Geometry Developing Formulas for Circles and Regular Polygons

CONFIDENTIAL 1

Geometry

Developing Formulas for Circles and

Regular Polygons

CONFIDENTIAL 2

Warm up

Find the each measurement:

1) d2 of a kite if A = 14 cm2 and d1 = 20 cm.

2) the area of a trapezoid in which b1 = 3 yd, b2 = 6 yd and h = 4 yd

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A circle is the locus of a point in a plane that are a fixed distance from a point called the center of the circle. A circle is named by the symbol and its center. A has

a radius r= AB and diameter d = CD.

Formulas for Circle

The irrational number ∏ is defined as the ratio of the circumference C to the

diameter d, or ∏ = C. d

Solving for C gives the formula C= ∏d.Also d = 2r, so C= 2∏r. C D

B

A

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You can use the circumference of a circle to find its area. Divide the circle and rearrange the pieces to make a

shape that resembles a parallelogram.

The base of the parallelogram is about half the circumference, or ∏r, and the height is close to the radius r.

So A ≈ ∏r. r = ∏r2.

∏r

r

Formulas for Circle

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The more pieces you divide the circle into, the more accurate the estimate will be.

Formulas for Circle

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A circle with diameter d and radius r has circumference C= ∏d or C= 2∏r and area A = ∏r2.

Circumference and Area of a Circle

diameterra

diu

s

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Finding measurements of of Circle

16 cm

P

Find each measurement:

A = ∏r2

A = ∏(8)2

A = 64∏ cm2

Area of a circle.

Divide the diameter by 2 to find the radius, 8.

Simplify.

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Finding measurements of of CircleFind each measurement:

B) the radius of X in which C = 24∏ in.

C = 2∏r

24A = 2∏r

r = 12 in

Circumference of a circle.

Substitute 24∏ for C.

Divide both sides by 2∏.

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Finding measurements of of Circle

C) the circumference of S in which A = 9x2∏ cm2.

A = ∏r2

9x2∏ = ∏r2

9x2= 2r2

3x= r

Area of a circle.Substitute 9x2 for A.Divide both sides by 2.Take square root of both sides.

Step 1: Use the given area to solve for r.

C = 2∏rC = 2∏(3x)C = 6x∏

Substitute 3x for r.Divide both sides by 2.Simplify.

Step 2: Use the value of r to find the circumference.

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Now you try!

1) Find the area of A in terms of ∏ in which C = (4x - 6)∏ m.

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Music Application

10 in. diameterA = ∏(5)2 since, r = 10 2 A≈ 78.5 in2

14 in. diameterA = ∏(7)2 since, r = 12 2 A≈ 153.9 in2

A drum kit contains three drums with diameters of 10 in., 12 in. and 14 in. Find the area of the top of each drum.

Round to the nearest tenth.

12 in. diameterA = ∏(6)2 since, r = 10 2 A≈ 113.1 in2

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Now you try!

2) A drum kit contains three drums with diameters of 10 in., 12 in. and 14 in. Find the circumference of the

top of each drum. Round to the nearest tenth.

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Formulas for a Regular Polygon

The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a

side. A central angle of a regular polygon has its vertex at the center, and its side pass through consecutive vertices. Each central angle measure of a regular n - gon is 360°.

n

C

H

B

DG

F E

Regular pentagon DEFGH has center C, apothem BC,

and central angle /DCE.

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To find the area of a regular n - gon with side length s and apothem a, divide it into n congruent isosceles

triangles.

area of each triangle : 1as 2

total area of the polygon: A = n. 1as 2

or A= 1aP 2

The perimeter is P = ns

Formulas for a Regular Polygon

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The area of a regular polygon with apothem a and perimeter P is

A = 1aP 2

Formulas for a Regular Polygon

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A) a regular hexagon with side length 6 m

The perimeter is 6(6)= = 36 m .

Finding the area of a Regular PolygonFind the area of each regular polygon. Round to the nearest tenth.

Area of a regular polygon

Substitute 3√3 for a and 36 for P

Simplify

6 m

3m3√3m

A= 1aP 2

A= 1(3√3)(36) 2

A= 54√3≈ 93.5 m2

The hexagon can be divided into 6 equilateral triangles with side length 6 m. By the 30°-60°- 90° Triangle theorem, the apothem is

3√3.

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B) a regular pentagon with side length 6 in.

36°

4 in

a

Step 1: Draw the pentagon. Draw an isosceles triangle with its vertex at the centre of the pentagon. The

central angle is 360° = 72 ° . 5

Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.

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36°

4 in

a

Step 2: Use the tangent ratio to find the apothem.

The tangent of an angle is opp. leg adj. leg

Solve for a.

tan 36° = 4 a a = 4 tan36°

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Step 3 : Use the apothem and the given side length to find the area .

Area of a regular polygon.

The perimeter is 8(5) = 40 in.

Simplify. Round to the nearest tenth.

A= 1aP 2 A = 1. 4 .(40) 2 tan36°

A ≈ 101.1 in2

36°

4 in

a

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Now you try!

3) Find the area of a regular octagon with a side of 4 cm.

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Now some problems for you to practice !

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Find each measurement:

Assessment

1) the circumference of C. 10 cm∏C

2) the area of A in terms of ∏.3x inA

3) the circumference of Pin which A = 36∏ ft2.

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4) a food pizza parlor offers pizzas with diameters of 8 in., 10 in. and 12 in. Find the area of each pizza size.

Round to the nearest tenth.

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5) 6)

Find the area of each regular polygon. Round to the nearest tenth:

10 in

3 cm

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7) An equilateral triangle with an apothem length of 2 ft.

8) An regular dodecagon with an side length of 5 ft.

Find the area of each regular polygon. Round to the nearest tenth:

CONFIDENTIAL 26

A circle is the locus of a point in a plane that are a fixed distance from a point called the center of the circle. A circle is named by the symbol and its center. A has

a radius r= AB and diameter d = CD.

Formulas for Circle

The irrational number ∏ is defined as the ratio of the circumference C to the

diameter d, or ∏ = C. d

Solving for C gives the formula C= ∏d.Also d = 2r, so C= 2∏r. C D

B

A

Let’s review

CONFIDENTIAL 27

A circle with diameter d and radius r has circumference C= ∏d or C= 2∏r and area A = ∏r2.

Circumference and Area of a Circle

diameterra

diu

s

CONFIDENTIAL 28

Finding measurements of of Circle

C) the circumference of S in which A = 9x2∏ cm2.

A = ∏r2

9x2∏ = ∏r2

9x2= 2r2

3x= r

Area of a circle.Substitute 9x2 for A.Divide both sides by 2.Take square root of both sides.

Step 1: Use the given area to solve for r.

C = 2∏rC = 2∏(3x)C = 6x∏

Substitute 3x for r.Divide both sides by 2.Simplify.

Step 2: Use the value of r to find the circumference.

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Music Application

10 in. diameterA = ∏(5)2 since, r = 10 2 A≈ 78.5 in2

14 in. diameterA = ∏(7)2 since, r = 12 2 A≈ 153.9 in2

A drum kit contains three drums with diameters of 10 in., 12 in. and 14 in. Find the area of the top of each drum.

Round to the nearest tenth.

12 in. diameterA = ∏(6)2 since, r = 10 2 A≈ 113.1 in2

CONFIDENTIAL 30

Formulas for a Regular Polygon

The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a

side. A central angle of a regular polygon has its vertex at the center, and its side pass through consecutive vertices. Each central angle measure of a regular n - gon is 360°.

n

C

H

B

DG

F E

Regular pentagon DEFGH has center C, apothem BC,

and central angle /DCE.

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The area of a regular polygon with apothem a and perimeter P is

A = 1aP 2

Formulas for a Regular Polygon

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A) a regular hexagon with side length 6 m

The perimeter is 6(6)= = 36 m .

Finding the area of a Regular PolygonFind the area of each regular polygon. Round to the nearest tenth.

Area of a regular polygon

Substitute 3√3 for a and 36 for P

Simplify

6 m

3m3√3m

A= 1aP 2

A= 1(3√3)(36) 2

A= 54√3≈ 93.5 m2

The hexagon can be divided into 6 equilateral triangles with side length 6 m. By the 30°-60°- 90° Triangle theorem, the apothem is

3√3.

CONFIDENTIAL 33

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