10-2 Developing Formulas Circles and Regular Polygons · Developing Formulas Circles and Regular...
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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
Holt McDougal Geometry
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
circle
center of a circle
center of a regular polygon
apothem
central angle of a regular polygon
Vocabulary
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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.
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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.
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
The key gives the best possible approximation for on your calculator. Always wait until the last step to round.
Helpful Hint
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
Check It Out! Example 2
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.
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE.
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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:
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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 .
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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.
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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!
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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.
Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.
Step 1 Draw the dodecagon. Draw an isosceles
triangle with its vertex at the center of the
dodecagon. The central angle is .
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
Example 3B Continued
Solve for a.
The tangent of an angle is . opp. leg
adj. leg
Step 2 Use the tangent ratio to find the apothem.
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
Example 3B Continued
Step 3 Use the apothem and the given side length to find the area.
Area of a regular polygon
The perimeter is 5(12) = 60 ft.
Simplify. Round to the nearest tenth.
A 279.9 cm2
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
Check It Out! Example 3
Find the area of a regular octagon with a side length of 4 cm.
Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.
Step 1 Draw the octagon. Draw an isosceles triangle
with its vertex at the center of the octagon. The
central angle is .
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
Step 2 Use the tangent ratio to find the apothem
Solve for a.
Check It Out! Example 3 Continued
The tangent of an angle is . opp. leg
adj. leg
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
Step 3 Use the apothem and the given side length to find the area.
Check It Out! Example 3 Continued
Area of a regular polygon
The perimeter is 4(8) = 32cm.
Simplify. Round to the nearest tenth.
A ≈ 77.3 cm2
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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
Holt McDougal Geometry
10-2 Developing Formulas
Circles and Regular Polygons
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