Question 1

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

A hexagonal pyramid has a regular hexagonal base with area 72 cm² and height 10 cm. What is the volume of the pyramid?

Accepted Answer

Step 1: Apply pyramid volume formula: V = (1/3) × Base Area × Height. Step 2: Substitute: V = (1/3) × 72 × 10 = (1/3) × 720 = 240 cm³. Therefore, the volume is 240 cm³.

Question 6

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 = 7 : 8.

Question 7

A pyramid with a square base has a base side of 12 cm and a height of 8 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: Base area = 12² = 144 cm². Step 3: Volume = (1/3) × 144 × 8 = (1/3) × 1152 = 384 cm³.

Question 8

A triangular pyramid (tetrahedron) has a triangular base with area 20 cm² and a perpendicular height of 9 cm from the apex to the base. 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) × 20 × 9 = (1/3) × 180 = 60 cm³.

Question 9

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

Accepted Answer

Step 1: Identify the formula for volume of a pyramid: V = (1/3) × Base Area × Height

Step 2: Calculate the base area. The base is rectangular with dimensions 12 cm × 8 cm.
Base Area = 12 × 8 = 96 cm²

Step 3: Apply the volume formula.
V = (1/3) × 96 × 9
V = (1/3) × 864
V = 288 cm³

Therefore, the volume of the pyramid is 288 cm³.

Question 10

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

Accepted Answer

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

Question 11

A rectangular prism has a length of 8 cm, width of 5 cm, and height of 6 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(8×5 + 8×6 + 5×6) = 2(40 + 48 + 30) = 2(118) = 236 cm². Therefore, the answer is 236 cm².

Question 12

A square pyramid has a base side length of 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 = 6 × 6 = 36 cm². Step 3: Volume = (1/3) × 36 × 8 = (1/3) × 288 = 96 cm³. Therefore, the answer is 96 cm³.

Question 13

A rectangular prism has length 8 cm, width 6 cm, and height 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: Calculate each product: lw = 8×6 = 48, lh = 8×5 = 40, wh = 6×5 = 30. Step 3: Sum = 48 + 40 + 30 = 118. Step 4: SA = 2 × 118 = 236 cm². Therefore, the surface area is 236 cm².

Question 14

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

Accepted Answer

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

Question 15

A triangular prism has a triangular base with area 24 cm² and height (length of prism) 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 16

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

Accepted Answer

Step 1: Volume of triangular prism = Base area × Height. Step 2: Volume = 24 × 15 = 360 cm³. Therefore, the answer is 360 cm³.

Question 17

A pyramid has a rectangular base with length 12 cm and width 8 cm. If the height of the pyramid is 9 cm, what is its volume?

Accepted Answer

Step 1: Volume of pyramid = (1/3) × Base area × Height. Step 2: Base area = 12 × 8 = 96 cm². Step 3: Volume = (1/3) × 96 × 9 = (1/3) × 864 = 288 cm³. Therefore, the answer is 288 cm³.

Question 18

A pyramid has a square base with side 14 cm. The height of the pyramid is 24 cm. What is the slant height of the pyramid in cm?

Accepted Answer

Step 1: The slant height is the distance from the apex to the midpoint of a base edge. Step 2: Distance from center to midpoint of edge = side/2 = 14/2 = 7 cm. Step 3: Using Pythagoras theorem: Slant height² = Height² + (Distance from center)² = 24² + 7² = 576 + 49 = 625. Step 4: Slant height = √625 = 25 cm. Therefore, the answer is 25 cm.

Question 19

A pyramid with a rectangular base of dimensions 12 cm × 8 cm has a height of 9 cm. What is the volume of the pyramid in cm³?

Accepted Answer

Step 1: Volume of pyramid = (1/3) × Base area × Height. Step 2: Base area = 12 × 8 = 96 cm². Step 3: Volume = (1/3) × 96 × 9 = (1/3) × 864 = 288 cm³. Therefore, the answer is 288 cm³.

Question 20

A rectangular prism has dimensions 12 cm × 8 cm × 5 cm. A smaller rectangular prism with dimensions 6 cm × 4 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 = 12 × 8 × 5 = 480 cm³. Step 2: Volume of smaller prism = 6 × 4 × 2.5 = 60 cm³. Step 3: Volume of remaining solid = 480 − 60 = 420 cm³. Step 4: Ratio = 420 : 480 = 7 : 8.

Question 21

A pyramid with a square base has a base side of 14 cm and a height of 24 cm. What is the volume of the pyramid (in cm³)?

Accepted Answer

Step 1: The volume of a pyramid = (1/3) × base area × height. Step 2: Base area of square = 14² = 196 cm². Step 3: Volume = (1/3) × 196 × 24 = (196 × 24) / 3 = 4704 / 3 = 1568 cm³.

Question 22

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 value in cm²?

Accepted Answer

Step 1: Lateral surface area of a rectangular prism = 2h(l + b), where h is height, l is length, b is breadth. Step 2: LSA = 2 × 5 × (8 + 6) = 2 × 5 × 14 = 140 cm². Therefore, the answer is 140 cm².

Question 23

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

Accepted Answer

Step 1: Lateral surface area of prism = Perimeter of base × Height. Step 2: Perimeter of regular hexagon = 6 × side = 6 × 4 = 24 cm. Step 3: LSA = 24 × 10 = 240 cm². Therefore, the answer is 240 cm².

Question 24

A rectangular prism has dimensions 8 cm × 6 cm × 5 cm. A smaller rectangular prism is removed from one corner such that its dimensions are 4 cm × 3 cm × 2.5 cm. What is the volume of the remaining solid in cm³?

Accepted Answer

Step 1: Volume of the original prism = 8 × 6 × 5 = 240 cm³. Step 2: Volume of the removed prism = 4 × 3 × 2.5 = 30 cm³. Step 3: Volume of the remaining solid = 240 - 30 = 210 cm³.

Question 25

A hexagonal prism has a regular hexagonal base with side 5 cm. The height of the prism is 10 cm. The area of the regular hexagon is 64.95 cm². A plane cuts the prism horizontally at a height of 4 cm from the base. What is the lateral surface area of the bottom portion (frustum-like section) in cm²?

Accepted Answer

Step 1: The lateral surface area of a prism is the perimeter of the base × height. Step 2: Perimeter of regular hexagon with side 5 cm = 6 × 5 = 30 cm. Step 3: The bottom portion has height = 4 cm. Step 4: Lateral surface area of bottom portion = 30 × 4 = 120 cm².

Question 26

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: Calculate the volume of the rectangular prism = length × width × height = 8 × 6 × 4 = 192 cm³. Step 2: Calculate the volume of the pyramid with base 8 cm × 6 cm and height 12 cm. 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 27

A pentagonal pyramid has a regular pentagonal base with side 6 cm. The height of the pyramid is 12 cm. If the area of the regular pentagon is 61.94 cm², what is the volume of the pyramid in cm³?

Accepted Answer

Step 1: For a pyramid, Volume = (1/3) × Base Area × Height. Step 2: Base area of the regular pentagon = 61.94 cm². Step 3: Height = 12 cm. Step 4: Volume = (1/3) × 61.94 × 12 = (1/3) × 743.28 = 247.76 ≈ 248 cm³.

Question 28

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

Accepted Answer

Step 1: For an equilateral triangle with side a = 8 cm, the area = (√3/4) × a² = (√3/4) × 64 = 16√3 cm². Step 2: Height of pyramid = 6√3 cm. Step 3: Volume = (1/3) × Base Area × Height = (1/3) × 16√3 × 6√3 = (1/3) × 16 × 6 × 3 = (1/3) × 288 = 96 cm³.

RRB NTPC Prism & Pyramid

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