10-2 Developing Formulas Circles and Regular Polygons · Holt McDougal Geometry 10-2 Developing Formulas Circles and Regular Polygons Warm Up Find the unknown side lengths in each

Holt McDougal Geometry

10-2 Developing Formulas

Circles and Regular Polygons 10-2

Developing Formulas

Circles and Regular Polygons

Holt Geometry

Warm Up

Lesson Presentation

Lesson Quiz





Warm Up Find the unknown side lengths in each special

right triangle.

1. a 30°-60°-90° triangle with hypotenuse 2 ft

2. a 45°-45°-90° triangle with leg length 4 in.

3. a 30°-60°-90° triangle with longer leg length 3m




Develop and apply the formulas for the area and circumference of a circle.

Develop and apply the formula for the area of a regular polygon.

Objectives




circle

center of a circle

center of a regular polygon

apothem

central angle of a regular polygon

Vocabulary




A circle is the locus of points 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 radius r = AB and diameter d = CD.

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

The irrational number

is defined as the ratio of

the circumference C to

the diameter d, or




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.

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







Find the area of K in terms of .

Example 1A: Finding Measurements of Circles

A = r2 Area of a circle.

Divide the diameter by 2 to find the radius, 3.

Simplify.

A = (3)2

A = 9 in2




Find the radius of J if the circumference is (65x + 14) m.

Example 1B: Finding Measurements of Circles

Circumference of a circle

Substitute (65x + 14) for C.

Divide both sides by 2.

C = 2r

(65x + 14) = 2r

r = (32.5x + 7) m




Find the circumference of M if the area is 25 x2 ft2

Example 1C: Finding Measurements of Circles

Step 1 Use the given area to solve for r.

Area of a circle

Substitute 25x2 for A.

Divide both sides by .

Take the square root of both sides.

A = r2

25x2 = r2

25x2 = r2

5x = r




Example 1C Continued

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

Substitute 5x for r.

Simplify.

C = 2(5x)

C = 10x ft

C = 2r




Check It Out! Example 1

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

A = r2 Area of a circle.

A = (2x – 3)2 m

A = (4x2 – 12x + 9) m2

Divide the diameter by 2 to find the radius, 2x – 3.

Simplify.




The key gives the best possible approximation for on your calculator. Always wait until the last step to round.

Helpful Hint




A pizza-making kit contains three circular baking stones with diameters 24 cm, 36 cm, and 48 cm. Find the area of each stone. Round to the nearest tenth.

Example 2: Cooking Application

24 cm diameter 36 cm diameter 48 cm diameter

A = (12)2 A = (18)2 A = (24)2

≈ 452.4 cm2 ≈ 1017.9 cm2 ≈ 1809.6 cm2





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

10 in. diameter 12 in. diameter 14 in. diameter

C = d C = d C = d

C = (10) C = (12) C = (14)

C = 31.4 in. C = 37.7 in. C = 44.0 in.




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 sides

pass through consecutive vertices. Each central

angle measure of a regular n-gon is




Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE.




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

The perimeter is P = ns.

area of each triangle:

total area of the polygon:







Find the area of regular heptagon with side length 2 ft to the nearest tenth.

Example 3A: Finding the Area of a Regular Polygon

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

Step 1 Draw the heptagon. Draw an isosceles

triangle with its vertex at the center of the

heptagon. The central angle is .




Example 3A Continued

tan 25.7

1

a

Solve for a.

The tangent of an angle is . opp. leg

adj. leg

Step 2 Use the tangent ratio to find the apothem.




Example 3A Continued

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

Area of a regular polygon

The perimeter is 2(7) = 14ft.

Simplify. Round to the nearest tenth.

A 14.5 ft2




The tangent of an angle in a right triangle is the ratio of the opposite leg length to the adjacent leg length. See page 525.

Remember!




Example 3B: Finding the Area of a Regular Polygon

Find the area of a regular dodecagon with side length 5 cm to the nearest tenth.


Step 1 Draw the dodecagon. Draw an isosceles

triangle with its vertex at the center of the

dodecagon. The central angle is .




Example 3B Continued

Solve for a.


adj. leg

Step 2 Use the tangent ratio to find the apothem.




Example 3B Continued



The perimeter is 5(12) = 60 ft.


A 279.9 cm2





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


Step 1 Draw the octagon. Draw an isosceles triangle

with its vertex at the center of the octagon. The

central angle is .




Step 2 Use the tangent ratio to find the apothem

Solve for a.

Check It Out! Example 3 Continued


adj. leg





Check It Out! Example 3 Continued


The perimeter is 4(8) = 32cm.


A ≈ 77.3 cm2




Lesson Quiz: Part I

Find each measurement.

1. the area of D in terms of

A = 49 ft2

2. the circumference of T in which A = 16 mm2

C = 8 mm




Lesson Quiz: Part II

Find each measurement.

3. Speakers come in diameters of 4 in., 9 in., and 16 in. Find the area of each speaker to the nearest tenth.

A1 ≈ 12.6 in2 ; A2 ≈ 63.6 in2 ; A3 ≈ 201.1 in2

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

4. a regular nonagon with side length 8 cm

A ≈ 395.6 cm2

5. a regular octagon with side length 9 ft

A ≈ 391.1 ft2

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10-2 Developing Formulas Circles and Regular Polygons · Holt McDougal Geometry 10-2 Developing Formulas Circles and Regular Polygons Warm Up Find the unknown side lengths in each