Question 1

A triangle has sides of length 5 cm, 12 cm, and 13 cm. What is its area using Heron's formula?

Accepted Answer

Step 1: Calculate the semi-perimeter: s = (5 + 12 + 13) / 2 = 30 / 2 = 15 cm. Step 2: Apply Heron's formula: Area = √[s(s-a)(s-b)(s-c)] = √[15 × 10 × 3 × 2] = √[900] = 30 cm². Therefore, the area is 30 cm².

Question 2

A trapezium has parallel sides of 12 cm and 18 cm, and a height of 8 cm. What is its area?

Accepted Answer

Step 1: Recall the formula for area of a trapezium: A = [(a + b) / 2] × h, where a and b are the parallel sides and h is the height. Step 2: Substitute a = 12 cm, b = 18 cm, and h = 8 cm. Step 3: A = [(12 + 18) / 2] × 8 = [30 / 2] × 8 = 15 × 8 = 120 cm². Therefore, the area is 120 cm².

Question 3

An equilateral triangle has a side length of 10 cm. What is its perimeter?

Accepted Answer

Step 1: An equilateral triangle has all three sides equal. Step 2: Perimeter = 3 × side length = 3 × 10 = 30 cm. Therefore, the perimeter is 30 cm.

Question 4

The area of a triangle is 24 cm² and its base is 6 cm. What is the height of the triangle?

Accepted Answer

Step 1: Use the area formula: Area = (1/2) × base × height. Step 2: Rearrange to find height: height = (2 × Area) / base. Step 3: Substitute: height = (2 × 24) / 6 = 48 / 6 = 8 cm. Therefore, the height is 8 cm.

Question 5

A right-angled triangle has legs of length 6 cm and 8 cm. What is the length of its hypotenuse?

Accepted Answer

Step 1: Use the Pythagorean theorem: c² = a² + b², where c is the hypotenuse and a, b are the legs. Step 2: Substitute: c² = 6² + 8² = 36 + 64 = 100. Step 3: Take the square root: c = √100 = 10 cm. Therefore, the hypotenuse is 10 cm.

Question 6

The perimeter of a rhombus is 80 cm. What is the length of each side of the rhombus?

Accepted Answer

Step 1: Recall that a rhombus has all four sides equal in length. Step 2: If the perimeter is 80 cm, then 4 × side = 80. Step 3: side = 80 / 4 = 20 cm. Therefore, each side of the rhombus is 20 cm.

Question 7

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 8

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 9

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 10

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 11

A triangle has a base of 12 cm and a height of 8 cm. What is its area?

Accepted Answer

Step 1: Use the formula for the area of a triangle: Area = (1/2) × base × height. Step 2: Substitute the values: Area = (1/2) × 12 × 8 = (1/2) × 96 = 48 cm². Therefore, the area is 48 cm².

Question 12

In a triangle, the sum of any two sides must be greater than the third side. If two sides of a triangle are 7 cm and 10 cm, which of the following can be the length of the third side?

Accepted Answer

Step 1: Apply the triangle inequality theorem. For sides a = 7, b = 10, and c (unknown): (1) 7 + 10 > c → c < 17, (2) 7 + c > 10 → c > 3, (3) 10 + c > 7 (always true for positive c). Step 2: Combine conditions: 3 < c < 17. Step 3: Check the options: 5 cm satisfies 3 < 5 < 17. Therefore, the third side can be 5 cm.

Question 13

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 14

A rectangle has a length of 24 cm and a breadth of 16 cm. What is the perimeter of the rectangle?

Accepted Answer

Step 1: Recall the formula for perimeter of a rectangle: P = 2(l + b). Step 2: Substitute l = 24 cm and b = 16 cm. Step 3: P = 2(24 + 16) = 2(40) = 80 cm. Therefore, the perimeter is 80 cm.

Question 15

The area of a square is 144 cm². What is the length of its diagonal?

Accepted Answer

Step 1: If area = 144 cm², then side² = 144, so side = 12 cm. Step 2: For a square, diagonal = side × √2. Step 3: Diagonal = 12√2 cm ≈ 12 × 1.414 = 16.97 cm, or exactly 12√2 cm. Step 4: If numerical form is required: 12√2 ≈ 17 cm (rounded). However, the exact answer is 12√2 cm.

Question 16

A parallelogram has a base of 20 cm and a height of 15 cm. What is its area?

Accepted Answer

Step 1: Recall the formula for area of a parallelogram: A = base × height. Step 2: Substitute base = 20 cm and height = 15 cm. Step 3: A = 20 × 15 = 300 cm². Therefore, the area is 300 cm².

Question 17

A rhombus has diagonals of length 18 cm and 24 cm. What is its area?

Accepted Answer

Step 1: Recall the formula for area of a rhombus: A = (d₁ × d₂) / 2, where d₁ and d₂ are the diagonals. Step 2: Substitute d₁ = 18 cm and d₂ = 24 cm. Step 3: A = (18 × 24) / 2 = 432 / 2 = 216 cm². Therefore, the area is 216 cm².

Question 18

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 19

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 20

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.

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