Geometry

Volume & Surface Area

Volume and surface area of cuboids, prisms, cylinders, spheres and cones — plus working backwards

Explanation & Worked Examples

Jump to section:

Cuboids & Prisms

Volume is the space inside a solid, measured in cubic units (cm³, m³). Surface area is the total area of all its faces, measured in square units (cm², m²).

Cuboid: Volume = length × width × height

Surface area = 2(lw + lh + wh)

Cube (side s): Volume = s³ | Surface area = 6s²

Any Prism

A prism has the same cross-section all the way through. Its volume is always:

Volume of a prism = area of cross-section × length

Worked Example — Triangular Prism

A prism has a triangular cross-section of area 12 cm² and length 10 cm.

1Volume = cross-section area × length

2= 12 × 10 = 120 cm³

Units matter: volume is always cubed (cm³) and surface area squared (cm²). If your answer's units don't match what's asked, you've used the wrong formula.

Cylinders, Spheres & Cones

Keep answers in terms of π unless told to round.

Solid	Volume	Surface area
Cylinder (radius r, height h)	πr²h	2πr² + 2πrh
Sphere (radius r)	⁴⁄₃πr³	4πr²
Cone (radius r, height h, slant l)	⅓πr²h	πr² + πrl
Pyramid	⅓ × base area × height	sum of faces

The curved surface area alone is 2πrh for a cylinder and πrl for a cone (l is the slant height, not the vertical height).

Worked Example — Cylinder Volume

A cylinder has radius 2 cm and height 5 cm.

1Volume = πr²h = π × 2² × 5

2= π × 4 × 5 = 20π cm³

Cone vs cylinder: a cone is exactly one third of the cylinder with the same base and height — that's where the ⅓ comes from. The same ⅓ appears in the pyramid formula.

Easy Examples

Volume and surface area of cuboids and cubes.

Example 1 — Volume of a Cuboid

A cuboid measures 2 cm × 3 cm × 4 cm.

1Volume = l × w × h = 2 × 3 × 4

2= 24 cm³

Example 2 — Volume of a Cube

A cube has side 5 cm.

1Volume = s³ = 5 × 5 × 5

2= 125 cm³

Example 3 — Surface Area of a Cube

Find the surface area of a cube with side 3 cm.

1A cube has 6 equal square faces: SA = 6s²

2= 6 × 3² = 6 × 9 = 54 cm²

Medium Examples

Cylinders, prisms, and the surface area of a cuboid.

Example 1 — Cylinder Volume

A cylinder has radius 3 cm and height 10 cm. Give the volume in terms of π.

1Volume = πr²h = π × 3² × 10

2= π × 9 × 10 = 90π cm³

Example 2 — Surface Area of a Cuboid

Find the surface area of a cuboid 5 cm × 4 cm × 3 cm.

1SA = 2(lw + lh + wh) = 2(5×4 + 5×3 + 4×3)

2= 2(20 + 15 + 12) = 2 × 47 = 94 cm²

Example 3 — Find a Missing Length

A cuboid has volume 60 cm³ and a base measuring 4 cm × 3 cm. Find its height.

1Volume = base area × height, base area = 4 × 3 = 12

2Height = 60 ÷ 12 = 5 cm

Working backwards: if you're given the volume and asked for a length, divide the volume by the product of the known dimensions.

Complex Examples

Spheres, cones, and working backwards from a volume.

Example 1 — Volume of a Sphere

Find the volume of a sphere with radius 3 cm in terms of π.

1Volume = ⁴⁄₃πr³ = ⁴⁄₃ × π × 3³

2= ⁴⁄₃ × π × 27 = 36π cm³

Example 2 — Volume of a Cone

A cone has radius 3 cm and height 4 cm. Find its volume.

1Volume = ⅓πr²h = ⅓ × π × 3² × 4

2= ⅓ × π × 9 × 4 = ⅓ × 36π = 12π cm³

Example 3 — Working Backwards (sphere)

A sphere has volume 36π cm³. Find its radius.

1⁴⁄₃πr³ = 36π → divide both sides by π: ⁴⁄₃r³ = 36

2r³ = 36 × ¾ = 27 → r = ∛27 = 3 cm

Doubling a radius: volume depends on r³, so doubling the radius makes the volume 2³ = 8 times bigger — not twice as big. Surface area depends on r², so it grows by a factor of 4.

All Topics

Practice Questions

Ready to test yourself? Pick a level — 30 questions each.

Easy Medium Complex