Surface Area of Prisms and Cuboids: KS3 Maths

The surface area of a prism is the total area of all its faces. The clearest way to work it out is to imagine unfolding the prism into a flat net, calculate the area of each face separately, then add them all together. Always state the answer in square units (cm², m²).

What is a prism?

A prism is a 3-D solid with two identical parallel ends (the cross-sections) joined by flat rectangular faces. The cross-section can be any 2-D shape — rectangle, triangle, hexagon — and it stays the same all the way through the length of the prism. A cuboid is the most familiar prism, with a rectangular cross-section.

Prism type | Cross-section shape | Number of faces
Cuboid | Rectangle | 6
Triangular prism | Triangle | 5
Hexagonal prism | Hexagon | 8

How do you find the surface area of a cuboid?

A cuboid has three pairs of identical faces:

Top and bottom (each = length × width)

Front and back (each = length × height)

Left and right (each = width × height)

Formula: SA = 2(lw + lh + wh)

Worked example: A cuboid is 8 cm long, 5 cm wide and 3 cm tall.

Top/bottom pair: 2 × (8 × 5) = 2 × 40 = 80 cm²

Front/back pair: 2 × (8 × 3) = 2 × 24 = 48 cm²

Left/right pair: 2 × (5 × 3) = 2 × 15 = 30 cm²

Total: 80 + 48 + 30 = **158 cm²**

How do you find the surface area of a triangular prism?

A triangular prism has 5 faces: two triangular ends and three rectangular side faces.

Worked example: A triangular prism has a right-angled triangle as its cross-section, with base 6 cm, height 4 cm and hypotenuse 7.21 cm. The prism is 10 cm long.

Two triangular ends: area of one triangle = ½ × base × height = ½ × 6 × 4 = 12 cm². Both ends = 2 × 12 = 24 cm².

Rectangle on the base (6 cm × 10 cm): 6 × 10 = 60 cm²

Rectangle on the height (4 cm × 10 cm): 4 × 10 = 40 cm²

Rectangle on the hypotenuse (7.21 cm × 10 cm): 7.21 × 10 = 72.1 cm²

Total: 24 + 60 + 40 + 72.1 = **196.1 cm²**

What is the net method and why is it useful?

A net is what you get when you unfold a 3-D shape flat. Drawing or imagining the net:

Helps you see each face clearly — you are much less likely to miss a face.

Lets you label every dimension directly onto the flat shape.

Works for any prism, no matter how unusual the cross-section.

If the question asks you to "draw the net", sketch each face connected along shared edges. The total area of the net equals the surface area.

How do you handle the rectangular side faces of any prism?

For any prism, the side faces are always rectangles. Each rectangle's width equals the length of one edge of the cross-section, and its height equals the length of the prism. So the total area of all side faces = (perimeter of cross-section) × (length of prism). This shortcut saves time once you know the perimeter of the end face.

Total side area = perimeter of cross-section × length

What mistakes do students commonly make?

Counting faces wrong: Triangular prisms have 5 faces, not 4. Always list each face before starting.

Forgetting to double the ends: There are always two identical cross-section faces — a top and a bottom (or a front and a back).

Mixing up area and perimeter: Surface area uses area (cm²); if you find yourself multiplying three lengths together, you are calculating volume instead.

Wrong units: Surface area is always in square units. If lengths are in cm, the answer is in cm².

Frequently asked questions

What is the formula for the surface area of a cuboid?

The formula is SA = 2(lw + lh + wh), where l is length, w is width, and h is height. It accounts for all three pairs of identical faces. Alternatively, find the area of each of the six faces individually and add them together.

How many faces does a triangular prism have?

A triangular prism has 5 faces: 2 triangular ends and 3 rectangular side faces. List them before you calculate to avoid missing any. The three rectangles may all have different dimensions if the triangle is scalene.

Can I use the formula (perimeter × length) for the side faces?

Yes — for any prism, multiply the perimeter of the cross-section by the prism's length to get the total area of all the rectangular side faces. Add the two cross-section areas to complete the surface area.

How is surface area different from volume?

Surface area measures the total 2-D area covering the outside of a solid, in square units (cm²). Volume measures the 3-D space inside, in cubic units (cm³). For a cuboid, volume = l × w × h, which is very different from the surface area formula.

For Socratic 3-D shape practice at KS3, see aitutors.me.