What is volume in mathematics?

A measure of space inside a 3D object

What is surface area in mathematics?

The total area of the surface of a 3D object

Quizzes / CBSE / Class 7 / Mathematics / Volume and Surface Area

Introduction Quiz - CBSE Class 7 Mathematics

Free practice quiz: Introduction (CBSE Class 7 Mathematics). 12 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 7 Mathematics practice quiz has 12 multiple-choice questions on Introduction, and 12 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

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Volume versus surface area

Volume is the amount of space a solid takes up. It is measured in cubic units like cm³ or m³.
Surface area is the total area of all outer faces of a solid. It is measured in square units like cm² or m².

Units of volume

\[\text{cm}^3,\ \text{m}^3\]

Units of area

\[\text{cm}^2,\ \text{m}^2\]

Common mistake: Writing volume in square units or surface area in cubic units. Volume is always cubic and area is always square.

Cube and cuboid formulas

A cube has six equal square faces with side a. A cuboid has length l, breadth b and height h.
A rectangular tank is a cuboid, so its volume is length times breadth times height.

Volume of cube

\[V = a^3\]

Surface area of cube

\[S = 6a^2\]

Volume of cuboid

\[V = l \times b \times h\]

Surface area of cuboid

\[S = 2(lb + bh + hl)\]

Common mistake: For a cuboid, adding only three faces (lb + bh + hl) and forgetting to multiply by 2 for the opposite faces.

Sphere and hemisphere formulas

A sphere is a perfectly round ball with radius r. A hemisphere is exactly half of a sphere.
A hemisphere's curved surface is half the sphere's surface. A solid hemisphere also has a flat circular base.

Volume of sphere

\[V = \frac{4}{3}\pi r^3\]

Surface area of sphere

\[S = 4\pi r^2\]

Curved surface of hemisphere

\[S = 2\pi r^2\]

Total surface of solid hemisphere

\[S = 3\pi r^2\]

Common mistake: Leaving out or wrongly including the flat circular base of a hemisphere. First check whether the base is also painted or covered.

Volume of a cylinder

A cylinder has two equal circular ends of radius r and height h.
Its volume is the base area times the height. If r stays the same, volume changes in the same ratio as h.

Volume of cylinder

\[V = \pi r^2 h\]

Volume proportional to height

\[\frac{V_1}{V_2} = \frac{h_1}{h_2}\]

Common mistake: Using the diameter in place of the radius, or forgetting to square r in the formula.

Scaling lengths changes area, volume

If every length is multiplied by k, the surface area becomes k² times as large.
The volume becomes k³ times as large. This rule works for cubes, spheres and other solids.

Area scale factor

\[S_{\text{new}} = k^2 S\]

Volume scale factor

\[V_{\text{new}} = k^3 V\]

Equal volumes, find r

\[\frac{4}{3}\pi r^3 = a^3 \Rightarrow r = \sqrt[3]{\frac{3a^3}{4\pi}}\]

Common mistake: Thinking volume grows by k times when the side grows by k times. Volume actually grows by k³ times.

Exam tips

Write the formula first, then substitute the values. Always give units in the final answer.
Use π ≈ 22/7 when the radius is a multiple of 7. Otherwise use π ≈ 3.14.
For scaling questions, find k, then use k² for area and k³ for volume.

Beta: made with AI and checked by a second AI. A teacher has not reviewed it yet.

Sample questions

What is volume in mathematics?
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Answer: A. A measure of space inside a 3D object
Volume is the amount of space inside a three-dimensional object, like a cube or a cylinder.
What is surface area in mathematics?
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Answer: A. The total area of the surface of a 3D object
Surface area is the total area that the surface of an object occupies.
Find the surface area of a cube with side length of 5 cm.
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Answer: A. 150 cm²
The surface area of a cube is 6 times the area of one face. So, 6 × (5 cm × 5 cm) = 150 cm².
A cube with a side length of 6 cm and a sphere have equal volumes. Find the sphere's approximate radius.
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Answer: A. 3.6 cm
Volume of cube = 6³ = 216 cm³. Use sphere volume formula 4/3 πr³ = 216 to find r ≈ 3.6 cm.
By what factor does the surface area change if a sphere's radius triples?
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Answer: A. 9 times
Surface area is proportional to r². Tripling radius increases surface area by 3² = 9 times.