Holt Geometry
9-2 Developing Formulas for Circles and Regular Polygons9-2 Developing Formulas for
Circles and Regular Polygons
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Geometry
9-2 Developing Formulas for Circles and Regular Polygons
Lesson Quiz: Part IFind each measurement.
1. the height of the parallelogram, in whichA = 182x2 mm2
h = 9.1x mm
2. the perimeter of a rectangle in which h = 8 in. and A = 28x in2
P = (16 + 7x) in.
Holt Geometry
9-2 Developing Formulas for Circles and Regular Polygons
Lesson Quiz: Part II
3. the area of the trapezoid
A = 16.8x ft2
4. the base of a triangle in which h = 8 cm and A = (12x + 8) cm2
b = (3x + 2) cm
5. the area of the rhombus
A = 1080 m2
Holt Geometry
9-2 Developing Formulas for Circles and Regular Polygons
Lesson Quiz: Part III
6. The wallpaper pattern shown is a rectangle with a base of 4 in. and a height of 3 in. Use the grid to find the area of the shaded kite.
A = 3 in2
Holt Geometry
9-2 Developing Formulas for Circles and Regular Polygons
Warm UpFind 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 Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for Circles and Regular Polygons
circlecenter of a circlecenter of a regular polygonapothemcentral angle of a regular polygon
Vocabulary
Holt Geometry
9-2 Developing Formulas for 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 formulaC = d. Also d = 2r, so C = 2r.
The irrational number is defined as the ratio ofthe circumference C tothe diameter d, or
Holt Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for Circles and Regular Polygons
Holt Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for Circles and Regular Polygons
Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE.
Holt Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for Circles and Regular Polygons
Holt Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for Circles and Regular Polygons
Example 3A Continued
tan 25.7 1
a
Solve for a.
The tangent of an angle is . opp. legadj. leg
Step 2 Use the tangent ratio to find the apothem.
a ≈ 2.078 but they don’t want usTo round until the last step
Holt Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for Circles and Regular Polygons
Example 3B Continued
Solve for a.
The tangent of an angle is . opp. legadj. leg
Step 2 Use the tangent ratio to find the apothem.
Holt Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for 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. legadj. leg
Holt Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for 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 Geometry
9-2 Developing Formulas for 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
Holt Geometry
9-2 Developing Formulas for Circles and Regular Polygons