1. If the total surface area of a cube is 216 square centimeters, what is its volume?

(a) 216 tcubic cm (b) 212 cubic cm (c) 316 cubic cm (d) 256

2. If the total area of the 6 faces of a cube is 216 square centimeters, calculate and write the volume of the cube.

3. If the total surface area of a cube is 150 square centimeters, then its volume will be 150 cubic centimeters.

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. 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 \(Sd = 6\sqrt{3}V\).

6. The total surface area of a right square prism is equal to half the square of its diagonal. Show that the prism is a cube.

7. 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

8. The dimensions of a right rectangular prism are in the ratio 6:5:4. If its total surface area is 3700 square centimeters, what is its volume in cubic centimeters?

(a) 1500 cubic cm (b) 51000 cubic cm (c) 50100 cubic cm (d) 15000 cubic cm

9. If the length, breadth, and height of a right rectangular prism are in the ratio 3:2:1 and its total surface area is 88 square centimeters, then what is its volume?

(a) 120 cubic cm (b) 64 cubic cm (c) 48 cubic cm (d) 24 cubic cm

10. 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.

11. 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."

12. The curved surface area of a right circular cylinder is 440 square centimeters. If the height of the cylinder is 10 centimeters, find its volume.

13. If the ratio of the volumes of two cubes is 8:27, then what will be the ratio of their total surface areas?

14. An open cylindrical vessel has a total surface area of 2002 square centimeters. If the radius of its base is 7 cm, how many liters of water can it hold? (1 liter = 1 cubic decimeter)

15. The length, breadth, and height of a cuboid are in the ratio 4:3:2. If its total surface area is 468 square meters, find the volume of the cuboid.

16. If the ratio of the volumes of two cubes is 1:27, then the ratio of their total surface areas will be —

(a) 1:3 (b) 1:8 (c) 1:9 (d) 1:18

17. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) will be –

(a) \(s=6d^2\) (b) \(3s=7d\) (c) \(s^3=d^2\) (d) \(d^2 = \cfrac{s}{2}\)

18. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) is \[ s = 6d^2 \]

19. The numerical value of the volume of a cube is equal to the numerical value of the sum of its edges. The total surface area of the cube is - -

(a) 12 square unit (b) 36 square unit (c) 72 square unit (d) 144 square unit

20. 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.

21. If the ratio of the volumes of two cubes is 4:343, determine the ratio of their total surface areas.

22. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) is \(s^3 = d^2\).

23. If the ratio of the volumes of two cubes is 1:27, then the ratio of their total surface areas will be _____.

24. If the length of the diagonal of a cube is \(4\sqrt{3}\) cm, determine the total surface area of the cube.

25. The space diagonal of a cube is 4√3 cm. Find the total surface area of the cube.

26. If the area of one face of a cube is 64 square meters, calculate and write the volume of the cube.

27. If the length of a diagonal of a cube is \(4\sqrt{3}\) cm, calculate and write the total surface area of the cube.

28. The lateral surface area of a cube is 256 square meters. What is the volume of the cube?

(a) 64 cubic m. (b) 216 cubic m. (c) 256 cubic m (d) 512 cubic m

29. If the ratio of the volumes of two cubes is 1:27, then what is the ratio of their total surface areas?

(a) 1:3 (b) 1:8 (c) 1:9 (d) 1:18

30. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, what is the relationship between \(s\) and \(d\)?

(a) s=6d\(^2\) (b) 3s=7d (c) s\(^3\)=d2\(^2\) (d) d\(^2\)=s/2