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Areas of Regular Polygons and Circles Chapter 9 – Lesson 2 Areas of Regular Polygons and Circles
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.
Vocabulary circle center of a circle center of a regular polygon apothem central angle of a regular polygon
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, since d = 2r, then C = 2r. The formula for Area of a Circle is A = r2.
Example 1A: Finding Measurements of Circles Find the area of K in terms of . A = r2 Area of a Circle A = (3)2 A = 9 in2
Example 1B: Finding Measurements of Circles Find the radius of J if the circumference is (65x + 14) m. C = 2r (65x + 14) = 2r r = (32.5x + 7) m J
Example 1C: Finding Measurements of Circles Find the circumference of M if the area is 25 x2 ft2 Step 1 Use the given area to solve for r. A = r2 Area of a Circle 25x2 = r2 M 25x2 = r2 5x = r Step 2 Use the value of r to find the circumference. C = 2r Circumference of a Circle C = 2(5x) C = 10x ft
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
Always wait until the last step to round. The key gives the best possible approximation for on your calculator. Always wait until the last step to round. Helpful Hint
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 36 cm diameter 48 cm diameter A = (12)2 A = (18)2 A = (24)2 ≈ 452.4 cm2 ≈ 1017.9 cm2 ≈ 1809.6 cm2
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 to the nearest tenth of an inch. 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 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
Definition Apothem [of a regular polygon] A segment from the center of the figure which is perpendicular to the side is called the apothem apothem Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE.
Area of a Regular Polygon Area A of a regular polygon with perimeter P and apothem a is given by:
The tangent of an angle in a right triangle is the ratio of the opposite leg length to the adjacent leg length. Remember! See page 525. Soh-Cah-Toa
Example 3: Finding the Area of a Regular Polygon Find the area of regular heptagon with side length 2 ft to the nearest tenth. Area of a regular polygon Step 1 Draw the heptagon. Step 2 Draw an isosceles triangle with its vertex at the center of the heptagon. Central angle measure is 25.7o Step 3 Draw a segment that bisects the central angle and the side of the polygon to form a right triangle. 2 a 1 c This segment is the apothem. Step 4 Use the tangent ratio to find the apothem.
Step 4 Use the tangent ratio to find the apothem. Example 3 Continued Step 4 Use the tangent ratio to find the apothem. Tangent of an angle is . opp. leg adj. leg a Step 5 Use the apothem and the given side length to find the area. Area of a regular polygon A 14.5 ft2
Step 1 Draw the dodecagon. Check It Out! Example 3 Find the area of a regular dodecagon with side length 5 cm to the nearest tenth. Step 1 Draw the dodecagon. Step 2 Draw an isosceles with its vertex at the center of the dodecagon. c 15o The central angle is . 5 a c 2.5 Step 3 Draw a segment that bisects the central angle and the side of the polygon to form a right . This segment is the apothem. Step 4 Use the tangent ratio to find the apothem.
Check It Out! Example 3 Continued Step 4 Use the tangent ratio to find the apothem. Tangent of an angle is . opp. leg adj. leg Step 5 Use the apothem and the given side length to find the area. Area of a regular polygon A 279.9 cm2
30-60-90 Triangle 45-45-90 Triangle
Kahoot!
Lesson Summary: 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.
Preview of the Next Lesson: Objectives Use the Area Addition Postulate to find the areas of composite figures. Use composite figures to estimate the areas of irregular shapes.
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