Problem Set for Plane and Solid Geometry Midterm Coverage:

1. If the lateral area of the right circular cylinder is 88 and its volume is 220, find its radius. Ans. 5
2. A prism has an equilateral triangle with 20 cm on a side for its base, and an altitude of 30 cm. Determine its lateral area. Ans. 1800 sq. cm.
3. How many sides have a regular polygon, if each interior angle is 165 deg? Ans. 24
4. The area of a hexagon inscribed in a circle is 374.11 sq cm. Determine the area of the circle. Ans. 452.39 sq. cm.
5. A polygon has 170 diagonals. How many sides does it have? Ans. 20
6. The circumference of the base of a right circular cylinder is 48 cm and its altitude is 15 cm. Determine its total surface area. Ans. 1086.69 sq. cm.
7. A rectangle is inscribed in a circle whose radius is 5 inches. The base of the rectangle is 8 inches. Find the area of the rectangle. Ans. 48 sq. in.
8. Find the length of the side of a pentagon if the line perpendicular to its side is 12 units from the center. Ans. 17.44 units
9. A regular dodecagon is inscribed in a circle of radius 24. Find the perimeter of the dodecagon. Ans. 149.08
10. Find the area of the rhombus if one diagonal has length 30 and a side has length 17. Ans. 240
11. The area of the rhombus is 168 sq. m. If one of the diagonal is 12 m, find the length of the sides of a rhombus. Ans. 15.23 cm
12. The sum of the interior angles of a polygon is 540 deg. Find the number of sides. Ans. 5
13. How many diagonals are there in a polygon if it has 16 sides? Ans. 104
14. In a square, find the length of the apothem if the radius of the circumscribing circle is 16. Ans. 11.31
15. The bases of a right prism are regular hexagons with each side equal to 6 cm. The bases are 10 cm apart. What is the volume of the prism? Ans. 935.31 sq. cm.
16. What is the volume of a right circular cylinder inscribed in a cube of edge 10 cm? Ans. 785.4 sq. cm.
17. The volume of the prism with an altitude of 15 m and having an equilateral triangle as its base is equal to 234 sq. m. Determine the length of the side of the triangular base. Ans. 6 m
18. The area of a hexagon is 54√3. Determine the length of each side. Ans. 6
19. The wall at one end of an attic takes the shape of a trapezoid because of a slanted ceiling. The wall is 8 ft high at one end, 10 ft wide and only 3 ft high on the other end. Determine the area of the wall in sq. ft. Ans. 55
20. The bases of a right prism are a hexagon with one side 6 cm long. If the volume of the prism is 450 cu cm, how apart are the bases? Ans. 4.81 cm
21. Find the length of the diagonal of a cube whose volume is 729 cu. cm. Ans. 15.59 cm
22. If the sides of a parallelogram and an included angle are 6, 10 and 100, respectively, find the length of the shorter diagonal. Ans. 10.73
23. Three circles C1, C2, and C3 are externally tangent to each other. Center to center distances are 10 cm between C1 and C2, 8 cm between C2 and C3 and 6 cm between C3 and C1. Determine the total areas of the circles. Ans. 175.93 sq. cm.
24. Find the area of a regular pentagon whose side is 25 cm and apothem is 17.2 cm. Ans. 1075 sq. cm.
25. For a regular polygon of heptagon sides, find the number of degrees contained in each central angle. Ans. 51.43 deg
26. A regular octagon is inscribed in a circle. What is the radius of the circle if the length of each side of the octagon is 12.6 inches? Ans. 16. 46 in
27. A circle with radius 6 cm has half its area removed by cutting off a border of uniform width. Find the width of the border. Ans. 1.76 cm
28. Find the number of degrees in the measure of each exterior angle of a regular polygon which has 12 sides. Ans. 150
29. The altitude of a parallelepiped is 20 cm and the base is a rhombus with diagonals 14 cm and 48 cm. Find the volume of the parallelepiped. Ans. 6720 cu. cm.
30. The diagonal of the face of the cube is 3√2. Find the diagonal of the cube. Ans. 3√3.
31. A regular dodecagon is inscribed in a circle of radius 24. Find the perimeter of the dodecagon. Ans. 149.08
32. The volume of the rectangular solid is 480 cu. cm. If the base is 8 cm by 10 cm, find the total area of the solid. Ans. 376 sq. cm.
33. A cylindrical tank with horizontal axis has a radius of 1m and 6 m long. If it is filled with gasoline to a depth of 1.3m, determine the volume of the gasoline in the tank. Ans. 12.97 cu. m.
34. An open box with a square base is to be constructed from a square piece of tin by cutting out a 3 – inch square from each corner and folding up the sides. If the box is to hold 48 cubic inches, what must be the length of the side of tin? Ans.10 in
35. The volume of a rectangular parallelepiped is 1944 cu. cm. The edges are in the ratio 3:4:6. Determine the length of the shortest edge? Ans. 9
36. The area of a circle circumscribing a hexagon is 452. 39 sq. m. Find the area of the hexagon. Ans. 374. 12
37. Find the area of the inscribed circle of a regular hexagon whose side has a length of 8 cm. Ans. 201. 06 sq. cm.
38. A regular hexagonal prism is inscribed in a right circular cylinder whose lateral edge is 10 cm and the radius of base is 3cm. Compute the volume of the prism. Ans. 234 cu. cm.
39. The length of the diagonal of the rhombus are 30 cm and 40 cm respectively. Find the perimeter of the rhombus. Ans. 100 cm.
40. The base of a right prism is an equilateral triangle 3 cm on a side. If its altitude is 8 cm, find its volume. Ans. 31.18 cu. cm.
41. A piece of wire is shaped to enclose a square whose area is 169 sq. in. It is then reshaped to form a rectangle whose length is 15 in. Determine the area of the rectangle. Ans. 165 sq. in.
42. The base area of a circular cylinder is 12 sq. cm. and an element of the cylinder is 10 cm. Find the volume of the cylinder if the element is inclined to the base an angle of 60 deg. 103.92 cu. cm.
43. The corner of a 2 - meter square is cut off to form a regular octagon. Determine the length of the resulting side of the octagon. Ans. 0.828 m
44. The area of the rectangular lot is increased by 25 sq. m. if it longer by 10 m and narrower by 2.5 m. Also when it is shorter by 10 m and wider by 5 m, its area is reduced by 50 sq. m. Solve the original area. Ans. 200 sq. m.
45. A right hexagonal prism is inscribed in a right circular cylinder whose height is 20 cm. The difference between the circumference of the circle and the perimeter of the hexagon 4 cm. Determine the volume of the prism. Ans. 10367.18 cu. cm.
46. An irrigation canal is 200 m long and 2 m deep. It is 2 m wide at the top and 1 m at the bottom. Find the volume excavated to make the canal. Ans. 600 cu. m.
47. A room is 12 ft wide, 15 ft long and 8 ft high. If an air conditioner changes the air once every five minutes, how many cubic feet of air does it change per hour? Ans. 17280
48. The volume of a rectangular parallelepiped is 15, 000 cu. m. and its total surface area is 3,700 sq. cm. If its altitude is 20 cm, find the dimension of the shorter side of the base. Ans. 25 cm
49. The circumference of the base of a closed right circular cylinder is 48 cm and its altitude is 15 cm. Determine its total surface area. 1086. 69 sq. cm.
50. A circular cylinder with a volume of 6.54 cubic meters is circumscribed about a right prism whose base is an equilateral triangle of side 1.25 m. What is the altitude of the cylinder? Ans. 4m