ICSE Maths: Surface Area & Volume of Cylinders, Cones & Spheres

Why bother with cylinders, cones and spheres?

Ever wondered how a soda can, an ice‑cream cone or a basketball is measured? Those everyday objects are perfect examples of the solids you’ll meet in the ICSE Mensuration chapter.

💡 In Simple Words: The surface area tells you how much material you need to cover a shape, while the volume tells you how much space it can hold. For a cylinder, cone or sphere, just plug the radius and height (or just radius for a sphere) into the right formula.

Surface Area of a Cylinder

A cylinder looks like a soup can. It has two circular faces (top and bottom) and a curved side called the lateral surface.

Key terms

• Radius (r): distance from the centre of the circular face to its edge.
• Height (h): distance between the two circular faces.

Formula

Surface Area (SA) = 2πr (r + h). Think of it as two circles (2πr²) plus the side that rolls out into a rectangle (2πrh).

Worked example

Find the total surface area of a cylinder with radius 4 cm and height 10 cm.

1. Calculate the area of one circle: πr² = 3.14 × 4² = 3.14 × 16 = 50.24 cm².
2. Two circles give 2 × 50.24 = 100.48 cm².
3. Side area: 2πrh = 2 × 3.14 × 4 × 10 = 251.2 cm².
4. Add them: 100.48 + 251.2 = 351.68 cm².

So, the can needs about 352 cm² of metal.

Volume of a Cylinder

Volume tells you how much liquid a can hold.

Formula

Volume (V) = πr²h. Imagine filling the base circle with layers of thickness h.

Worked example

Using the same cylinder (r = 4 cm, h = 10 cm):

1. Base area = πr² = 50.24 cm².
2. Multiply by height: 50.24 × 10 = 502.4 cm³.

That’s about half a litre.

Surface Area of a Cone

A cone is like an ice‑cream scoop on a wafer. It has a circular base and a slant side.

Key terms

• Slant height (l): the distance from the tip of the cone down the side to the edge of the base. It’s longer than the vertical height.

Formula

Surface Area = πr (r + l). The first part (πr²) is the base, the second part (πrl) is the curved side.

Worked example

Radius = 3 cm, height = 4 cm. First find slant height using Pythagoras: l = √(r² + h²) = √(9 + 16) = √25 = 5 cm.

1. Base area = πr² = 3.14 × 9 = 28.26 cm².
2. Side area = πrl = 3.14 × 3 × 5 = 47.1 cm².
3. Total SA = 28.26 + 47.1 = 75.36 cm².

Volume of a Cone

Volume is one‑third the volume of a cylinder with the same base and height.

Formula

Volume = (1/3)πr²h.

Worked example

Using r = 3 cm, h = 4 cm:

1. Base area = πr² = 28.26 cm² (as before).
2. Multiply by height: 28.26 × 4 = 113.04 cm³.
3. Take one‑third: 113.04 ÷ 3 ≈ 37.68 cm³.

Surface Area of a Sphere

A sphere is a perfect ball – like a basketball or a marble.

Formula

Surface Area = 4πr². Imagine four circles stitched together to cover the whole ball.

Worked example

Radius = 5 cm.

1. r² = 25.
2. 4πr² = 4 × 3.14 × 25 = 314 cm².

Volume of a Sphere

Volume tells you how much air is inside a ball.

Formula

Volume = (4/3)πr³. The factor 4/3 comes from integrating tiny disks that make up the sphere.

Worked example

Radius = 5 cm.

1. r³ = 125.
2. (4/3)πr³ = (4/3) × 3.14 × 125 ≈ 523.33 cm³.

Quick Comparison Table

How to Solve Any Mensuration Problem

📝 Likely Exam Questions

• Q1. A cylindrical tank has radius 7 cm and height 14 cm. Find its total surface area.
  Answer: SA = 2πr(r + h) = 2×3.14×7(7+14)=2×3.14×7×21≈923 cm².
• Q2. The slant height of a cone is 13 cm and its base radius is 5 cm. Find its curved surface area.
  Answer: CSA = πrl = 3.14×5×13≈204 cm².
• Q3. Find the volume of a sphere whose diameter is 12 cm.
  Answer: r = 6 cm; V = (4/3)πr³ = (4/3)×3.14×216 ≈ 904 cm³.
• Q4. A cone and a cylinder have the same base radius (3 cm) and the same height (9 cm). What is the ratio of their volumes?
  Answer: V_cone = (1/3)πr²h = (1/3)π×9×9 = 27π/3 = 9π; V_cyl = πr²h = π×9×9 = 81π; Ratio = 9π : 81π = 1 : 9.
• Q5. A sphere is inscribed in a cylinder of height 10 cm. If the cylinder’s radius is also 5 cm, find the volume of the sphere.
  Answer: For an inscribed sphere, r = 5 cm; V = (4/3)πr³ = (4/3)×3.14×125 ≈ 523 cm³.