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Areas of Trapezoids, Rhombuses, and Kites

94.5 3 in. 2. 100 2 m 2. Areas of Trapezoids, Rhombuses, and Kites. Lesson 10-2. Lesson Quiz. 1. Find the area of a trapezoid with bases 3 cm and 19 cm and height 9 cm. 2. Find the area of a trapezoid in a coordinate plane with

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Areas of Trapezoids, Rhombuses, and Kites

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  1. 94.5 3 in.2 100 2 m2 Areas of Trapezoids, Rhombuses, and Kites Lesson 10-2 Lesson Quiz 1. Find the area of a trapezoid with bases 3 cm and 19 cm and height 9 cm. 2. Find the area of a trapezoid in a coordinate plane with vertices at (1, 1), (1, 6), (5, 9), and (5, 1). Find the area of each figure in Exercises 3–5. Leave your answers in simplest radical form. 3. trapezoid ABCD 4. kite with diagonals 20 m and 10 2 m long 5. rhombus MNOP 99 cm2 26 square units 840 mm2 10-3

  2. Areas of Regular Polygons Lesson 10-3 Check Skills You’ll Need (For help, go to Lesson 8-2.) Find the area of each regular polygon. If your answer involves a radical, leave it in simplest radical form. 1. 2. 3. 4. a hexagon with sides of 4 in. 5. an octagon with sides of 2 3 cm Find the perimeter of the regular polygon. Check Skills You’ll Need 10-3

  3. Solutions 1. The triangle is equilateral and equiangular, so each of its angles is 60°. The altitude divides the triangle into two 30°-60°-90° triangles. Since the short leg of the 30°-60°-90° triangle is 5cm, the long leg, which is the altitude of the equilateral triangle, is 5 3 cm. The base is 10 cm and the height is 5 3 cm. The area A = bh = (10)(5 3) = 25 3 cm2. 2. The diagonal is 10 cm and divides the square into two 45°-45°-90° triangles. The legs are each , or 5 2 ft. The base is 5 2 and the height is 5 2. The area A = bh = (5 2)(5 2) = (25)(2) = 50 ft2. 1 2 1 2 10 2 Areas of Regular Polygons Lesson 10-3 Check Skills You’ll Need 10-3

  4. 20 3 100 3 100 3 3 Areas of Regular Polygons Lesson 10-3 Check Skills You’ll Need Solutions (continued) 3. The triangle is equilateral and equiangular, so each of its angles is 60°. The altitude divides the triangle into two 30°-60°-90° triangles. The altitude is 10 m. Since the long leg of the 30°-60°-90° triangle is 10m the short leg is m and the hypotenuse is m. Thus the base of the equilateral triangle is m. The area A = bh = ( )(10) = , or m2. 4. The perimeter of a polygon is the sum of the lengths of its sides. A regular hexagon has six sides of the same length. Since each side has length 4 in., the perimeter is = (6)(4) = 24 in. 5. The perimeter of a polygon is the sum of the lengths of its sides. A regular octagon has eight sides of the same length. Since each side has length 2 3 cm, the perimeter is (8)(2 3) = 16 3 cm. 10 3 20 3 20 3 1 2 1 2 10-3

  5. Areas of Regular Polygons Lesson 10-3 Notes The center of a regular polygonis equidistant from the vertices. The radius of a regular polygon is the distance from the center to a vertex. The apothemis the distance from the center to a side. A central angle of a regular polygonhas its vertex at the center, and its sides pass through consecutive vertices. Each central angle measure of a regular n-gon is 10-3

  6. Areas of Regular Polygons Lesson 10-3 Notes Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE. 10-3

  7. area of each triangle: total area of the polygon: Areas of Regular Polygons Lesson 10-3 Notes 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. 10-3

  8. Areas of Regular Polygons Lesson 10-3 Notes 10-3

  9. 360 6 m 1 = = 60 Divide 360 by the number of sides. 1 2 m 2 = (60) = 30 Substitute 60 for m 1. m 2 = m 1The apothem bisects the vertex angle of the isosceles triangle formed by the radii. 1 2 m 3 = 180 – (90 + 30) = 60 The sum of the measures of the angles of a triangle is 180. m 1 = 60, m 2 = 30, and m 3 = 60. Areas of Regular Polygons Lesson 10-3 Additional Examples Finding Angle Measures A portion of a regular hexagon has an apothem and radii drawn. Find the measure of each numbered angle. Quick Check 10-3

  10. Find the area of a regular polygon with twenty 12-in. sides and a 37.9-in. apothem. 1 2 A = apArea of a regular polygon 1 2 A = (37.9)(240) Substitute 37.9 for a and 240 for p. Areas of Regular Polygons Lesson 10-3 Additional Examples Finding the Area of a Regular Polygon p = nsFind the perimeter. p = (20)(12) = 240 Substitute 20 for n and 12 for s. A = 4548 Simplify. The area of the polygon is 4548 in.2 Quick Check 10-3

  11. A library is in the shape of a regular octagon. Each side is 18.0 ft. The radius of the octagon is 23.5 ft. Find the area of the library to the nearest 10 ft2. Consecutive radii form an isosceles triangle, as shown below, so an apothem bisects the side of the octagon. 1 2 To apply the area formula A = ap, you need to find a and p. Areas of Regular Polygons Lesson 10-3 Additional Examples Real-World Connection 10-3

  12. Step 1: Find the apothem a. a2 + (9.0)2 = (23.5)2Pythagorean Theorem a2 + 81 = 552.25 Solve for a. a2 = 471.25 a 21.7 Areas of Regular Polygons Lesson 10-3 Additional Examples (continued) Step 2: Find the perimeter p. p = nsFind the perimeter. p = (8)(18.0) = 144 Substitute 8 for n and 18.0 for s, and simplify. 10-3

  13. Step 3: Find the area A. A = apArea of a regular polygon A (21.7)(144) Substitute 21.7 for a and 144 for p. A 1562.4 Simplify. 1 2 1 2 Areas of Regular Polygons Lesson 10-3 Additional Examples (continued) To the nearest 10 ft2, the area is 1560 ft2. Quick Check 10-3

  14. Textbook

  15. 6 3 in.2 48 3 m2 Areas of Regular Polygons Lesson 10-3 Lesson Quiz 1. Find m 1. 2. Find m 2. 3. Find m 3. 4. Find the area of a regular 9-sided figure with a 9.6-cm apothem and 7-cm side. For Exercises 5 and 6, find the area of each regular polygon. Leave your answer in simplest radical form. 5.6. Use the portion of the regular decagon for Exercises 1–3. 36 18 72 302.4 cm2 10-3

  16. Textbook

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