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Holt McDougal Geometry Developing Formulas Circles and Regular Polygons Developing Formulas Circles and Regular Polygons Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Geometry

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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

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Holt McDougal Geometry 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

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Holt McDougal Geometry Developing Formulas Circles and Regular Polygons circle center of a circle center of a regular polygon apothem central angle of a regular polygon Vocabulary

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Holt McDougal Geometry 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

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Holt McDougal Geometry 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 = r 2. The more pieces you divide the circle into, the more accurate the estimate will be.

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Holt McDougal Geometry Developing Formulas Circles and Regular Polygons

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

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Holt McDougal Geometry 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

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

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Holt McDougal Geometry 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

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

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Holt McDougal Geometry 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

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Holt McDougal Geometry 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 diameter36 cm diameter48 cm diameter A = (12) 2 A = (18) 2 A = (24) 2 ≈ cm 2 ≈ cm 2 ≈ cm 2

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Holt McDougal Geometry 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 = (10)C = (12)C = (14) C = 31.4 in.C = 37.7 in.C = 44.0 in.

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Holt McDougal Geometry 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

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Holt McDougal Geometry Developing Formulas Circles and Regular Polygons Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE.

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Holt McDougal Geometry 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:

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Holt McDougal Geometry Developing Formulas Circles and Regular Polygons

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Holt McDougal Geometry 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 .

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

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Holt McDougal Geometry 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 ft 2

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Holt McDougal Geometry 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!

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Holt McDougal Geometry 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.

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Holt McDougal Geometry 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.

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Holt McDougal Geometry 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 cm 2

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Holt McDougal Geometry 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.

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Holt McDougal Geometry 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

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Holt McDougal Geometry 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 cm 2

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Holt McDougal Geometry Developing Formulas Circles and Regular Polygons Lesson Quiz: Part I Find each measurement. 1. the area of D in terms of A = 49 ft 2 2. the circumference of T in which A = 16 mm 2 C = 8 mm

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Holt McDougal Geometry 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. A 1 ≈ 12.6 in 2 ; A 2 ≈ 63.6 in 2 ; A 3 ≈ in 2 Find the area of each regular polygon to the nearest tenth. 4. a regular nonagon with side length 8 cm A ≈ cm 2 5. a regular octagon with side length 9 ft A ≈ ft 2

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