Class 9 Maths Chapter 11: Surface Areas and Volumes

Surface Areas and Volumes is Chapter 11 of NCERT Class 9 Maths. It teaches you how to find the curved surface area, total surface area and volume of a right circular cone, a sphere and a hemisphere, and how to apply these formulas to everyday objects. Watch the five video lessons below, then use the formulas and solved examples to practise.

What you will learn

The parts of a right circular cone and the link between radius, height and slant height
Curved and total surface area of a cone
Surface area of a sphere and a hemisphere
Volume of a cone, a sphere and a hemisphere
Converting units such as cm³ to litres in word problems

Video Lessons

Surface Areas and Volumes - Part 1

Surface Areas and Volumes - Part 2

Surface Areas and Volumes - Part 3

Surface Areas and Volumes - Part 4

Surface Areas and Volumes - Revision

Key Concepts and Formulas

Parts of a Right Circular Cone

A right circular cone is formed when a right-angled triangle is rotated about one of the sides containing the right angle. Think of an ice-cream cone or a birthday cap. The radius, height and slant height form a right triangle.

l² = r² + h²
l = √(r² + h²)
where r = base radius, h = vertical height,
l = slant height

Right circular cone with radius 7 cm, height 24 cm and slant height 25 cm, Class 9 Surface Areas — Radius, height and slant height of a cone

Surface Area of a Cone

If you cut a hollow cone along its slant edge and open it out, you get a sector of a circle with radius l and arc length 2πr. Its area gives the curved surface.

Sector area = ½ × 2πr × l = πrl
CSA of cone = πrl
TSA of cone = πrl + πr² = πr(l + r)

Use CSA for open objects like a tent or a cap, and TSA for a solid cone with its base.

Volume of a Cone

If a cone and a cylinder have the same base radius and the same height, the cone holds exactly one-third of the water the cylinder holds.

Volume of cone = ⅓πr²h

The formula uses the vertical height h, never the slant height l.

Surface Area of a Sphere and Hemisphere

A sphere is a perfectly round solid like a ball, and a hemisphere is exactly half of a sphere.

Surface area of sphere = 4πr²
CSA of hemisphere = 2πr²
TSA of solid hemisphere = 2πr² + πr² = 3πr²

For a hollow spherical shell, use the outer radius for the outside surface and the inner radius for the inside surface.

Volume of a Sphere and Hemisphere

Volumes grow much faster than surface areas when the radius increases.

Volume of sphere = (4/3)πr³
Volume of hemisphere = (2/3)πr³
1 litre = 1000 cm³
1 m³ = 1000 litres

Sphere and hemisphere compared with surface area and volume formulas 4πr², 3πr² and (2/3)πr³, Class 9 Surface Areas and Volumes — Sphere and hemisphere formulas compared

If the radius is doubled, the surface area becomes 4 times and the volume becomes 8 times.

Solved Examples

Example 1: A cone has radius 7 cm and height 24 cm. Find its slant height and curved surface area.

l = √(7² + 24²) = √(49 + 576)
l = √625 = 25 cm
CSA = πrl = 22/7 × 7 × 25
∴ l = 25 cm, CSA = 550 cm²

Example 2: Find the total surface area of the cone in Example 1.

TSA = πr(l + r)
= 22/7 × 7 × (25 + 7)
= 22 × 32
∴ TSA = 704 cm²

Example 3: Find the volume of a cone of radius 6 cm and height 14 cm.

V = ⅓πr²h
= ⅓ × 22/7 × 6² × 14
= ⅓ × 22 × 36 × 2
= ⅓ × 1584
∴ V = 528 cm³

Example 4: Find the surface area of a sphere of radius 3.5 cm.

SA = 4πr²
= 4 × 22/7 × 3.5 × 3.5
= 4 × 38.5
∴ SA = 154 cm²

Example 5: Find the volume of a sphere of radius 21 cm.

V = (4/3)πr³
= 4/3 × 22/7 × 21 × 21 × 21
= 4 × 22 × 441
∴ V = 38808 cm³

Example 6: A hemispherical bowl has an inner radius of 10.5 cm. How many litres of soup can it hold?

V = (2/3)πr³
= 2/3 × 22/7 × 10.5³
= 2/3 × 22/7 × 1157.625
= 2425.5 cm³
= 2425.5 ÷ 1000 litres
∴ Capacity = 2.4255 litres

Example 7: A conical tent has base radius 6 m and slant height 10 m. Find the canvas needed and the height of the tent (take π = 3.14).

Canvas = πrl = 3.14 × 6 × 10
= 188.4 m²
h = √(l² − r²) = √(10² − 6²)
= √64 = 8 m
∴ Canvas = 188.4 m², height = 8 m

Common Mistakes to Avoid

Using h in place of l in the curved surface area formula. CSA and TSA need the slant height, while volume needs the vertical height.
Forgetting the ⅓ in the volume of a cone. Without it, you are finding the volume of a cylinder.
Taking the total surface area of a solid hemisphere as 2πr². The flat circular top adds πr², making it 3πr².
Not checking whether the question gives the diameter or the radius, and forgetting to convert cm³ to litres when the question asks for capacity.

Quick Revision

l = √(r² + h²)
Cone: CSA = πrl, TSA = πr(l + r)
Cone: V = ⅓πr²h
Sphere: SA = 4πr², V = (4/3)πr³
Hemisphere: CSA = 2πr², TSA = 3πr²
Hemisphere: V = (2/3)πr³
1 litre = 1000 cm³, 1 m³ = 1000 litres