Question 1

A pyramid has a square base with side length 6 cm and a height of 8 cm. What is the volume of the pyramid?

Accepted Answer

Step 1: Volume of pyramid = (1/3) × Base Area × Height. Step 2: Base area = 6² = 36 cm². Step 3: Volume = (1/3) × 36 × 8 = (1/3) × 288 = 96 cm³. Therefore, the answer is 96 cm³.

Question 2

A rectangular prism has a length of 12 cm, width of 8 cm, and height of 5 cm. What is the total surface area of the prism?

Accepted Answer

Step 1: Surface area of rectangular prism = 2(lw + lh + wh). Step 2: Substitute values: 2(12×8 + 12×5 + 8×5) = 2(96 + 60 + 40) = 2(196) = 392 cm². Therefore, the answer is 392 cm².

Question 3

A pyramid with a rectangular base measuring 8 cm × 6 cm has a height of 9 cm. What is the volume of the pyramid?

Accepted Answer

Step 1: Volume of pyramid = (1/3) × Base Area × Height. Step 2: Base area = 8 × 6 = 48 cm². Step 3: Volume = (1/3) × 48 × 9 = (1/3) × 432 = 144 cm³. Therefore, the answer is 144 cm³.

Question 4

A triangular pyramid (tetrahedron) has an equilateral triangular base with side 6 cm and a height of 8 cm. What is the volume of the pyramid in cm³? (Use √3 ≈ 1.73)

Accepted Answer

Step 1: For an equilateral triangle with side a = 6 cm, area = (√3/4) × a² = (√3/4) × 36 = 9√3 cm². Step 2: Using √3 ≈ 1.73, area ≈ 9 × 1.73 = 15.57 cm². Step 3: Volume of pyramid = (1/3) × base area × height = (1/3) × 15.57 × 8 ≈ (1/3) × 124.56 ≈ 41.52 cm³. Step 4: Rounding to nearest integer: 42 cm³ (or 41.6 cm³ depending on rounding convention used in exam).

Question 5

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 the 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 6

A hexagonal prism has a regular hexagonal base with side 4 cm and a height (length) of 10 cm. What is the lateral surface area of the prism in cm²?

Accepted Answer

Step 1: A regular hexagon has 6 equal sides. Step 2: Perimeter of the hexagonal base = 6 × 4 = 24 cm. Step 3: Lateral surface area of a prism = perimeter of base × height = 24 × 10 = 240 cm².

Question 7

A rectangular prism (cuboid) has dimensions 8 cm × 6 cm × 5 cm. A smaller rectangular prism with dimensions 4 cm × 3 cm × 2.5 cm is removed from one corner. What is the volume of the remaining solid in cm³?

Accepted Answer

Step 1: Volume of the original cuboid = length × width × height = 8 × 6 × 5 = 240 cm³. Step 2: Volume of the smaller cuboid removed = 4 × 3 × 2.5 = 30 cm³. Step 3: Volume of the remaining solid = 240 − 30 = 210 cm³.

Question 8

A triangular pyramid (tetrahedron) has a right-angled triangular base with legs 9 cm and 12 cm. The height of the pyramid from the apex to the base is 20 cm. If a plane parallel to the base cuts the pyramid at 2/3 of the height from the apex, what is the volume of the frustum (lower portion) formed (in cm³)?

Accepted Answer

Step 1: Area of right-angled triangular base = (1/2) × 9 × 12 = 54 cm². Step 2: Volume of original pyramid = (1/3) × 54 × 20 = 360 cm³. Step 3: The plane cuts at 1/2 of the height from apex, so height of smaller pyramid = 10 cm. Step 4: Linear scale factor = 10/20 = 1/2. Volume of smaller pyramid = (1/2)³ × 360 = (1/8) × 360 = 45 cm³. Step 5: Volume of frustum = 360 - 45 = 315 cm³.

Question 9

A hexagonal prism has a regular hexagonal base with side length 8 cm. The lateral surface area of the prism is 960 cm². What is the volume of the prism (in cm³)?

Accepted Answer

Step 1: Lateral surface area of prism = perimeter of base × height. Perimeter of regular hexagon = 6 × 8 = 48 cm. Step 2: 48 × h = 960, so h = 20 cm. Step 3: Area of regular hexagon = (3√3/2) × side² = (3√3/2) × 64 = 96√3 cm². Step 4: Volume = base area × height = 96√3 × 20 = 1920√3 cm³.

Question 10

A pentagonal pyramid has a regular pentagonal base with side 6 cm and apothem 4.1 cm. The total surface area is 205 cm². What is the slant height of the pyramid (in cm)?

Accepted Answer

Step 1: Area of regular pentagonal base = (1/2) × perimeter × apothem = (1/2) × 30 × 4 = 60 cm². Step 2: Lateral surface area = 210 - 60 = 150 cm². Step 3: Lateral surface area = (1/2) × perimeter × slant height = (1/2) × 30 × l. Step 4: 150 = 15l, so l = 10 cm.

Question 11

A rectangular prism has dimensions 8 cm × 6 cm × 4 cm. A smaller rectangular prism is cut from one corner such that its dimensions are exactly half those of the original prism. What is the volume of the remaining solid (in cm³)?

Accepted Answer

Step 1: Volume of original prism = 8 × 6 × 4 = 192 cm³. Step 2: Dimensions of smaller prism = 4 cm × 3 cm × 2 cm. Step 3: Volume of smaller prism = 4 × 3 × 2 = 24 cm³. Step 4: Volume of remaining solid = 192 - 24 = 168 cm³.

Question 12

A square pyramid has a base side of 12 cm and a slant height of 10 cm. If the pyramid is cut by a plane parallel to the base at a height of 6 cm from the apex, what is the volume of the smaller pyramid (top portion) formed (in cm³)?

Accepted Answer

Step 1: Find the height of the original pyramid using Pythagoras. Distance from apex to centre of base = √(10² - 6²) = √(100 - 36) = √64 = 8 cm. Step 2: The smaller pyramid is cut at height 6 cm from apex, so its height = 6 cm. Step 3: By similarity, if height ratio = 6/8 = 3/4, then side ratio = 3/4. Smaller base side = 12 × (3/4) = 9 cm. Step 4: Volume of smaller pyramid = (1/3) × 9² × 6 = (1/3) × 81 × 6 = 162 cm³.

Question 13

A right triangular prism has a triangular base with sides 13 cm, 14 cm, and 15 cm. The height of the prism is 20 cm. What is the total surface area of the prism (in cm²)?

Accepted Answer

Step 1: Find the area of the triangular base using Heron's formula. Semi-perimeter s = (13 + 14 + 15)/2 = 21 cm. Area = √[21(21-13)(21-14)(21-15)] = √[21 × 8 × 7 × 6] = √7056 = 84 cm². Step 2: Find the lateral surface area = perimeter × height = (13 + 14 + 15) × 20 = 42 × 20 = 840 cm². Step 3: Total surface area = 2 × base area + lateral surface area = 2 × 84 + 840 = 168 + 840 = 1008 cm².

SBI PO Prism & Pyramid

Prism & Pyramid— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Prism & Pyramid — Practice Questions

60-Second Revision — Prism & Pyramid