Indicate whether the statement is true or false.
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1 Indicate whether the statement is true or false. 1. For a circle, the constant ratio of the circumference C to length of diameter d is represented by the number. 2. For an isosceles triangle with length of leg s and length of base b, the perimeter P is given by P = 2s + b. 3. The area of the trapezoid with base lengths and and altitude h is given by A = h( + ). 4. The area of an equilateral triangle with sides of length s is given by A =. 5. If the base of the pyramid shown is a square, then the pyramid is a regular square pyramid. 6. Prisms and pyramids are types of polyhedrons. 7. If two triangles have the same areas and the same length of altitude, these triangles must be congruent. 8. An octagonal prism has 9 faces. Copyright Cengage Learning. Powered by Cognero. Page 1
2 9. For a square with apothem of length a, the perimeter equals 4a. 10. For a circle of radius r, the area is given by A = 2 r. Indicate the answer choice that best completes the statement or answers the question. 11. If the area of a regular octagon is 25, find the area of the regular octagon whose sides are twice those of the first octagon. a. 50 b. 25 c. 50 d. None of These 12. For a triangle whose perimeter measures 36 units, the radius of the inscribed circle is 3.Find the area of the triangle. a. 27 b. 54 c. 108 d. None of These 13. A food container is in the shape of a right circular cylinder. To the nearest cubic inch, find the exact volume of the food container if the radius of the base measures 2.3 inches and the height of the container is 6.2 inches. a. 52 b. 76 c. 92 d For a right circular cylinder with radius of base r and length of altitude h, the formula for the lateral area L is: a. b. c. d. None of These 15. Find the lateral area for a regular hexagonal prism in which the base edges are 4 inches long and the altitude is h = 6 inches. a. 36 b. 72 c. 144 d. None of These 16. In kite HJKL, HJ = HL and JK = LK. If m = 90, HJ = 12, and JK = 5, find the length of diagonal. a. b. c. d. Copyright Cengage Learning. Powered by Cognero. Page 2
3 17. For the right triangular prism shown, the base is a right triangle with sides of lengths 3 in, 4 in, and 5 in. If the prism has a height of 6 inches, find its volume. a. 24 b. 36 c. 72 d Each lateral face of a regular hexagonal pyramid has lateral edges of length 10 inches and a base edge of 12 inches. What is the lateral area of the pyramid? a. 48 b. 288 c. 384 d. 19. To the nearest tenth of square inch, find the area of the regular pentagon whose apothem measures 4.2 inches and whose sides measure 6.2 inches. a b c d In, m = 45 and OA = 6 cm. Find. a. 1.5 cm b. 3 cm c. 6 cm d. 12 cm 21. For the triangle with sides of lengths a, b, and c, find an expression for the semiperimeter s of the triangle. 22. In, m = 90. What fraction represents the part of the circumference that is the length of? Copyright Cengage Learning. Powered by Cognero. Page 3
4 23. A right circular cone has a radius of 6 inches and an altitude of 8 inches. Use in order to find the exact total area of the cone. 24. and have the same length of base and the same length of altitude. If the area of is 25, find the area of. 25. Write the formula for the area A of any quadrilateral that has perpendicular diagonals of lengths and. 26. A square has an area of 13. What is the exact length of each side of the square? 27. An equilateral triangle has an inscribed circle with the length of radius 3 inches. Find the exact length of each side of the equilateral triangle For, radii and and arc bound a sector. If m = 100 and PX = 5.6 inches, find the area of the sector correct to the nearest tenth of a square inch. 29. In the form, verify Euler s Formula for the pyramid shown. 30. For a square whose length of apothem is a, find an expression containing a that represents the area A of the square. Copyright Cengage Learning. Powered by Cognero. Page 4
5 31. A square is inscribed in a circle. If the area of the circle is and the area of the square is 18, find the area of the four-part region that lies outside the square but inside the circle. 32. For regular pentagon ABCDE, diagonals and are drawn to form. Find m. 33. The area of a circle is To the nearest tenth of a square inch, find the area of the square that is circumscribed about this circle. 34. If m = 45 and OA = 4.3 cm, find l correct to the nearest tenth of a centimeter. 35. For a triangle with sides of lengths 13 cm, 14 cm, and 15 cm, the altitude to the 14 cm side separates that side into segments of lengths 5 cm and 9 cm. For the solid of revolution determined by revolving the triangular region about the 14 cm side, find its volume. 36. In, chord is 12 inches long and m = 90. Find the exact area of the segment bounded by and. Copyright Cengage Learning. Powered by Cognero. Page 5
6 37. For, the radius has length r and m = 180. In terms of r, write an expression for the perimeter of the sector bounded by diameter and. 38. Like the pyramid, the volume formula for a right circular cone can be expressed as. State another form of the formula for the volume of the cone. 39. A right rectangular prism has a base that measures 6 in by 8 in. If the volume of the box is 360, find the height of the prism. 40. Given that 1 foot = 12 inches, what is the number of square inches in 1 square foot? 41. Use the drawing provided to explain the following theorem. The area of any quadrilateral with perpendicular diagonals of lengths and is given by. Given: Quadrilateral with at point F; and Prove: 42. Consider a circle with diameter length d, radius length r, and circumference C. Given that formula for the circumference of a circle is given by., explain why the 43. Using the drawing provided and fact that the area of a parallelogram is given by, show that the area of a triangle is given by. Copyright Cengage Learning. Powered by Cognero. Page 6
7 44. Use the drawing provided to explain the following theorem. The area A of a regular polygon whose apothem has length a and whose perimeter is P is given by. Given: Regular polygon apothem so that Prove: with center O and length s for each side; 45. Where is the degree measure for the arc of a sector of a circle, the ratio of the area of the sector to that of the area of the circle is given by. Use this ratio to explain why the area of the sector is given by. [Note: In the figure, the sector with arc measure is bounded by radii,, and.] Copyright Cengage Learning. Powered by Cognero. Page 7
8 Answer Key 1. True 2. True 3. False 4. False 5. False 6. True 7. False 8. False 9. False 10. False 11. d 12. b 13. d 14. b 15. c 16. d 17. b 18. b 19. b 20. a 21. s = ( ) Copyright Cengage Learning. Powered by Cognero. Page 8
9 25. A = 26. inches 27. inches = A = cm (18-36) 37. 2r + r or (2 + )r in To box the quadrilateral, we draw auxiliary lines as follows: through point D, we draw ; through point B, we draw ; through point A, we draw ; and through point C, we draw. The quadrilateral formed is a parallelogram that can be shown to have a right angle; this follows from the fact that is a parallelogram that contains a right angle at vertex F... so the opposite angle (at vertex R) must also be a right angle. Because is a diagonal of (actually rectangle, ; that is, a diagonal of a parallelogram separates the parallelogram into 2 congruent. Similarly,,, and. Thus, the area of quadrilateral is one half of that of rectangle. But the area of is, so is given by. 42. Given that, we use the Multiplication Property of Equality to obtain. Because the length of the diameter of a circle is twice that of a radius,. By substitution, or. Copyright Cengage Learning. Powered by Cognero. Page 9
10 43. For the parallelogram ( ) with base length b and altitude length h, the area is given by. By the Area- Addition Postulate,. But diagonal of separates the parallelogram into 2 congruent triangles that have equal areas. By substitution, and by division (or multiplication),. 44. From center O, we draw radii,,,, and. Because the radii are congruent to each other and the sides of the regular polygon are all congruent to each other as well,,,,, and are all congruent to each other by SSS. Each of the congruent triangles has an altitude length of. Further, the length of each base of a triangle is s, the length of side of the polygon. Therefore, the area of the regular polygon is Because the sum of the sides equals perimeter P, we have. 45. Given that, it follows that. Where r is the length of radius of the circle, the area of the circle is given by. By substitution, it follows that. Copyright Cengage Learning. Powered by Cognero. Page 10
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