1. If the total length of the edges of a solid cube is 36 cm, then its volume will be –?
(a) 9 cubic cm (b) 54 cubic cm (c) 27 cubic cm (d) 36 cubic cm
2. If the sum of the lengths of the edges of a cube is 60 cm, calculate and write the volume of the cube.
3. The dimensions of a cuboid are 12 cm, 6 cm, and 3 cm respectively. Calculate the length of each edge of a cube whose volume is equal to that of the cuboid.
4. Let the length of each edge of the cube be \(a\) cm \(\therefore\) The total surface area of the cube = \(6a^2\) square cm If the edge length is increased by 20%, the new edge length = \(a + a \times \cfrac{20}{100}\) cm \(= a + \cfrac{a}{5} = \cfrac{6a}{5}\) cm \(\therefore\) New total surface area of the cube = \(6\left(\cfrac{6a}{5}\right)^2\) square cm = \(\cfrac{216a^2}{25}\) square cm \(\therefore\) Percentage increase in surface area \(= \cfrac{\cfrac{216a^2}{25} - 6a^2}{6a^2} \times 100\%\) \(= \cfrac{(216a^2 - 150a^2) \times 100}{25 \times 6a^2}\%\) \(= \cfrac{4 \times 66a^2}{6a^2}\%\) \(= 44\%\)
5. The sum of the edges of a cube is 60 cm. What is the volume of the cube?
6. From a cuboid measuring 16 cm × 4 cm × 2 cm, several cubes with edge length 2 cm are cut out. What is the ratio of the total surface area of the original cuboid to the total surface area of all the cubes?
(a) 11:48 (b) 12:50 (c) 13:36 (d) 14:28
7. If the length of a cube’s diagonal is \(4\sqrt{3}\) cm, then its volume is—
(a) 60 cubic cm (b) 64 cubic cm (c) 72 cubic cm (d) 80 cubic cm
8. The sum of the lengths of the edges of a solid cube is 36 cm. Find the volume of the cube.
9. The sum of the edge lengths of a solid cube is 36 cm. Find the volume of the cube.
(a) 27 cubic cm (b) 36 cubic cm (c) 9 cubic cm (d) 54 cubic cm
10. What is the volume of the largest right circular cone that can be obtained from a cube with an edge length of 28 cm?
11. A rectangular container has a base in the shape of a rectangle with a length of 60 cm and a width of 45 cm. The container has a height of 20 cm and is half-filled with water. Determine the side length of a metallic cube that, when placed in the container, will cause the water level to reach the brim.
12. A solid cuboid has a length, width, and height ratio of \(4:3:2\), and its total surface area is \(468\) square cm. Determine the volume of the cuboid.
13. The volume of a rectangular prism is 432 cubic cm. If two equal-sized cubes are created from it, determine the edge length of each cube.
14. The sum of the lengths of the edges of a solid cube is 36 cm; the volume of the cube –
(a) 27cm\(^3\) (b) 36cm\(^3\) (c) 9cm\(^3\) (d) 54cm\(^3\)
15. If a cube has a side length of \(a\) units and a diagonal length of \(d\) units, then the relationship between \(a\) and \(d\) is—?
(a) \(\sqrt2a=d\) (b) \(\sqrt3a=d\) (c) \(a=\sqrt3d\) (d) \(a=\sqrt2d\)
16. A circle with center O has a radius of 10 cm. PQ is a chord with a length of 16 cm. The length of the perpendicular drawn from O to PQ is —
(a) 8 cm (b) 10 cm (c) 16 cm (d) 6 cm
17. The volume of a right rectangular prism is 960 cubic centimeters. If the ratio of its length, width, and height is 6:5:4, what is the total surface area of the prism?
(a) 590 square cm (b) 592 square cm (c) 295 square cm (d) 596 square cm
18. The volume of a cube is equal to the area of one of its faces. What is the length of its edge?
(a) 4 units (b) 5 units (c) 6 units (d) 8 units
19. If the length of the diagonal of a cube is \(\sqrt{12}\) cm, what is its volume?
(a) 18 cubic cm (b) 8 cubic cm (c) 6 cubic cm (d) 16 cubic cm
20. If the length of the diagonal of a cube is \(8\sqrt{3}\) cm, then what is the length of its edge?
(a) 8 cm (b) 4 cm (c) 5 cm (d) 7 cm
21. The diagonal of a cuboid is √725 cm and its volume is 3000 cubic cm. The total surface area of the cuboid is 1300 square cm. Find the length, breadth, and height of the cuboid.
22. If the volume of a cube is \(V\) cubic centimeters, the total surface area is \(S\) square centimeters, and the length of the diagonal is \(d\) centimeters, then prove that \(Sd = 6\sqrt{3}V\).
23. Here’s the English translation of your sentence: "A right rectangular prism has its length twice the breadth and its height half the breadth. If the total surface area of the prism is 448 square centimeters, find its volume."
24. If the side length of a cube is equal to the radius of a sphere, then which one has the greater volume?
25. In a right-angled triangle, the lengths of the two sides adjacent to the right angle are 4 cm and 3 cm respectively. If the triangle is rotated once completely around the longer of the two adjacent sides, find the total surface area and the volume of the solid formed.
26. If the length of the diagonal of a cube is \(5\sqrt{3}\) cm, find its volume.
27. A hollow vertical cylindrical iron pipe has an outer radius of 5 cm and an inner radius of 4 cm. If the total surface area of the pipe is 1188 square cm, what is the length of the pipe?
28. If the length of each edge of a cube is increased by 50%, by what percentage will its volume increase?
29. If each edge of a cube is doubled in length, then the volume of the new cube will be eight times that of the original cube.
30. A right circular cylinder has a height-to-base-radius ratio of 3:1. If the volume of the cylinder is \(1029\pi\) cubic cm, find the total surface area of the cylinder.
