Question 1

A triangular prism has a triangular base with area 24 cm² and height (length) of the prism is 10 cm. What is the volume of the prism?

Accepted Answer

Step 1: Volume of a prism = Base Area × Height. Step 2: Volume = 24 × 10 = 240 cm³. Therefore, the answer is 240 cm³.

Question 2

A square pyramid has a base with side 8 cm. If the volume of the pyramid is 256 cm³, what is its height?

Accepted Answer

Step 1: Area of square base = 8 × 8 = 64 cm². Step 2: Volume = (1/3) × Base Area × Height, so 256 = (1/3) × 64 × h. Step 3: 256 = (64h)/3, therefore 768 = 64h, so h = 12 cm. Therefore, the answer is 12 cm.

Question 3

A rectangular prism (cuboid) has dimensions 5 cm × 4 cm × 6 cm. What is its total surface area?

Accepted Answer

Step 1: Surface Area of cuboid = 2(lw + wh + lh). Step 2: Surface Area = 2(5×4 + 4×6 + 5×6) = 2(20 + 24 + 30) = 2(74) = 148 cm². Therefore, the answer is 148 cm².

Question 4

A triangular prism has a triangular base with area 24 cm² and height (length) of the prism is 15 cm. What is the volume of the prism?

Accepted Answer

Step 1: Identify the formula for volume of a prism: Volume = Base Area × Height (length of prism). Step 2: Substitute values: Volume = 24 × 15 = 360 cm³. Therefore, the answer is 360 cm³.

Question 5

A pentagonal prism has a pentagonal base with area 50 cm² and the prism's length is 8 cm. What is the volume of the prism?

Accepted Answer

Step 1: Volume of prism = Base Area × Length. Step 2: Volume = 50 × 8 = 400 cm³. Therefore, the answer is 400 cm³.

Question 6

A triangular pyramid (tetrahedron) has a triangular base with area 15 cm² and height 12 cm. What is its volume?

Accepted Answer

Step 1: Volume of pyramid = (1/3) × Base Area × Height. Step 2: Volume = (1/3) × 15 × 12 = (1/3) × 180 = 60 cm³. Therefore, the answer is 60 cm³.

Question 7

A square pyramid has a base side length of 10 cm and height of 12 cm. What is the volume of the pyramid?

Accepted Answer

Step 1: Find the area of the square base: Base Area = 10² = 100 cm². Step 2: Apply pyramid volume formula: Volume = (1/3) × Base Area × Height. Step 3: Volume = (1/3) × 100 × 12 = (1200)/3 = 400 cm³. Therefore, the answer is 400 cm³.

Question 8

A rectangular prism has dimensions 8 cm × 6 cm × 5 cm. What is its volume?

Accepted Answer

Step 1: Apply the formula for volume of a rectangular prism: Volume = length × width × height. Step 2: Substitute values: Volume = 8 × 6 × 5 = 240 cm³. Therefore, the answer is 240 cm³.

Question 9

A triangular pyramid (tetrahedron) has a triangular base with area 36 cm² and height 9 cm. What is the volume of the pyramid?

Accepted Answer

Step 1: Apply the pyramid volume formula: Volume = (1/3) × Base Area × Height. Step 2: Substitute values: Volume = (1/3) × 36 × 9 = (324)/3 = 108 cm³. Therefore, the answer is 108 cm³.

Question 10

A rectangular prism has dimensions 8 cm × 6 cm × 5 cm. If the lateral surface area (excluding top and bottom) is calculated, what is the perimeter of the base multiplied by the height?

Accepted Answer

Step 1: Identify base dimensions: length = 8 cm, width = 6 cm, height = 5 cm. Step 2: Lateral surface area formula = Perimeter of base × height. Step 3: Perimeter of base = 2(8 + 6) = 2(14) = 28 cm. Step 4: Lateral surface area = 28 × 5 = 140 cm². Therefore, the answer is 140 cm².

Question 11

A pyramid with a rectangular base of dimensions 8 m × 6 m has a height of 12 m. What is the volume of the pyramid?

Accepted Answer

Step 1: Base area = length × width = 8 × 6 = 48 m². Step 2: Volume of pyramid = (1/3) × base area × height. Step 3: Volume = (1/3) × 48 × 12 = (1/3) × 576 = 192 m³. Therefore, the answer is 192 m³.

Question 12

A hexagonal prism has a regular hexagonal base with side 4 cm and the height of the prism is 9 cm. What is the lateral surface area of the prism?

Accepted Answer

Step 1: A regular hexagon has 6 equal sides. Step 2: Perimeter of hexagonal base = 6 × 4 = 24 cm. Step 3: Lateral surface area = perimeter × height. Step 4: Lateral SA = 24 × 9 = 216 cm². Therefore, the answer is 216 cm².

Question 13

A pentagonal pyramid has a regular pentagonal base with side 5 cm. The slant height of the pyramid is 8 cm. What is the lateral surface area of the pyramid?

Accepted Answer

Step 1: A regular pentagon has 5 equal sides. Step 2: Perimeter of pentagonal base = 5 × 5 = 25 cm. Step 3: Lateral surface area of pyramid = (1/2) × perimeter × slant height. Step 4: Lateral SA = (1/2) × 25 × 8 = (1/2) × 200 = 100 cm². Therefore, the answer is 100 cm².

Question 14

A rectangular prism has dimensions 8 cm × 6 cm × 5 cm. A smaller rectangular prism with dimensions 4 cm × 3 cm × 2.5 cm is removed from it. What is the ratio of the volume of the remaining solid to the volume of the original prism?

Accepted Answer

Step 1: Volume of original prism = 8 × 6 × 5 = 240 cm³. Step 2: Volume of smaller prism = 4 × 3 × 2.5 = 30 cm³. Step 3: Volume of remaining solid = 240 - 30 = 210 cm³. Step 4: Ratio = 210 : 240. Step 5: Simplify by dividing both by 30: 210 ÷ 30 = 7 and 240 ÷ 30 = 8. Therefore, ratio = 7 : 8.

Question 15

A pyramid with a square base of side 12 cm has a height of 8 cm. What is the volume of the pyramid in cm³?

Accepted Answer

Step 1: Area of square base = 12² = 144 cm². Step 2: Volume of pyramid = (1/3) × base area × height = (1/3) × 144 × 8 = (1/3) × 1152 = 384 cm³.

Question 16

A pentagonal prism has a regular pentagonal base with side 8 cm. The height of the prism is 15 cm. The area of a regular pentagon with side a is (a²√25+10√5)/4. What is the lateral surface area of the prism in cm²?

Accepted Answer

Step 1: Lateral surface area of a prism = perimeter of base × height of prism. Step 2: Perimeter of regular pentagon = 5 × 8 = 40 cm. Step 3: Lateral surface area = 40 × 15 = 600 cm². Note: The formula for the pentagonal base area is provided but is NOT needed for lateral surface area calculation—it is a distractor.

Question 17

A right circular cone is inscribed in a cylinder of radius 5 cm and height 12 cm such that the base of the cone coincides with the base of the cylinder and the apex of the cone touches the top circular face of the cylinder. If a plane parallel to the base cuts the cone at 1/3 of its height from the apex, what is the ratio of the volume of the smaller cone (above the plane) to the volume of the original cone?

Accepted Answer

Step 1: The original cone has radius 5 cm and height 12 cm. Volume of original cone = (1/3)π(5²)(12) = 100π cm³. Step 2: The cutting plane is at 1/3 of height from apex, meaning it is at height 4 cm from apex. Step 3: By similarity, the radius at height 4 cm from apex = 5 × (4/12) = 5 × (1/3) = 5/3 cm. Step 4: Volume of smaller cone = (1/3)π(5/3)²(4) = (1/3)π(25/9)(4) = (100π/27) cm³. Step 5: Ratio = (100π/27) ÷ (100π) = 1/27.

Question 18

A rectangular prism has dimensions 8 cm × 6 cm × 4 cm. A pyramid with the same rectangular base (8 cm × 6 cm) and height 12 cm is placed on top of the prism. What is the total volume of the combined solid in cm³?

Accepted Answer

Step 1: Volume of rectangular prism = length × width × height = 8 × 6 × 4 = 192 cm³. Step 2: Volume of pyramid = (1/3) × base area × height = (1/3) × (8 × 6) × 12 = (1/3) × 48 × 12 = (1/3) × 576 = 192 cm³. Step 3: Total volume = 192 + 192 = 384 cm³.

SSC CHSL Prism & Pyramid

Prism & Pyramid— Rules & Concept

Key Points to Remember

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Prism & Pyramid — Practice Questions

60-Second Revision — Prism & Pyramid