Chapter 11 · NCERT Class 9 Mathematics

Surface Areas and Volumes

How much paper wraps a box, and how much water fills a tank? Surface area measures the outside of a solid; volume measures the space inside. Learn the formulas for the cuboid, cube, cylinder, cone, sphere and hemisphere — and never mix the two up. Tap each solid to see exactly what it means.

📐 3 topics⏱ ~25 min📝 12-question quiz

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Six solids, two questions each

Every solid in this chapter answers two questions: how much surface does it have, and how much space does it hold? Tap each shape to reveal its volume and surface-area formula.

Explore · Solids & their formulastap a solid

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The three big ideas

Surface area is the total area of all the faces (or the curved boundary) of a solid — always in square units.

Cuboid (length l, breadth b, height h): Total surface area = 2(lb + bh + hl); lateral surface area = 2h(l + b).
Cube (side a): Total surface area = 6a² — six equal square faces; lateral surface area = 4a².
Cylinder (radius r, height h): Curved surface area = 2πrh; total surface area = 2πr(r + h).

Volume is the amount of space a solid occupies (its capacity) — always in cubic units.

Cuboid: Volume = l × b × h.
Cube (side a): Volume = a³.
Cylinder (radius r, height h): Volume = πr²h.

Worked example. Find the volume of a cube of side 4 cm.

Volume of a cube = a³ = 4³ = 4 × 4 × 4 = 64 cm³.

Common mistake: mixing up area and volume. Surface area is measured in square units (cm²) and volume in cubic units (cm³) — keep their formulas and units separate. If your answer to a volume question comes out in cm², you have used a surface-area formula by accident.

Cone (radius r, height h, slant height l = √(r² + h²)): Volume = ⅓πr²h; Curved surface area = πrl; total surface area = πr(l + r).
Sphere (radius r): Volume = 4⁄3πr³; Surface area = 4πr².
Hemisphere (radius r): Volume = ⅔πr³; Curved surface area = 2πr²; total surface area = 3πr².

Notice the pattern: a cone is exactly one-third of the cylinder on the same base and height, and a hemisphere is exactly half of a sphere.

Where you'll meet it

Mensuration, all around you

Packaging & box design

To make a carton you need its surface area — that decides how much cardboard or wrapping paper is used and what it costs. To know what the carton can hold, you need its volume — the capacity. Designers balance the two: least material for the most space.

Tanks, drums & cylinders

A water tank, an oil drum or a gas cylinder is sized by volume — how much it can hold (1 m³ = 1000 litres). The curved surface area tells engineers how much sheet metal or paint the curved side will need.