume of one cube.
The top layer of this prism has 4 rows of 4 cubes, making a total of 16 cubes per layer.
The prism has 4 layers, 16 cubes per layer, so has 64 small cubes total.
Therefore the total volume of the prism is:
Example
Volume Example
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Example You can also use the formula to find the volume of the same prism.
The length, width, and the height of this prism is four small cubes.
Remember each small cube has a length, width, and height of 1/4 inch.
Therefore, you can find the total volume finding the total length, width, and height of the prism and multiplying them together.
Volume Example
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ExampleHow would you find the volume of the rectangular prism with side lengths of 1/2 cm, 1/8 cm, and 1/4 cm?
Volume Practice
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Try This Every cube in the rectangular prism has a length, width and height of 1/5 inch.
Find the total volume of the rectangular prism.
Method 1: Find volume of one small cube and multiply it by the number of cubes.
One cube: Total Volume:
Method 2: Find the length, width, and height of the rectangular prism and use the formula.
Click to Reveal
Volume Example
Click to Reveal
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90 Find the volume of the given figure.
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91 Find the volume of the given figure.
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92 Find the volume of the given figure.
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93 Find the volume of the given figure.The length,
width, and height of one small cube is .
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94 Find the volume of the given figure. The length, width,
and height of one cube is .
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95 A student filled a right rectangular prism-shaped box with one inch cubes to find the volume, in cubic inches. The student's work is shown.
Part A
Explain why the student's reasoning is incorrect. Provide the correct volume, in cubic inches, of the box.
From PARCC PBA sample test calculator #9
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96 Part B
A second box also has a base of 63 square inches, but it has a volume of 756 cubic inches. What is the height, in inches, of the second box? Explain or show how you determined the height.
From PARCC PBA sample test calculator #9
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97 A rectangular storage box is 12 1/4 in wide, 15 3/5 in long and 9 in high. How many square inches of colored paper are needed to cover the surface area of the box?
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98 A teacher made 2 pair of foam dice to use in math games. Each cube measured 10 2/3 in on each side. How many square inches of fabric were needed to cover the 2 cubes?
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99 A company is packaging their cereal in two rectangular-shaped containers. Container A is 5.5in x 7.25in x 10 3/4 in. Container B is 8 1/2in x 3 1/4 in x 12in. Which container will hold more cereal? Input your answer, then explain your answer in a sentence on your paper.
A Container A
B Container B
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100 A company is packaging their cereal in two rectangular-shaped containers. Container A is 5.5in x 7.25in x 10 3/4 in. Container B is 8 1/2in x 3 1/4 in x 12in. Which container will require more cardboard to make the box? Input your answer, then explain your answer in a sentence on your paper.
A Container A
B Container B
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101 A toy company manufactured a new set of toy blocks. The packaging manager insists that the cubes be arranged to form a rectangular prism and that the package be designed to hold the blocks exactly, with no leftover packaging. Each block measures 1 in. x 1 in. x 1 in. There are 24 toy blocks to be sold in a box. What are all of the possible box dimensions in inches? (Select all that apply.)
A 1 x 1 x 24
B 1 x 2 x 12
C 1 x 3 x 8
D 2 x 2 x 8
E 2 x 3 x 6
F 1 x 3 x 6
G 1 x 4 x 6
H 2 x 2 x 6
I 2 x 4 x 8
(Problem derived from )
J 2 x 3 x 4
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102 (Cont. from previous slide) Which toy block box design will use the least amount of cardboard for packaging? Select one measurement (in inches) for each dimension of the box.
A 1
B 2
C 1
D 2
E 3
F 4
G 4
H 6
I 8
Height Width Length
(Problem derived from )
J 12
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103 A 250 in3 box needs to be packaged for shipment. One shipping container has a length of 7 inches, a height of 5 inches, and a width of 6 inches. The other container has a length of 8 in, a height of 4 inches, and a width of 9 inches. Which container can the package be shipped in? Explain.
A Container A: 7 in x 6 in x 5 in
B Container B: 8 in x 4 in x 9 in
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104 Small cubes with edge lengths of 1/4 inch will be packed into the right rectangular prism shown.
How many small cubes are needed to completely fill the right rectangular prism?
cubes
From PARCC EOY sample test non-calculator #6
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105 The right rectangular prism is built with small cubes.
Part A
What is the volume, in cubic inch(es), of the right rectangular prism? Enter your fraction.
From PARCC EOY sample test calculator #10
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106 Part B
What is the volume, in cubic inch(es), of 1 of the small cubes? Enter your fraction.
From PARCC EOY sample test calculator #10
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107 Draw a polygon in the coordinate plane using the given coordinates.
(4, -4)
(6, -2)
(8, -6)
What is the area of
the polygon?
Students type their answers here
(Problem from )
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108 A surveyor is mapping a city block on a coordinate grid. The square-shaped block has vertices at (-4,1), (-4, -4), and (1, -4). What are the coordinates of the remaining vertex?
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109 What is the area of the square block described in the previous problem?
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Instruction
3-D FiguresAn object with three different dimensions: length, width (or depth or breadth), and
height. Also called a solid figure.
One-Dimensional Three-DimensionalTwo-Dimensional
lengthlength
lengthwidthwidth
height
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Instruction
AltitudeA line segment from a vertex of the triangle and perpendicular to the opposite side. The
height.
h
bb
h
b
h
This is not the height. It is not perpendicular to
the base.
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Instruction
Base & HeightBase- the surface that a solid object
stands on
Height- the distance from the base to the top of a solid object.
base
height
base
height
The base and height
always form a right angle.
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ConeA 3-dimensional figure with one circular
base, a vertex at the top, and one curved surface connecting the two.
1 Circular Base
1 Curved Surface 1 Vertex
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CubeA 3-dimensional figure with 3 pairs
of parallel, congruent, square bases.
12 Edges8 vertices 6 faces
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CylinderA 3-dimensional figure with two-
congruent, circular bases, and one curved surface connecting them.
2 congruent, parallel, circular
bases
1 curved surface
No vertices
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DiagonalA line that goes from one non-
adjacent vertex to another.
Cannot draw a diagonal, because
all vertices are adjacent.
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DimensionsA measurement of
length in one direction.
1 dimension
2 dimensions
3 dimensions length
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EdgeThe line segment
where two faces meet.
10 edges
edge
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Euler's FormulaF + V - 2 = E
For any polyhedron that doesn't intersect itself, the number of edges is 2 less than
the sum of the faces and vertices.
Faces: 6Vertices: 8
6+8-2=12 Edges: 12
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Face
A flat surface of a 3-d figure.
There is still debate over
whether curved surfaces are
faces.Face6 faces
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Net
= =
A 2-dimensional pattern which can be folded into a 3-
dimensional figure.
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Perimeter
The distance around an object.
side 1
side 2side 3
P= side 1 + side 2 + side 3
To fence in this rectangular yard,
you would measure the perimeter.
l wP=2l+2w
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Polyhedron
"Polyhedra" is the
singular form of
polyhedron
A three dimensional figure with all flat faces.
non-polyhedraPolyhedra
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PrismA 3-dimensional figure with two congruent, parallel bases, and all other faces are rectangles.
Prisms are named by the
Pentagonal Prism
shape of their bases. 2 triangular bases
Triangular Prism
3 rectangular faces
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PyramidA 3-dimensional figure with one base, a vertex at the top, and all other faces are
triangles.
1 baseAll other faces are triangles
A vertex at the top
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Instruction
Square Units1
unit
1 unit
1 unit x 1 unit = 1 square unit
A measurement in the shape of a square with side lengths
that are one unit long.
Notation:sq unit
unit2
u2
3 units
3 un
its
3 units x 3 units = 9 units2
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6 u2
6 u2
12 u2
8 u2
8 u2
12 u2
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Instruction
Surface AreaThe total area of the surface
of a 3-dimensional figure.
2lw+2lh+2wh2 12+2 6+2 8SA=24+12+16
SA=52u2
Surface Area=
+ + +
+ +
=SA
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TrapezoidA quadrilateral with one
pair of parallel sides.
There are no // sides.
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Instruction
VertexA point where two or more
straight lines meet.
The plural of vertex is "vertices"
APoint A or
vertex A
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Instruction
Volume
4
3
3
v=lwhv= 4 3 3v= 36 u3
The amount of space within a 3-dimensional object. Measured in cubic units.
11
1
V=1 1 1V= 1 cubic unit
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Throughout this unit, the Standards for Mathematical Practice are used.
MP1: Making sense of problems & persevere in solving them.MP2: Reason abstractly & quantitatively.MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics.MP5: Use appropriate tools strategically.MP6: Attend to precision.MP7: Look for & make use of structure.MP8: Look for & express regularity in repeated reasoning.
Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used.
If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.
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