Q: A sector of a circle has a central angle of 150° and radius 10 cm. Another sector of the same circle has a central angle of $\frac{5\pi}{6}$ radians. What is the ratio of the area of the first sector to that of the second?

1: 1. The first angle is 150°. The second angle $\frac{5\pi}{6}$ radians equals 150°. Since both sectors are from the same circle and have equal central angles, their areas are equal.

Q: What is the area of a sector of a circle with radius 9 cm and central angle 80°?

18π cm². The area of a sector is $\frac{\theta}{360^\circ}\pi r^2$. Substituting $\theta=80^\circ$ and $r=9$ gives $\frac{80}{360}\pi\times81=18\pi$ cm².

Q: The area of a sector of a circle of radius 6 cm with central angle $120^\circ$ is:

12 π cm². The area of the full circle is $\pi r^2=36\pi$. A $120^\circ$ sector is one-third of the circle, so its area is $\frac{120}{360}\times36\pi=12\pi\,\text{cm}^2$.

Q: An annular ring-shaped disc has outer and inner radii of 12 cm and 8 cm. What percentage of the full circle is this ring?

55.56%. The ring’s area is $\pi(R^2-r^2)$ and the full outer circle’s area is $\pi R^2$. So percentage = $\frac{12^2-8^2}{12^2}\times 100 = \frac{80}{144}\times 100 = 55.56\%$.

Q: A garden is circular with a border 3 m wide around it. The total area including the border is 794 m², and the garden area alone is 379.94 m². What is the radius of the garden?

11 m. The garden area is 379.94 m², so $\pi r^2 = 379.94$. Using $\pi \approx 3.14$, $r^2 = 379.94/3.14 = 121$, hence $r = 11$ m.

Q: The circumferences of two circular paths are in the ratio 2:7. If the smaller path has radius 8 m, what is the radius of the larger path?

28 m. Since circumference $C = 2\pi r$, the ratio of circumferences equals the ratio of radii. So if radii are in ratio 2:7 and the smaller radius is 8 m, the larger radius is $8 \times \frac{7}{2} = 28$ m.

Question 1

A right circular cone has a radius of 9 cm and a height of 40 cm. What is its slant height?

Accepted Answer

41 cm. In a right circular cone, the slant height $l$ is found using $l^2=r^2+h^2$. Substituting $r=9$ cm and $h=40$ cm gives $l^2=81+1600=1681$. Therefore, $l=41$ cm.

Question 2

A rectangular prism has a base area of 36 cm². If its height is increased by 25% from the original height of 12 cm, what is the new volume?

Accepted Answer

540 cm³. The original height is 12 cm, so after a 25% increase it becomes 15 cm. Volume of a rectangular prism = base area × height = 36 × 15 = 540 cm³. Hence the new volume is 540 cm³.

Question 3

A solid metal sphere of radius 6 cm is completely immersed in a vertical cylindrical vessel containing water, causing the water level to rise by 4 cm. What is the radius of the cylinder?

Accepted Answer

6 √ 2 cm. When the sphere is immersed, it displaces water equal to its own volume. So, volume of sphere = volume of cylindrical water rise: $\frac{4}{3}\pi(6)^3 = \pi r^2 	imes 4$. Solving gives $r^2=72$, hence $r=6\sqrt{2}$ cm.

Question 4

A gardener has 80 m of fencing wire. He can use this wire to enclose either a circular lawn or a square lawn. If he uses the entire wire in each case, what is the approximate ratio of the area of the circular lawn to the area of the square lawn?

Accepted Answer

1.27:1. For the same perimeter, a circle encloses more area than a square. With perimeter 80 m, the circle's area is 400/ and the square's area is 400, so the ratio is : 1 1.27 : 1. This is a standard perimeter-to-area comparison.

Question 5

A sector of a circle has a central angle of 150° and radius 10 cm. Another sector of the same circle has a central angle of $\frac{5\pi}{6}$ radians. What is the ratio of the area of the first sector to that of the second?

Accepted Answer

1: 1. The first angle is 150°. The second angle $\frac{5\pi}{6}$ radians equals 150°. Since both sectors are from the same circle and have equal central angles, their areas are equal.

Question 6

What is the area of a sector of a circle with radius 9 cm and central angle 80°?

Accepted Answer

18π cm². The area of a sector is $\frac{\theta}{360^\circ}\pi r^2$. Substituting $\theta=80^\circ$ and $r=9$ gives $\frac{80}{360}\pi\times81=18\pi$ cm².

Question 7

The area of a sector of a circle of radius 6 cm with central angle $120^\circ$ is:

Accepted Answer

12 π cm². The area of the full circle is $\pi r^2=36\pi$. A $120^\circ$ sector is one-third of the circle, so its area is $\frac{120}{360}\times36\pi=12\pi\,\text{cm}^2$.

Question 8

An annular ring-shaped disc has outer and inner radii of 12 cm and 8 cm. What percentage of the full circle is this ring?

Accepted Answer

55.56%. The ring’s area is $\pi(R^2-r^2)$ and the full outer circle’s area is $\pi R^2$. So percentage = $\frac{12^2-8^2}{12^2}\times 100 = \frac{80}{144}\times 100 = 55.56\%$.

Question 9

A garden is circular with a border 3 m wide around it. The total area including the border is 794 m², and the garden area alone is 379.94 m². What is the radius of the garden?

Accepted Answer

11 m. The garden area is 379.94 m², so $\pi r^2 = 379.94$. Using $\pi \approx 3.14$, $r^2 = 379.94/3.14 = 121$, hence $r = 11$ m.

Question 10

The circumferences of two circular paths are in the ratio 2:7. If the smaller path has radius 8 m, what is the radius of the larger path?

Accepted Answer

28 m. Since circumference $C = 2\pi r$, the ratio of circumferences equals the ratio of radii. So if radii are in ratio 2:7 and the smaller radius is 8 m, the larger radius is $8 \times \frac{7}{2} = 28$ m.

Question 11

How many diagonals does a pentagon have?

Accepted Answer

5. The number of diagonals in an n-sided polygon is n(n-3)/2. For a pentagon, n = 5, so the number of diagonals is 5(5-3)/2 = 5.

Question 12

A pump fills 0.625 liters of water every second. How much does it fill in 6 seconds?

Accepted Answer

3.75 L. The pump fills 0.625 L each second. In 6 seconds, it fills 0.625 × 6 = 3.75 L.

Question 13

A container holds 4,000 millilitres of liquid. If its full capacity is 0.008 kilolitres, what percentage is empty?

Accepted Answer

50%. The full capacity is 0.008 kL = 8 L, while the liquid present is 4,000 mL = 4 L. So the empty part is 8 − 4 = 4 L, which is 50% of the total capacity.

Question 14

A hexagonal prism has a regular hexagonal base with side 8 cm and height 12 cm. What is its volume?

Accepted Answer

1152√3 cm³. The volume of a prism is base area × height. For a regular hexagon of side 8 cm, area = $\frac{3\sqrt{3}}{2}\times 8^2 = 96\sqrt{3}$ cm². Multiplying by height 12 cm gives $96\sqrt{3}\times 12 = 1152\sqrt{3}$ cm³.

Question 15

A solid cylinder of radius 6 cm has five cylindrical holes of radius 1.5 cm each drilled throughout its length. Approximately what percent of the volume is removed?

Accepted Answer

31.25%. The removed volume is proportional to the total area of the five holes. Main cylinder area = $\pi\times 6^2 = 36\pi$. Area of one hole = $\pi\times 1.5^2 = 2.25\pi$, so five holes remove $11.25\pi$. Percentage removed = $\frac{11.25}{36}\times 100 = 31.25\%$.

Question 16

A sector has a central angle of 144° and radius 7 cm. Another sector with the same radius has a central angle of $\frac{4\pi}{5}$ radians. What is the ratio of their areas?

Accepted Answer

1:1. The area of a sector is proportional to its central angle when the radius is the same. Here, $144^\circ = \frac{144\pi}{180} = \frac{4\pi}{5}$ radians, so both sectors have equal angles and equal areas. Therefore, the ratio is 1:1.

Question 17

A circular park with diameter 8 m is surrounded by a 0.5 m wide path. What is the percentage increase in area?

Accepted Answer

26.56%. The park radius is 4 m and the outer radius is 4.5 m. Increase in area = $\pi(4.5^2-4^2)=\pi(20.25-16)=4.25\pi$. Percentage increase = $\frac{4.25\pi}{16\pi}\times100=26.5625\%$, approximately 26.56%.

Question 18

A sector with radius 12 cm has area 132 cm². Find the central angle.

Accepted Answer

105°. For a sector, $\text{Area} = \frac{\theta}{360}\pi r^2$. With $r=12$, we get $132 = \frac{\theta}{360}\pi(144)$. Using $\pi=\frac{22}{7}$, this gives $\theta \approx 105^\circ$.

Question 19

A circular board has radius 3 m. Painting costs ₹50 per m². If 15% is unpainted, what is the total cost?

Accepted Answer

₹ 1202. Area of the board = $\pi \times 3^2 = 9\pi$ m². Painted area = 85% of $9\pi$ = $7.65\pi$ m². At ₹50 per m², cost = $7.65\pi \times 50 \approx ₹1202$.

Question 20

A cone has its base area equal to its lateral surface area. If the radius is $r$, find the slant height $l$.

Accepted Answer

2r. Base area of a cone is $\pi r^2$ and lateral surface area is $\pi r l$. Since they are equal, $\pi r^2 = \pi r l$, which gives $l = r$. However, the provided options do not include this correct result, so the intended option appears inconsistent.

Question 21

A square field has side 35 m and contains a circular pond. If the area of the pond is 770 m², what is the remaining area of the field?

Accepted Answer

455 m². Area of the square = $35^2 = 1225$ m². Subtract the pond area $770$ m² to get remaining area $1225 - 770 = 455$ m².

Question 22

If the height of a right prism is increased by 60% while the base area remains the same, by what percentage does its volume increase?

Accepted Answer

60%. For a prism, volume = base area × height. If the base area stays constant and height increases by 60%, the volume also increases by 60%.

Question 23

A wall clock of radius 18 cm has a chord that, together with two radii from the centre, forms an equilateral triangle. If the smaller segment is painted, what is its area as a percentage of the total area of the clock (approximately)?

Accepted Answer

3%. The chord with two radii forms an equilateral triangle, so the central angle is 60°. The smaller segment area equals the area of the 60° sector minus the area of the equilateral triangle. For radius 18 cm, this comes to about 30.6 cm², which is roughly 3% of the total circle area.

Question 24

A cylinder of radius 5 cm and height 12 cm is bored through its full height by a cylindrical hole of radius 3 cm. What percentage of the original volume is removed?

Accepted Answer

36%. Original volume = $\pi r^2 h = \pi\cdot 5^2\cdot 12$. Removed volume = $\pi\cdot 3^2\cdot 12$. The ratio removed/original = $9/25$, which is 36%.

Question 25

A sector of a circle of radius 20 cm has a central angle of 60°. Another sector of the same circle has an angle of $\pi/3$ radians. What is the ratio of their areas?

Accepted Answer

1:1. The two sectors have the same radius and the same central angle, just expressed in different units. Since sector area is proportional to the angle when radius is fixed, their areas are equal.

Question 26

A circular path 2 m wide surrounds a park of radius 10 m. What is the area of the path?

Accepted Answer

138.2 m². The inner radius is 10 m and the outer radius is 12 m. Area of the path = $\pi(12^2-10^2)=\pi(144-100)=44\pi$. Using $\pi\approx 3.14$, this is $44	imes 3.14=138.16\approx 138.2$ m².

Question 27

Find the area of a sector of a circle with radius 12 cm and central angle $\pi/5$ radians.

Accepted Answer

45.24 cm². For a sector with angle in radians, area = $\frac{1}{2}r^2\theta$. Substituting $r=12$ and $\theta=\pi/5$ gives $\frac{1}{2}\times 144 \times \frac{\pi}{5} = 14.4\pi \approx 45.24\,\text{cm}^2$.

Question 28

What is the maximum possible length of a chord in a circle with a diameter of 28 cm?

Accepted Answer

28cm. The maximum possible chord in a circle is the diameter. Since the diameter is 28 cm, the longest chord is also 28 cm.

Question 29

How many smaller hemispheres of radius 4 cm can be formed by melting a hemisphere of radius 16 cm?

Accepted Answer

64. The number of smaller hemispheres equals the ratio of the volume of the large hemisphere to the volume of one small hemisphere. Since volume is proportional to r^3, the ratio is (16/4)^3 = 4^3 = 64. So 64 smaller hemispheres can be formed.

Question 30

A cone and a hemisphere have the same radius. Their combined height is 24 cm. If both have equal volumes, find the radius.

Accepted Answer

8 cm. Let the common radius be r. The hemisphere’s height is r, so the cone’s height is 24 - r. Since their volumes are equal, $\frac{1}{3}\pi r^2(24-r)=\frac{2}{3}\pi r^3$. Solving gives r = 8 cm.

Question 31

A cone has a base area of 78.5 cm² and height 12 cm. What is its volume? (Use $\pi \approx 3.14$)

Accepted Answer

314 cm³. The volume of a cone is $V=\frac{1}{3}Ah$, where $A$ is the base area and $h$ is the height. Substituting $A=78.5$ cm² and $h=12$ cm gives $V=\frac{1}{3}\times 78.5\times 12=314$ cm³.

Question 32

A solid metal sphere of radius 15 cm is melted and recast into 27 identical smaller spheres. What is the ratio of the surface area of the original sphere to the total surface area of all 27 smaller spheres?

Accepted Answer

1: 3. Since volume is conserved, 27 smaller spheres each have radius $15/3=5$ cm. Surface area is proportional to the square of radius, so each smaller sphere has $1/9$ the surface area of the original; for 27 spheres, total surface area becomes 3 times the original.

Question 33

A sector of a circle with radius 8 cm has a central angle of 45°. What is the area of the corresponding segment?

Accepted Answer

(8 π - 16 √ 2) cm². The area of the sector is c0r²θ/360 = c0×64×45/360 = 8c0 cm². The triangle formed by the two radii has area 1/2 × 8 × 8 × sin 45° = 16√2 cm². So the segment area is 8c0 - 16√2 cm².

Question 34

If the lateral surface area of a cylinder is 320 cm² and its height is 10 cm, what is the radius?

Accepted Answer

16/ π cm. The lateral surface area of a cylinder is 2πrh. Substituting 320 = 2π × r × 10 gives r = 16/π cm. So the correct option is 16/π cm.

Question 35

If the perimeter of a regular hexagon is 72 cm, what is the side length?

Accepted Answer

12 cm. In a regular hexagon, all 6 sides are equal. So each side = 72 ÷ 6 = 12 cm.

Question 36

A right circular cone has radius 8 cm and height 20 cm. A sphere is inscribed in the cone such that it touches the base and the lateral surface. Find the radius of the inscribed sphere.

Accepted Answer

5.42 cm. The axial section of the cone is an isosceles triangle with base $16$ cm, equal sides $\sqrt{8^2+20^2}=\sqrt{464}$, and height 20 cm. The inradius of this triangle equals the radius of the inscribed sphere, and using the triangle inradius formula gives approximately 5.42 cm. Hence the sphere’s radius is 5.42 cm.

Question 37

The cost of fencing a circular garden at ₹180 per metre is ₹6,786. Find the radius of the garden.

Accepted Answer

6 m. Total fencing length = $6786/180 = 37.7$ m. For a circle, circumference is $2\pi r$, so $2\pi r = 37.7$. Using $\pi\approx 3.14$, $r\approx 37.7/6.28\approx 6$ m.

Question 38

A circular medallion of radius 28 cm is fitted perfectly inside a square box. Find the area of the unused square space around the circle.

Accepted Answer

672 cm². Since the circle fits perfectly inside the square, the square’s side is the diameter, i.e. 56 cm. Square area = $56^2=3136$ cm² and circle area = $\pi r^2 = \frac{22}{7}\times 28^2 = 2464$ cm². The unused area is $3136-2464=672$ cm².

Question 39

A circular wall clock has a radius of 28 cm. Calculate the distance covered by the tip of the hour hand in 9 hours, assuming it is positioned at the edge of the clock face. Use $\pi = 22/7$.

Accepted Answer

132 cm. The tip of the hour hand moves along a circle of radius 28 cm. In 12 hours it covers one full circumference, so in 9 hours it covers $9/12=3/4$ of the circle. Circumference = $2\pi r = 2\times \frac{22}{7}\times 28 = 176$ cm, and $3/4$ of this is 132 cm.

Question 40

A solid sphere of radius 6 cm is melted and recast into three identical cones of equal height. If each cone's radius is 3 cm, what is the height of each cone?

Accepted Answer

32 cm. Volume of sphere = $\frac{4}{3}\pi(6^3)=288\pi$. Let the height of each cone be $h$; total volume of 3 cones = $3\times \frac{1}{3}\pi(3^2)h = 9\pi h$. Equating gives $9\pi h = 288\pi$, so $h=32$ cm.

Question 41

If the radius of a cylinder is increased to 1.5 times and the height is doubled, what is the ratio of the new volume to the original volume?

Accepted Answer

4.5:1. For a cylinder, volume $V=\pi r^2 h$. If radius becomes 1.5 times and height becomes 2 times, new volume factor = $(1.5)^2 \times 2 = 2.25 \times 2 = 4.5$. So the ratio is 4.5:1.

Question 42

A cylinder and a cone have the same base radius $R$ and the same volume. Determine the ratio of the height of the cylinder to the height of the cone.

Accepted Answer

1:3. Since the base radius is the same, the base area is the same. Equal volumes give $\pi R^2 h_{cyl} = \frac{1}{3}\pi R^2 h_{cone}$, so $h_{cyl} = \frac{1}{3}h_{cone}$. Therefore, the ratio is 1:3.

Question 43

Two circular plates have radii in the ratio 3:5. If the smaller plate has an area of 108 cm², what is the area of the larger plate?

Accepted Answer

300 cm². The areas of circles are proportional to the squares of their radii. With radii in the ratio $3:5$, the area ratio is $9:25$. So the larger area is $108 \times \frac{25}{9} = 300$ cm².

Question 44

If the base perimeter is 48 cm and the slant height is 12 cm, what is the lateral surface area of a square pyramid?

Accepted Answer

288 cm². The lateral surface area of a pyramid is given by $\frac{1}{2} \times$ perimeter of base $\times$ slant height. Substituting the values gives $\frac{1}{2} \times 48 \times 12 = 288$ cm².

Question 45

A cylinder is melted and recast into 10 identical cones, each with radius 4 cm and height 6 cm. What was the radius of the original cylinder if its height was 8 cm?

Accepted Answer

6.32 cm. Since the cylinder is recast into 10 identical cones, their total volumes are equal. Equating $\pi R^2 \cdot 8 = 10 \times \frac{1}{3}\pi \cdot 4^2 \cdot 6$ gives $8R^2 = 320$, so $R^2 = 40$ and $R \approx 6.32$ cm.

Question 46

A tent designed as a rectangular prism features a rectangular base with an area of 28 m² and a height of 5 m. What is the volume of this tent?

Accepted Answer

140 m³. For a rectangular prism, volume = base area × height. Here, the base area is 28 m² and the height is 5 m, so the volume is 28 × 5 = 140 m³.

Question 47

A circular pond with a radius of 9 m is surrounded by a walking path that is 4.5 m wide. Determine the ratio of the area of the walking path to the area of the pond.

Accepted Answer

5:4. The pond radius is 9 m and the path width is 4.5 m, so the outer radius is 13.5 m. Path area = $\pi(13.5^2-9^2)=\pi(182.25-81)=101.25\pi$, while pond area = $81\pi$. Their ratio is $101.25:81 = 5:4$.

Question 48

A square pyramid has a base side of 12 cm and a slant height of 10 cm. Find its lateral surface area.

Accepted Answer

240 cm². The lateral surface area of a square pyramid is given by $\frac{1}{2} \times \text{perimeter of base} \times \text{slant height}$. Here, perimeter of the square base = $4 \times 12 = 48$ cm. So, LSA = $\frac{1}{2} \times 48 \times 10 = 240$ cm².

Question 49

If the side of a cube is tripled, what is the ratio of the new total surface area to the original total surface area, and the ratio of the new volume to the original volume?

Accepted Answer

9:1 and 27:1. For a cube, total surface area is proportional to side² and volume is proportional to side³. Tripling the side makes the surface area 3² = 9 times and the volume 3³ = 27 times.

Question 50

A cylinder and a cone have the same base radius and the same volume. What is the ratio of the height of the cylinder to the height of the cone?

Accepted Answer

1:3. Volume of cylinder = $\pi r^2 h_c$ and volume of cone = $\frac{1}{3}\pi r^2 h_k$. With equal volumes and same radius, $h_c = \frac{1}{3}h_k$, so the ratio of cylinder height to cone height is 1:3.

SSC CGL (Prelims) General: Mensuration questions with solutions

Questions