CONFIDENTIAL 1
Geometry
Developing Formulas for Triangles and
Quadrilaterals
CONFIDENTIAL 2
Warm up
Find the perimeter and area of each figure:
1) 2)
1) P = 6x + 4; A = 2x2 + 4x 2) P = 2x + 1; A = 7x/2
x + 2
2x
x
x + 17
CONFIDENTIAL 3
When a Figure is made from different shapes, the area of the figure is the sum of the areas of the pieces.
Area Addition Postulate
Postulate 1: The area of a region is equal to the sum of the areas of non-overlapping parts.
CONFIDENTIAL 4
Recall that a rectangle with base b and height h has an area of A = bh. You can use the Area Addition Postulate
to see that a parallelogram has the same area as a rectangle with the same base and height.
b
h
b
CONFIDENTIAL 5
The area of a Parallelogram with base b and height h is A = bh.
Area: Parallelogram
b
h
Remember that rectangles and squares are also Parallelograms. The area of a square with side s is
A = s2, and perimeter is P = 4s.
CONFIDENTIAL 6
Finding measurements of Parallelograms Find each measurement:
A) the area of a Parallelogram
6 in
h5 in
3 in
Step 1: Use Pythagorean Theorem to find the height h.
32 + h2 = 52
h = 4
Step 2: Use h to find the area of parallelogram.
A = bhA = 6(4)A = 24 in2
Area of a parallelogram.Substitute 6 for b and 4 for h.Simplify.
CONFIDENTIAL 7
B) the height of a rectangle in which b = 5 cm and A = (5x2 – 5x) cm2.
A = bh
5x2 – 5x = 5(h)
5(x2 – x) = 5(h)
x2 – x = h
h = (x2 – x) cm
Area of a rectangle .
Substitute (5x2 – 5x) for A and 5 for b.
Factor 5 out of the expression for A.
Divide both sides by 5.
Sym. Prop. of =.
CONFIDENTIAL 8
C) the perimeter of a rectangle in which A = 12x ft2.
A = bh
12x = 6(b)
2x = b
Area of a rectangle .
Substitute 12x for A and 6 for h.
Divide both sides by 6.
Perimeter of a rectangle
Substitute 2x for A and 6 for h.
Step 1: Use Pythagorean Theorem to find the height h.
P= 2b + 2h
P = 2(2x) +2(6)
P = (4x +12) ft
Step 2: Use the base and height to find the perimeter.
Simplify.
6x
CONFIDENTIAL 9
Now you try!
1) Find the base of a Parallelogram in which h = 65 yd and A = 28 yd2.
1) b = 0.5 yd
CONFIDENTIAL 10
To understand the formula for the area of a triangle or trapezoid, notice that the two congruent triangles or
two congruent trapezoids fit together to form a parallelogram. Thus the area of a triangle or a
trapezoid is half the area of the related parallelogram.
h h
b b
b1
b2
h
b2
b2
h
b1
CONFIDENTIAL 11
Area: Triangles and Trapezoids
The area of a Triangle with base b and height h is A = 1 bh.
2h
b
The area of a Trapezoid with bases b1 and b2 and height h is
A = 1 (b1 + b2 )h.b2
h
b1
2
CONFIDENTIAL 12
Finding measurements if Triangles and Trapezoids
Find each measurement:
A) the area of Trapezoid with b1 = 9 cm, b2 = 12 cm and h = 3 cm.
A = 1 (b1 + b2 )h 2
Area of a Trapezoid.
Substitute 9 for b1, 12 for b2 and 3 for h.
Simplify.
A = 1 (9 + 12 )3 2
A = 31.5 cm2
CONFIDENTIAL 13
B) the base of Triangle in which A = x2 in2.
A = 1 bh 2
Area of a Triangle.
Substitute x2 for A and x for h.
Divide both sides by x.
x2 = 1 bx 2
x = 1 b 2
x in
b
b = 2x in.
2x = b Multiply both sides by 2.
Sym. Prop. of =.
CONFIDENTIAL 14
C) b2 of the Trapezoid in which A = 8 ft2.
Multiply ½ by 2.8 = 3 + b2
b2 = 5 ft.
5 = b2Subtract 3 from both sides.
Sym. Prop. of =.
A = 1 (b1 + b2 )h 2
Area of a Trapezoid.
Substitute 8 for A1, 3 for b1 and 2 for h.8 = 1 (3 + b2)2 2
3 ft
2 ft
b2
CONFIDENTIAL 15
Now you try!
2) Find the area of the triangle.
2) b = 96 m2
20 m12 m
b
CONFIDENTIAL 16
A kite or a rhombus with diagonal d1 and d2 can be divided into two congruent triangles with a base d1
and height of ½ d2 .
d1
½ d2
d1
½ d2
Total area : A = 2(1 d1d2 ) = 1 d1d2
4 2
area of each triangle: A = 1 d1(½ d2 ) 2 = 1 d1d2
4
CONFIDENTIAL 17
Area: Rhombus and kites
The area of a rhombus or kite with diagonals d1 and d2 and height h is A = 1 d1d2 .
2
d1
½ d2
d1
½ d2
CONFIDENTIAL 18
Finding measurements of Rhombus and kites
Find each measurement:
A) d2 of a kite with d1 = 16 cm, and A = 48 cm2.
A = 1 (d1d2 ) 2
Area of a kite.
Substitute 48 for A, 16 for d1.
Simplify.
48 = 1 (16)d2
2
d2 = 6 cm
CONFIDENTIAL 19
B) The area of the rhombus .
A = 1 (d1d2 ) 2
Area of a kite.
Substitute (6x + 4) for d1 and (10x + 10) for d2.
Multiply the binomials.
A = 1 (6x + 4) (10x + 10) 2
d1 = (6x + 4)in.
d2 = (10x + 10)in.
A = 1 (6x2 + 100x + 40) 2
Simplify.A = (3x2 + 50x + 20)
41 ft9 ft
15 ftyx
CONFIDENTIAL 20
C) The area of the kite.
Step 1: The diagonal d1 and d2 form four right angles.
Use Pythagorean Theorem to find the x and y.
92 + x2 = 412
x2 = 1600
x = 40
92 + y2 = 152
y2 = 144
y = 12
CONFIDENTIAL 21
41 ft9 ft
15 ftyx
Step 2: Use d1 and d2 to find the area. d1 = (x + y) which is 52. Half of d2 = 9, so d2 = 18.
A = 1 (d1d2 ) 2
Area of a kite.
Substitute 52 for d1 and 18 for d2.A = 1 (52) (18) 2
Simplify.A = 468 ft2
CONFIDENTIAL 22
Now you try!
3) b = 96 m2
3) Find d2 of a rhombus with d1 = 3x m, and A = 12xy m2.
CONFIDENTIAL 23
Games ApplicationThe pieces of a tangram are arranged in a square in which s = 4 cm. Use the grid to find the perimeter and area of the
red square.
Perimeter:
Each side of the red square is the diagonal of the square grid. Each grid
square has a side length of 1 cm, so the diagonal is √2 cm. The perimeter of the
red square is P = 4s = 4 √2 cm.
CONFIDENTIAL 24
A = 1 (d1d2 ) = 1 (√2)(√2) = 2 cm2. 2 2
Area:
Method 1: d2 of a kite with d1 = 16 cm, and A = 48 cm2.
Method 2: The side length of the red square is √2 cm, so the area if A = (s2) = (√2)2 = 2 cm2.
CONFIDENTIAL 25
Now you try!
4) A = 4 cm2
P = 4 + 4√2 cm
4) Find the area and perimeter of the large yellow triangle in the figure given below.
CONFIDENTIAL 26
Now some problems for you to practice !
CONFIDENTIAL 27
Find each measurement:
Assessment
1) 120 cm2
2) 5x ft
12 cm
10 cm1) the area of the Parallelogram.
2x ft
2) the height of the rectangle in which A = 10x2 ft2.
CONFIDENTIAL 28
Find each measurement:
3) 240 m2
4) 13 in
3) the area of the Trapezoid.
4) the base of the triangle in which A = 58.5 in2.
20 m
9 m15 m
9 in
CONFIDENTIAL 29
5) 175 in2
6) 25 m
5) the area of the rhombus.
6) d2 of the kite in which A = 187.5 m2.
Find each measurement:
14 in
25 in
15 m
CONFIDENTIAL 30
7) The rectangle with perimeter of (26x + 16) cm and an area of (42x2 + 51x + 15) cm2. Find the dimensions
of the rectangle in terms of x.
7) (7x + 5) and (6x + 3)
CONFIDENTIAL 31
8) The stained-glass window shown ii a rectangle with a base of 4 ft and a height of 3 ft. Use the grid to find
the area of each piece.
8) √10 ft2
CONFIDENTIAL 32
Let’s review
When a Figure is made from different shapes, the area of the figure is the sum of the areas of the pieces.
Postulate 1: The area of a region is equal to the sum of the areas of non-overlapping parts.
CONFIDENTIAL 33
The area of a Parallelogram with base b and height h is A = bh.
Area: Parallelogram
b
h
Remember that rectangles and squares are also Parallelograms. The area of a square with side s is
A = s2, and perimeter is P = 4s.
CONFIDENTIAL 34
Finding measurements of Parallelograms Find each measurement:
A) the area of a Parallelogram
6 in
h5 in
3 in
Step 1: Use Pythagorean Theorem to find the height h.
32 + h2 = 52
h = 4
Step 2: Use h to find the area of parallelogram.
A = bhA = 6(4)A = 24 in2
Area of a parallelogram.Substitute 6 for b and 4 for h.Simplify.
CONFIDENTIAL 35
Area: Triangles and Trapezoids
The area of a Triangle with base b and height h is A = 1 bh.
2h
b
The area of a Trapezoid with bases b1 and b2 and height h is
A = 1 (b1 + b2 )h.b2
h
b1
2
CONFIDENTIAL 36
Finding measurements if Triangles and Trapezoids
Find each measurement:
A) the area of Trapezoid with b1 = 9 cm, b2 = 12 cm and h = 3 cm.
A = 1 (b1 + b2 )h 2
Area of a Trapezoid.
Substitute 9 for b1, 12 for b2 and 3 for h.
Simplify.
A = 1 (9 + 12 )3 2
A = 31.5 cm2
CONFIDENTIAL 37
A kite or a rhombus with diagonal d1 and d2 can be divided into two congruent triangles with a base d1
and height of ½ d2 .
d1
½ d2
d1
½ d2
Total area : A = 2(1 d1d2 ) = 1 d1d2
4 2
area of each triangle: A = 1 d1(½ d2 ) 2 = 1 d1d2
4
CONFIDENTIAL 38
Area: Rhombus and kites
The area of a rhombus or kite with diagonals d1 and d2 and height h is A = 1 d1d2 .
2
d1
½ d2
d1
½ d2
CONFIDENTIAL 39
Finding measurements of Rhombus and kites
Find each measurement:
A) d2 of a kite with d1 = 16 cm, and A = 48 cm2.
A = 1 (d1d2 ) 2
Area of a kite.
Substitute 48 for A, 16 for d1.
Simplify.
48 = 1 (16)d2
2
d2 = 6 cm
CONFIDENTIAL 40
Games ApplicationThe pieces of a tangram are arranged in a square in which s = 4 cm. Use the grid to find the perimeter and area of the
red square.
Perimeter:
Each side of the red square is the diagonal of the square grid. Each grid
square has a side length of 1 cm, so the diagonal is √2 cm. The perimeter of the
red square is P = 4s = 4 √2 cm.
CONFIDENTIAL 41
A = 1 (d1d2 ) = 1 (√2)(√2) = 2 cm2. 2 2
Area:
Method 1: d2 of a kite with d1 = 16 cm, and A = 48 cm2.
Method 2: The side length of the red square is √2 cm, so the area if A = (s2) = (√2)2 = 2 cm2.
CONFIDENTIAL 42
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