SSC CGL (Prelims) Maths: Mensuration questions with solutions

Q1. If each side of an equilateral triangle is $37\sqrt{3}$ cm, then its altitude (in cm) is equal to:

1. 37.5
2. 18.5
3. 60.5
4. 55.5

Answer: 55.5

In an equilateral triangle, altitude = $\frac{\sqrt{3}}{2}a$. Substituting $a=37\sqrt{3}$ gives altitude $=\frac{\sqrt{3}}{2}\times 37\sqrt{3}=\frac{37\times 3}{2}=55.5$ cm.

Q2. A cuboid of dimensions 50 cm, 150 cm, and 175 cm can be divided into how many identical largest cubes?

1. 75
2. 84
3. 85
4. 90

Answer: 84

The largest possible cube has side equal to the HCF of 50, 150, and 175, which is 25 cm. The number of such cubes is \((50/25)\times(150/25)\times(175/25)=2\times6\times7=84\).

Q3. The area of a square is $16x^2 + 40x + 25$ square units. Find the perimeter of the square.

1. 3(10x+8)
2. 3(8x+10)
3. 2(10x+8)
4. 2(8x+10)

Answer: 2(8x+10)

The area $16x^2 + 40x + 25$ can be written as $(4x+5)^2$. So the side of the square is $4x+5$, and the perimeter is $4(4x+5)=16x+20=2(8x+10)$.

Q4. A rectangular piece of paper is 50 cm long and 22 cm wide. If a cylinder is formed by rolling the paper along its breadth, then the volume of the cylinder is:

1. 2,125 cm³
2. 1,925 cm³
3. 2,025 cm³
4. 1,825 cm³

Answer: 1,925 cm³

If the paper is rolled along its breadth, the breadth 22 cm becomes the circumference of the base, so $2\pi r=22$. Thus $r=3.5$ cm and height $h=50$ cm. Volume is $\pi r^2 h = \frac{22}{7}\times 3.5^2 \times 50 = 1925$ cm³.

Q5. The side of an equilateral triangle is 28 cm. Taking each vertex as the centre, a circle is drawn with radius equal to half the length of the side of the triangle. Find the area of the part of the triangle that is not included in the circles (use \(\pi = \frac{22}{7}\) and \(\sqrt{3} = 1.73\)).

1. 30.89 cm²
2. 38.08 cm²
3. 31.08 cm²
4. 39.08 cm²

Answer: 31.08 cm²

The area of the equilateral triangle is \(\frac{\sqrt{3}}{4}a^2\). The three inside parts are 60° sectors of radius 14 cm, whose total area is \(3\times \frac{60}{360}\pi(14)^2\). Subtracting gives the required uncovered area.

Q6. If a right circular cone of height 24 cm has a volume of 1232 cm³, then the total surface area of the cone is (use \(\pi = \frac{22}{7}\)):

1. 806 cm²
2. 704 cm²
3. 904 cm²
4. 608 cm²

Answer: 704 cm²

Use \(V=\frac{1}{3}\pi r^2 h\) to find \(r\). With \(V=1232\) and \(h=24\), we get \(r=7\) cm. Then slant height \(l=\sqrt{r^2+h^2}=25\) cm, so total surface area \(=\pi r(l+r)=\frac{22}{7}\cdot 7\cdot(25+7)=704\text{ cm}^2\).

Q7. The perimeter of a square is 48 cm. What is the area of the square?

1. 144 cm²
2. 100 cm²
3. 121 cm²
4. 96 cm²

Answer: 144 cm²

A square has four equal sides, so each side is $48/4 = 12$ cm. Its area is $12^2 = 144$ cm².

Q8. What is the area of a triangle with base 10 cm and height 5 cm?

1. 25 cm²
2. 30 cm²
3. 50 cm²
4. 15 cm²

Answer: 25 cm²

The area of a triangle is given by \(\frac{1}{2} \times \text{base} \times \text{height}\). Substituting base = 10 cm and height = 5 cm gives 25 cm².

Q9. If the radius of a circle is 14 cm, find its circumference. (Use \(\pi = \frac{22}{7}\))

1. 88 cm
2. 86 cm
3. 90 cm
4. 84 cm

Answer: 88 cm

The circumference of a circle is given by \(2\pi r\). Substituting \(r=14\) cm and \(\pi=\frac{22}{7}\), we get \(2 \times \frac{22}{7} \times 14 = 88\) cm. So the correct answer is 88 cm.

Q10. What is the perimeter of a rectangle with length 12 cm and breadth 8 cm?

1. 40 cm
2. 48 cm
3. 42 cm
4. 36 cm

Answer: 40 cm

The perimeter of a rectangle is given by $2(l+b)$. Substituting $l=12$ cm and $b=8$ cm gives $2(12+8)=2\times 20=40$ cm. So the correct answer is 40 cm.