Question 1

A triangular pyramid (tetrahedron) has a triangular base with area 36 cm² and height 10 cm. What is the volume of the pyramid?

Accepted Answer

Step 1: Apply the pyramid volume formula: V = (1/3) × Base Area × Height. Step 2: Substitute values: V = (1/3) × 36 × 10 = (1/3) × 360 = 120 cm³. Therefore, the volume is 120 cm³.

Question 2

A rectangular prism has dimensions 8 cm × 6 cm × 5 cm. What is its total surface area?

Accepted Answer

Step 1: Apply surface area formula for rectangular prism: SA = 2(lw + lh + wh). Step 2: Substitute l=8, w=6, h=5: SA = 2(8×6 + 8×5 + 6×5) = 2(48 + 40 + 30) = 2(118) = 236 cm². Therefore, the surface area is 236 cm².

Question 3

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

Accepted Answer

Step 1: Use the prism volume formula: V = Base Area × Height. Step 2: Substitute the given values: V = 50 × 8 = 400 cm³. Therefore, the volume is 400 cm³.

Question 4

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

Accepted Answer

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

Question 5

A pyramid with a rectangular base of dimensions 12 cm × 8 cm has a height of 9 cm. What is its volume?

Accepted Answer

Step 1: Calculate the rectangular base area: A = 12 × 8 = 96 cm². Step 2: Apply pyramid volume formula: V = (1/3) × Base Area × Height. Step 3: V = (1/3) × 96 × 9 = (1/3) × 864 = 288 cm³. Therefore, the volume is 288 cm³.

Question 6

A rectangular prism has dimensions 5 cm × 4 cm × 3 cm. A smaller rectangular prism with dimensions 2 cm × 2 cm × 2 cm is removed from one corner. What is the volume of the remaining solid in cm³?

Accepted Answer

Step 1: Volume of the original rectangular prism = 5 × 4 × 3 = 60 cm³. Step 2: Volume of the smaller prism removed = 2 × 2 × 2 = 8 cm³. Step 3: Volume of remaining solid = 60 − 8 = 52 cm³.

Question 7

A pyramid with a square base of side 10 cm has a height of 12 cm. What is the volume of the pyramid in cm³?

Accepted Answer

Step 1: Area of square base = 10² = 100 cm². Step 2: Volume of pyramid = (1/3) × base area × height = (1/3) × 100 × 12 = (1/3) × 1200 = 400 cm³.

Question 8

A triangular pyramid (tetrahedron) has a triangular base with area 24 cm² and a height of 9 cm. What is the volume of the pyramid in cm³?

Accepted Answer

Step 1: Volume of a pyramid = (1/3) × base area × height. Step 2: Volume = (1/3) × 24 × 9 = (1/3) × 216 = 72 cm³.

Question 9

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.

Question 10

A triangular pyramid (tetrahedron) has an equilateral triangular base with side 10 cm. The height of the pyramid is 12 cm. What is the lateral surface area (excluding the base) in cm²?

Accepted Answer

Step 1: For equilateral triangle base with side 10 cm, the distance from centre to midpoint of a side (apothem) = 10/(2√3) = 5√3/3 cm. Step 2: Slant height = √(h² + apothem²) = √(144 + 25/3) = √(457/3) cm. Step 3: Each lateral face is a triangle with base 10 and height (slant height) √(457/3). Area of one face = (1/2) × 10 × √(457/3) = 5√(457/3) cm². Step 4: Lateral surface area = 3 × 5√(457/3) = 15√(457/3) ≈ 185.4 cm². For cleaner calculation: √(457/3) ≈ 12.36, so LSA ≈ 3 × (1/2) × 10 × 12.36 ≈ 185.4 cm². Using exact form: LSA = 15√(457/3) ≈ 185 cm² (rounded).

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60-Second Revision — Prism & Pyramid