9 2 Developing Formulas for Circles and Regular

9 -2 Developing Formulas for Circles and Regular Polygons Objectives 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. 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. The irrational number is defined as the ratio of the circumference C to the diameter d, or Solving for C gives the formula C = d. Also d = 2 r, so C = 2 r. Holt Geometry

9 -2 Developing Formulas for Circles and Regular Polygons Holt Geometry

9 -2 Developing Formulas for Circles and Regular Polygons Example 1 A: Finding Measurements of Circles Find the area of K in terms of . Holt Geometry A = r 2 Area of a circle. A = (3)2 Divide the diameter by 2 to find the radius, 3. A = 9 in 2 Simplify.

9 -2 Developing Formulas for Circles and Regular Polygons Example 1 B: Finding Measurements of Circles Find the radius of J if the circumference is (64 x) m. C = 2 r (64 x) = 2 r r = (32 x) m Holt Geometry Circumference of a circle Substitute (65 x + 14) for C. Divide both sides by 2.

9 -2 Developing Formulas for Circles and Regular Polygons Example 1 C: Finding Measurements of Circles Find the circumference of M if the area is 25 x 2 ft 2 Step 1 Use the given area to solve for r. A = r 2 Holt Geometry Area of a circle 25 x 2 = r 2 Substitute 25 x 2 for A. 25 x 2 = r 2 Divide both sides by . 5 x = r Take the square root of both sides.

9 -2 Developing Formulas for Circles and Regular Polygons Example 1 C Continued Step 2 Use the value of r to find the circumference. C = 2 r Holt Geometry C = 2 (5 x) Substitute 5 x for r. C = 10 x ft Simplify.

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 = (4 x – 6) m. A = r 2 A = (2 x – Area of a circle. 3)2 m Divide the diameter by 2 to find the radius, 2 x – 3. A = (4 x 2 – 12 x + 9) m 2 Simplify. Holt Geometry

9 -2 Developing Formulas for Circles and Regular Polygons Example 2: Cooking Application 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. 24 cm diameter A = (12)2 ≈ 452. 4 cm 2 Holt Geometry 36 cm diameter A = (18)2 ≈ 1017. 9 cm 2 48 cm diameter A = (24)2 ≈ 1809. 6 cm 2

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 = (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 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 Holt Geometry

9 -2 Developing Formulas for Circles and Regular Polygons Example 3 A: Finding the Area of a Regular Polygon Find the area of regular heptagon with side length 2 ft to the nearest tenth. Step 1 Draw the heptagon. Draw an isosceles triangle with its vertex at the center of the heptagon. The central angle is . Draw a segment that bisects the central angle and the side of the polygon to form a right triangle. Holt Geometry

9 -2 Developing Formulas for Circles and Regular Polygons Example 3 A Continued Step 2 Use the tangent ratio to find the apothem. The tangent of an angle is opp. leg. adj. leg Solve for a. Holt Geometry

9 -2 Developing Formulas for Circles and Regular Polygons Example 3 A 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) = 14 ft. A 14. 5 ft 2 Holt Geometry Simplify. Round to the nearest tenth.

9 -2 Developing Formulas for Circles and Regular Polygons Example 3 B: 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 . Draw a segment that bisects the central angle and the side of the polygon to form a right triangle. Holt Geometry

9 -2 Developing Formulas for Circles and Regular Polygons Example 3 B Continued Step 2 Use the tangent ratio to find the apothem. The tangent of an angle is Solve for a. Holt Geometry opp. leg. adj. leg

9 -2 Developing Formulas for Circles and Regular Polygons Example 3 B 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. A 279. 9 cm 2 Holt Geometry Simplify. Round to the nearest tenth.

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. Step 1 Draw the octagon. Draw an isosceles triangle with its vertex at the center of the octagon. The central angle is . Draw a segment that bisects the central angle and the side of the polygon to form a right triangle. Holt Geometry

9 -2 Developing Formulas for Circles and Regular Polygons Check It Out! Example 3 Continued Step 2 Use the tangent ratio to find the apothem The tangent of an angle is opp. leg. adj. leg Solve for a. Holt Geometry

9 -2 Developing Formulas for Circles and Regular Polygons Check It Out! Example 3 Continued Step 3 Use the apothem and the given side length to find the area. Area of a regular polygon The perimeter is 4(8) = 32 cm. A ≈ 77. 3 cm 2 Holt Geometry Simplify. Round to the nearest tenth.