What is the volume of a cuboid measuring 8 cm × 5 cm × 3 cm?

120 cm³. Volume = 8 5 3 = 120 cm³.

Find the surface area of a sphere of radius 7 cm. (Take = 227.)

616 cm². Surface area = 4 r^2 = 4 227 49 = 616 cm².

A cylinder has radius 7 cm and height 10 cm. Find its curved surface area. (Take = 227.)

440 cm². Curved surface area = 2 r h = 2 227 7 10 = 440 cm².

A solid metal sphere of radius 3 cm is melted and recast into small spheres of radius 1 cm. How many small spheres are formed?

27. Volume scales with the cube of the radius, so the number of small spheres is ( 31 )^3 = 27.

A cone of height 24 cm and base radius 7 cm is melted and recast into a cylinder of base radius 7 cm. What is the height of the cylinder?

8 cm. The cone's volume is 13 r^2 h = 13 (49)(24) = (49)(8). Equating to (49)H gives H = 8 cm.

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Quantitative Aptitude · Geometry & Mensuration

Surface Areas and Volume — Questions & Answers

These questions use the solid-geometry formulas for cuboids, cylinders, cones and spheres. 'Melting and recasting' keeps volume constant, which gives one equation linking the two shapes.

5 solved questions
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Key formulas included

Key formulas

Cuboid: volume $lbh$ , surface area $2(lb + bh + hl)$
Cylinder: volume $\pi r^2 h$ , curved surface $2\pi r h$
Cone: volume $\tfrac{1}{3}\pi r^2 h$ , curved surface $\pi r l$ . Sphere: volume $\tfrac{4}{3}\pi r^3$ , surface $4\pi r^2$

5 solved Surface Areas and Volume questions

Try each question first — tap an option to check your answer, or open Learn step by step for a guided walkthrough.

01
What is the volume of a cuboid measuring 8 cm × 5 cm × 3 cm?
4 steps · reveal one at a time
1. Step 1: Understand the Question
  We need to find the volume of a cuboid. The question provides the three dimensions: length, width, and height.
2. Step 2: Recall Volume Formula
  The formula for the volume of a cuboid is length $\times$ width $\times$ height. This calculates the total space it occupies.
3. Step 3: Calculate the Volume
  Substitute the given dimensions into the formula. Volume $= 8 \text{ cm} \times 5 \text{ cm} \times 3 \text{ cm}$ .
4. Step 4: The answer
  So the answer is (B) 120 cm³ because multiplying the dimensions $8 \times 5 \times 3$ gives $120$ , and the units are cubic centimeters.
02
Find the surface area of a sphere of radius 7 cm. (Take $\pi = \tfrac{22}{7}$ .)
5 steps · reveal one at a time
1. Step 1: Recall the formula
  To find the surface area of a sphere, we use the formula $A = 4\pi r^2$ , where $r$ represents the radius of the sphere. This is a standard formula you should remember for geometry problems.
2. Step 2: Note the given values
  The question provides us with the radius of the sphere, $r = 7$ cm. It also specifies that we should use the approximation $\pi = \frac{22}{7}$ for our calculation. These are all the values we need.
3. Step 3: Substitute and calculate
  Now, substitute the given values into the formula: $A = 4 \times \frac{22}{7} \times (7)^2$ . First, calculate the square of the radius: $(7)^2 = 49$ . So the expression becomes $A = 4 \times \frac{22}{7} \times 49$ .
4. Step 4: Simplify and find area
  We can simplify the calculation by cancelling out the 7 in the denominator with one of the 7s from 49. This gives $A = 4 \times 22 \times 7$ . Multiplying these numbers: $4 \times 22 = 88$ , and $88 \times 7 = 616$ . The surface area is $616$ cm $^2$ .
5. Step 5: The answer
  So the answer is (D) 616 cm². This matches our calculated surface area for the sphere with a radius of 7 cm.
03
A cylinder has radius 7 cm and height 10 cm. Find its curved surface area. (Take $\pi = \tfrac{22}{7}$ .)
4 steps · reveal one at a time
1. Step 1: Identify Given Values
  First, let's note the information given in the question. We have the radius $r = 7$ cm and the height $h = 10$ cm. We are also told to use $\pi = \frac{22}{7}$ .
2. Step 2: Recall the Formula
  The formula for the curved surface area of a cylinder is $2\pi r h$ . This formula calculates the area of the side of the cylinder, excluding the top and bottom circular faces.
3. Step 3: Calculate the Area
  Now, substitute the given values into the formula: Curved surface area $= 2 \times \frac{22}{7} \times 7 \times 10$ . We can simplify by cancelling out the 7s.
4. Step 4: The Answer
  So the answer is (C) 440 cm². Performing the multiplication, $2 \times 22 \times 10 = 44 \times 10 = 440$ . The curved surface area is $440$ cm².
04
A solid metal sphere of radius 3 cm is melted and recast into small spheres of radius 1 cm. How many small spheres are formed?
4 steps · reveal one at a time
1. Step 1: Understand the Goal
  We are melting a large sphere and recasting its metal into many smaller spheres. The total volume of metal must remain the same throughout this process.
2. Step 2: Recall Volume Scaling
  The volume of a sphere is proportional to the cube of its radius. This means if the radius changes by a factor, the volume changes by that factor cubed. The formula is $V = \frac{4}{3} \pi r^3$ .
3. Step 3: Calculate Volume Ratio
  The large sphere has a radius of $3$ cm, and the small spheres have a radius of $1$ cm. The ratio of the radii is $\frac{3}{1} = 3$ . Therefore, the ratio of their volumes is $\left(\frac{3}{1}\right)^3$ .
4. Step 4: The answer
  So the answer is (B) 27. The volume of the large sphere is $3^3 = 27$ times greater than the volume of one small sphere, meaning 27 small spheres can be formed.
05
A cone of height 24 cm and base radius 7 cm is melted and recast into a cylinder of base radius 7 cm. What is the height of the cylinder?
5 steps · reveal one at a time
1. Step 1: Volume remains constant
  When a 3D object is melted and recast into another shape, its volume stays the same. So, the volume of the original cone will be equal to the volume of the new cylinder.
2. Step 2: Calculate cone's volume
  First, let's find the volume of the cone. The formula for the volume of a cone is $V_c = \frac{1}{3}\pi r^2 h$ . With a radius of $7$ cm and height of $24$ cm, we get $V_c = \frac{1}{3}\pi (7^2) (24) = \frac{1}{3}\pi (49) (24) = \pi (49) (8)$ cubic cm.
3. Step 3: Set up cylinder's volume
  Now, let's consider the cylinder. Its base radius is also $7$ cm. The formula for the volume of a cylinder is $V_{cyl} = \pi r^2 H$ . So, $V_{cyl} = \pi (7^2) H = \pi (49) H$ , where $H$ is the unknown height of the cylinder.
4. Step 4: Equate volumes and solve
  Since the volumes are equal, we can set them up: $\pi (49) (8) = \pi (49) H$ . We can cancel out $\pi (49)$ from both sides, which leaves us with $H = 8$ cm.
5. Step 5: The answer
  So the answer is (C) 8 cm. The height of the cylinder is $8$ cm, as its volume must match the cone's volume.

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