Mensuration — Multiple Choice Questions

Area of a triangle is 30 cm². If its base is 10 cm, then its height is

1. 5 cm

2. 6 cm

3. 7 cm

4. 8 cm

Answer

Given,

Area of the triangle = 30 cm²

Base of the triangle = 10 cm

Let the height of the triangle be h cm.

By formula,

Area of a triangle = $\dfrac{1}{2}$ × base × height

⇒ $\dfrac{1}{2}$ x 10 x h = 30

⇒ 5 × h = 30

⇒ h = $\dfrac{30}{5}$

⇒ h = 6 cm

Hence, option 2 is the correct option.

If the perimeter of a square is 80 cm, then its area is

1. 800 cm²

2. 600 cm²

3. 400 cm²

4. 200 cm²

Answer

Given,

Perimeter of the square = 80 cm

Let the side of the square be a cm.

By formula,

Perimeter of a square = 4 × side

⇒ 4 × a = 80

⇒ a = $\dfrac{80}{4}$

⇒ a = 20 cm

By formula,

Area of a square = side²

= (20)² cm²

= 400 cm²

Hence, option 3 is the correct option.

Area of a parallelogram is 48 cm². If its height is 6 cm then its base is

1. 8 cm

2. 4 cm

3. 16 cm

4. none of these.

Answer

Given,

Area of the parallelogram = 48 cm²

Height of the parallelogram = 6 cm

Let the base of the parallelogram be b cm.

By formula,

Area of a parallelogram = base × height

⇒ b × 6 = 48

⇒ b = $\dfrac{48}{6}$

⇒ b = 8 cm

Hence, option 1 is the correct option.

If d is the diameter of a circle, then its area is

1. πd²

2. $\dfrac{\pi d^2}{2}$

3. $\dfrac{\pi d^2}{4}$

4. 2πd²

Answer

Let d be the diameter of the circle.

By formula,

Radius of a circle = $\dfrac{\text{diameter}}{2}$

So, radius of the circle = $\dfrac{d}{2}$

By formula,

Area of a circle = πr²

$= \pi \times \left(\dfrac{d}{2}\right)^2\\[1em] = \pi \times \dfrac{d^2}{4}\\[1em] = \dfrac{\pi d^2}{4}$

Hence, option 3 is the correct option.

If the area of a trapezium is 64 cm² and the distance between parallel sides is 8 cm, then sum of its parallel sides is

1. 8 cm

2. 4 cm

3. 32 cm

4. 16 cm.

Answer

Given,

Area of the trapezium = 64 cm²

Distance between the parallel sides = 8 cm

Let the sum of the parallel sides be a cm.

By formula,

Area of a trapezium = $\dfrac{1}{2}$ × sum of parallel sides × distance between the parallel sides

⇒ $\dfrac{1}{2}$ × a × 8 = 64

⇒ 4 × a = 64

⇒ a = $\dfrac{64}{4}$

⇒ a = 16 cm

Hence, option 4 is the correct option.

Area of a rhombus whose diagonals are 8 cm and 6 cm is

1. 48 cm²

2. 24 cm²

3. 12 cm²

4. 96 cm².

Answer

Given,

Diagonals of the rhombus = 8 cm and 6 cm

By formula,

Area of a rhombus = $\dfrac{1}{2}$ × product of its diagonals

= $\dfrac{1}{2}$ × 8 × 6 cm²

= $\dfrac{1}{2}$ × 48 cm²

= 24 cm²

Hence, option 2 is the correct option.

If the lengths of diagonals of a rhombus is doubled, then area of rhombus will be

1. doubled

2. tripled

3. four times

4. remains same.

Answer

Let the lengths of the diagonals of the rhombus be d₁ and d₂.

By formula,

Area of a rhombus = $\dfrac{1}{2}$ × product of its diagonals

Area of the given rhombus = $\dfrac{1}{2}$ × d₁ × d₂

When the lengths of the diagonals are doubled, the new diagonals are 2d₁ and 2d₂.

$\text{Area of the new rhombus} = \dfrac{1}{2} \times 2d_1 \times 2d_2\\[1em] = 4 \times \dfrac{1}{2} \times d_1 \times d_2\\[1em] = 4 \times \text{Area of the given rhombus}$

Hence, option 3 is the correct option.

If the length of a diagonal of a quadrilateral is 10 cm and lengths of the perpendiculars on it from opposite vertices are 4 cm and 6 cm, then area of quadrilateral is

1. 100 cm²

2. 200 cm²

3. 50 cm²

4. none of these.

Answer

Given,

Length of a diagonal of the quadrilateral, d = 10 cm

Lengths of the perpendiculars on it from the opposite vertices, h₁ = 4 cm and h₂ = 6 cm

By formula,

Area of a quadrilateral = $\dfrac{1}{2}$ × d × (h₁ + h₂)

Area of the quadrilateral = $\dfrac{1}{2}$ × 10 × (4 + 6) cm²

= $\dfrac{1}{2}$ × 10 × 10 cm²

= 50 cm²

Hence, option 3 is the correct option.

Area of a rhombus is 90 cm². If the length of one diagonal is 10 cm then the length of other diagonal is

1. 18 cm

2. 9 cm

3. 36 cm

4. 4.5 cm.

Answer

Given,

Area of the rhombus = 90 cm²

Length of one diagonal = 10 cm

Let the length of the other diagonal be d cm.

By formula,

Area of a rhombus = $\dfrac{1}{2}$ × product of its diagonals

⇒ $\dfrac{1}{2}$ × 10 × d = 90

⇒ 5 × d = 90

⇒ d = $\dfrac{90}{5}$

⇒ d = 18 cm

Hence, option 1 is the correct option.

If the volume of a cube is 729 cm³, then its surface area is

1. 486 cm²

2. 324 cm²

3. 162 cm²

4. none of these.

Answer

Given,

Volume of the cube = 729 cm³

Let the side of the cube be a cm.

By formula,

Volume of a cube = (side)³

⇒ a³ = 729

⇒ a = $\sqrt[3]{729}$

⇒ a = 9 cm

By formula,

Surface area of a cube = 6 × (side)²

= 6 × (9)² cm²

= 6 × 81 cm²

= 486 cm²

Hence, option 1 is the correct option.

If the lateral surface area of a cube is 100 cm², then its volume is

1. 25 cm³

2. 125 cm³

3. 625 cm³

4. none of these.

Answer

Given,

Lateral surface area of the cube = 100 cm²

Let the side of the cube be a cm.

By formula,

Lateral surface area of a cube = 4 × (side)²

⇒ 4 × a² = 100

⇒ a² = $\dfrac{100}{4}$

⇒ a² = 25

⇒ a = $\sqrt{25}$

⇒ a = 5 cm

By formula,

Volume of a cube = (side)³

= (5)³ cm³

= 125 cm³

Hence, option 2 is the correct option.

If the length of side of a cube is doubled, then the ratio of volumes of new cube and original cube is

1. 1 : 2

2. 2 : 1

3. 4 : 1

4. 8 : 1.

Answer

Let the side of the original cube be a units.

By formula,

Volume of a cube = (side)³

Volume of the original cube = a³ cubic units

When the length of the side is doubled, the side of the new cube = 2a units

Volume of the new cube = (2a)³

= 8a³ cubic units

Ratio of the volumes = $\dfrac{\text{Volume of the new cube}}{\text{Volume of the original cube}}$

= $\dfrac{8a^3}{a^3}$

= 8

Hence, option 4 is the correct option.

If the dimensions of a rectangular room are 10 m × 12 m × 9 m, then the cost of painting its four walls at the rate of ₹80 per m² is

1. ₹31860

2. ₹36180

3. ₹31680

4. none of these.

Answer

Given,

Dimensions of the rectangular room = 10 m × 12 m × 9 m

Here, length = 10 m, breadth = 12 m and height = 9 m.

By formula,

Area of four walls = 2(length + breadth) × height

= 2(10 + 12) × 9 m²

= 2 × 22 × 9 m²

= 396 m²

Cost of painting the four walls = Area of four walls × Rate

= 396 m² × ₹80/m²

= ₹31,680

Hence, option 3 is the correct option.

Volume of a cylinder is 1848 cm³. If the diameter of its base is 14 cm, then the height of the cylinder is

1. 12 cm

2. 6 cm

3. 3 cm

4. none of these.

Answer

Given,

Volume of the cylinder = 1848 cm³

Diameter of the base of the cylinder = 14 cm

Radius of the base = $\dfrac{14}{2}$ cm = 7 cm

Let the height of the cylinder be h cm.

By formula,

Volume of a cylinder = πr²h

$\Rightarrow \dfrac{22}{7} \times 7^2 \times h = 1848\\[1em] \Rightarrow \dfrac{22}{\cancel{7}} \times \cancel{7} \times 7 \times h = 1848\\[1em] \Rightarrow 154 \times h = 1848\\[1em] \Rightarrow h = \dfrac{1848}{154}\\[1em] \Rightarrow h = 12 \text{ cm}$

Thus, the height of the cylinder is 12 cm.

Hence, option 1 is the correct option.

If the radius of a cylinder is doubled and height is halved, then new volume is

1. same

2. 2 times

3. 4 times

4. 8 times.

Answer

Let the radius of the original cylinder be r and its height be h.

By formula,

Volume of a cylinder = πr²h

Volume of the original cylinder = πr²h

For the new cylinder, the radius is doubled and the height is halved.

Radius of the new cylinder = 2r

Height of the new cylinder = $\dfrac{h}{2}$

$\text{Volume of the new cylinder} = \pi \times (2r)^2 \times \dfrac{h}{2}\\[1em] = \pi \times 4r^2 \times \dfrac{h}{2}\\[1em] = 2\pi r^2 h\\[1em] = 2 \times \text{Volume of the original cylinder}$

Hence, option 2 is the correct option.