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Practice Quiz Polygons, Area Perimeter, Volume

Solve geometry problems involving polygons, angles, areas, and perimeters. Find measurements using formulas and the Pythagorean theorem.

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Practice Quiz Polygons, Area Perimeter, Volume

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  1. Practice Quiz Polygons, Area Perimeter, Volume

  2. Two angles of a hexagon measure 140° each. The other four angles are equal in measure. What is the measure of each of the other four equal angles, in degrees? 1 Step 1: Find the sum of interior angles in a hexagon. x x  Number of sides = 6 140 140  Number of sides – 2 = 6 – 2 = 4 x x  Multiply 4 by 180 = 4(180) = 720

  3. Two angles of a hexagon measure 140° each. The other four angles are equal in measure. What is the measure of each of the other four equal angles, in degrees? 1 Step 1: Find the sum of interior angles in a hexagon. x x 720 Step 2: Set up equation by letting sum of angles equal 720. 140 140 x + x + x + x + 140 + 140 = 720 4x + 280 = 720 x x – 280 – 280 4x = 440 Measure of each the four equal angles is 110 x = 110

  4. 2 In trapezoid ABCD, AB = CD. What is the value of x? Method #1 Isosceles Trapezoid AandB supplementary 75° + x = 180° –75 –75 x = 105°

  5. 2 In trapezoid ABCD, AB = CD. What is the value of x? Method #2 Sum of all angles = 360° x + x + 75 + 75 = 360 2x + 150 = 360 –150 –150 2x = 210 x = 105

  6. 3 In quadrilateral DEFG, is parallel to . What is the measure of F? Trapezoid FandG supplementary E F (x+10) x + 10 + x = 180 2x + 10 = 180 –10 –10 x D G 2x = 170 F = x + 10 x = 85 = 85 + 10 = 95

  7. 4 In the figure, what is the value of x ? Step 1: Solve for c using pythagorean theorem. a2 + b2 = c2 3 3 c 9 + 7 = c2 16 = c2 x 4 = c

  8. 4 In the figure, what is the value of x ? Step 2: Solve for x using pythagorean theorem. a2 + b2 = c2 3 32 + 42 = x2 3 4 9 + 16 = x2 25 = x2 x 5 = x

  9. 5 In the figure, what is the length of ? Step 1: Solve for ? using pythagorean theorem. A a2 + b2 = c2 4 32 + 42 = ?2 x D ? 9 + 16 = ?2 3 25 = ?2 C B 13 x is length of 5 = ?

  10. 5 In the figure, what is the length of ? Step 2: Solve for x using pythagorean theorem. A a2 + b2 = c2 4 x2 + 52 = 132 x D 5 x2 + 25 = 169 3 –25 –25 x2 = 144 C B 13 x is length of x = 12

  11. 6 Find the value of each interior angle for a regular polygon with 20 sides. Step 1: Find sum of the interior angles in a regular polygon with 20 sides.  Number of sides = 20  Number of sides – 2 = 20 – 2 = 18  Multiply 18 by 180 = 18(180) = 3240 Step 2: Find value of each interior angle. Divide sum by number of sides, 20. 3240  20 = 162

  12. 7 A regular octagon is shown. What is the measure, in degrees, of X? Step 1: Find sum of the interior angles in the regular octagon.  Number of sides = 8 X  Number of sides – 2 = 8 – 2 = 6  Multiply 6 by 180 = 6(180) = 1080 Step 2: Find value of each interior angle, X. Divide sum by number of sides, 8. 1080  8 = 135

  13. 8 In the figure, . What is the value of x. HCG = DCE Vertical Angles 50 HCG = 15 DCE = 15 15 K D D = 50 Corresponding Angles K + D = 180 K and D Supplementary Angles K + 50 = 180 K = 130

  14. 8 In the figure, . What is the value of x. 50 15 130 D The sum of the angles in ∆CDE is equal to 180 x + 15 + 130 = 180 x + 145 = 180 x = 35

  15. 9 In the figure, VW = WX = VX = XY = YZ = XZ. If VZ = 12, what is the perimeter of the triangle VWX? VZ = 12 VX = 6 VW = 6 WX = 6 Perimeter VWX = 6 + 6 + 6 = 18

  16. The perimeter of an isosceles triangle is 20 inches, its base measures 8 inches. Find the length of each of its equal sides in inches. 10 x = length of each equal side Perimeter = 20 inches x + x + 8 = 20 x x 2x + 8 = 20 –8 –8 2x = 12 8 x = 6

  17. 11 If each of the equal sides of an isosceles triangle is 10, and the base is 16, what is the area of the triangle? Find height (h) Use Pythagorean Theorem a2 + b2 = c2 10 10 h2 + 82 = 102 h h2 + 64 = 100 8 –64 –64 16 h2 = 36 Base (b) = 16 h = 6

  18. 11 If each of the equal sides of an isosceles triangle is 10, and the base is 16, what is the area of the triangle? Find height (h) h = 6 10 10 6 8 16 Base (b) = 16 = 48

  19. 12 In the figure, E is the midpoint of side CB of rectangle ABCD, and x = 45°. If AB is 3 centimeters, what is the area of rectangle ABCD, in square centimeters? ∆DCE is isosceles x =45 45 3 3 45 3 3 6 Area of rectangle ABCD = 6  3 = 18 Length  Width

  20. If the area of a right triangle is 16, the length of the legs could be 13 • 8 and 2 • 12 and 4 • 10 and 6 • 20 and 12 E. 32 and 1 = 8 h = 24 b = 30 = 120 = 16

  21. In the figure, right triangle ABC is contained within right triangle AED. What is the ratio of the area of AED to the area of ABC? 14 ∆ABC is Isosceles 45 ∆AED Big Triangle 45 A = 45 D = 90 E = 45 ∆AED is Isosceles 45 8

  22. In the figure, right triangle ABC is contained within right triangle AED. What is the ratio of the area of AED to the area of ABC? 14 Area of ∆ABC Small Triangle 45 = 18 45 Area of ∆AED Big Triangle = 32 45 8

  23. The figure above shows a square region divided into four rectangular regions, three of which have areas 5x, 5x, and x2, respectively. If the area of MNOP is 64, what is the area of square QROS? 15 x 5 Area of square QROS 5 5 Length  Width = 5  5 = 25 x x x 5

  24. In the figure, CDE is an equilateral triangle and ABCE is a square with an area of 1. What is the perimeter of polygon ABCDE? 16 ABCE is a square with an area of 1 1 1 Area = s2 1 1 1 = s2 1 1 = s 1 Perimeter of ABCDE 1 + 1 + 1 + 1 + 1 = 5

  25. 17 One-third of the area of a square is 12 square inches. What is the perimeter of the square, in inches? 6 6 6 6 A = 36 Perimeter = 4(6) s2 = 36 A = s2 s = 6 Perimeter = 24

  26. All the dimensions of a certain rectangular solid are integers greater than 1. If the volume is 126 cubic inches and the height is 6 inches, what is the perimeter of the base? 18 Perimeter of Base V = Volume V = lwh 2l + 2w 126 = l  w  6 = 2(7) + 2(3) h = 14 + 6 w 6 = 20 21 = l  w (Base) l Base l = 3 or 7 w = 3 or 7

  27. A rectangular solid has a square base. The volume is 360 cubic inches and the height is 10 inches. What is the perimeter of the base? 19 Perimeter of Base V = Volume V = lwh 4(s) 360 = l  w  10 = 4(6) = 24 h 10 36 = l  w (Base) w Base l = 6 and w = 6 l 360 l·w·10 = 10 10

  28. Cube A has an edge of 2. Each edge of cube A is increased by 50%, creating a second cube B. What is the ratio of the volume of cube A to cube B? 20 Cube A Cube B 2 V = side3 V = side3 3 V = 23 = 8 V = 33 = 27 2 2 3 3 50% of 2 = .50  2 = 1 2 + 1 = 3 50% increase =

  29. Cube A has an edge of 2. Each edge of cube A is increased by 50%, creating a second cube B. The surface area of cube B is how much greater than the surface area of cube A? 21 SA = Surface Area Cube A Cube B 2 SA = 6s2 SA = 6s2 3 = 622 = 632 2 2 = 64 = 69 3 = 24 = 54 3 – SA Cube B SA Cube A 54 – 24 = 30

  30. How many wooden toy cubes with a 3-inch edge can fit in a rectangular container with dimensions 3 inches by 21 inches by 15 inches? 22 V = Volume V = lwh V = 3  21  15 V = side3 15 V = 945 V = 33 = 27 3 3 21 Find number of toy cubes in rectangular (large) container Volume of Large Container Volume of One Toy Cube  945  27 = 35

  31. If you assume that there is no wasted ice, how many smaller rectangular block ice cubes, dimensions 234, can be cut from two large blocks of ice? The size of each block of ice is shown below. 23 V = Volume Ice Cube 4 V = l  w  h V = l  w  h 6 V = 4  10  6 10 V = 2  3  4 V = 240 V = 24 Find number of ice cubes in one large block of ice Ice cubes in two large blocks of ice 2(10) = 20 Volume of Large Block of Ice Volume of One Ice Cube  240  24 = 10

  32. If cube B has an edge three times that of cube A, the volume of cube B is how many times the volume of cube A? 24 Strategy: Substitute a number for each cube edge. V = Volume Cube B Cube A V = s3 V = s3 V = 33 V = 13 V = 27 V = 1 1 31= 3 Volume Cube B ÷ Volume Cube A = 27 ÷ 1 = 27

  33. The surface areas of the rectangular prism are given. If the lengths of the edges are integers, what is the volume in cubic inches? 25 Strategy: Use trial and error with different combinations of numbers to find the area of each face. NO Combination #1 24 sq in. 28 = 1  28 No integer factor for 42 42 = 28  ? 42 sq in. 28 sq in. 28 28 ? 1

  34. The surface areas of the rectangular prism are given. If the lengths of the edges are integers, what is the volume in cubic inches? 25 Strategy: Use trial and error with different combinations of numbers to find the area of each face. NO Combination #1 24 sq in. 28 = 1  28 No integer factor for 42 3 42 = 28  ? 8 42 sq in. Combination #2 NO 28 = 2  14 Parallel line segments not equal 28 sq in. 14 14 42 = 14  3 24 = 3  8 3 2

  35. The surface areas of the rectangular prism are given. If the lengths of the edges are integers, what is the volume in cubic inches? 25 Strategy: Use trial and error with different combinations of numbers to find the area of each face. Combination #3 24 sq in. YES 28 = 4  7 6 4 42 = 7  6 42 sq in. 24 = 6  4 28 sq in. 7 7 Volume = l · w · h Volume = 4 · 6 · 7 6 Volume = 168 cubic inches 4

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