Surface Area of a Triangular Prism –

Surface area of a triangular prism is a frequently tested GCSE Maths topic at both Foundation and Higher tiers. A triangular prism has five faces — two identical triangular ends and three rectangular faces. You need to calculate the area of each face and add them together. This guide explains the method clearly, works through examples at both levels, and gives you practice questions.

What Is the Surface Area of a Triangular Prism?

A triangular prism has a uniform triangular cross-section. Its five faces are:

• 2 identical triangular faces (the ends).
• 3 rectangular faces (the sides).

The total surface area is the sum of all five face areas.

Key Formulas

Each rectangular face has one dimension equal to the length (or depth) of the prism and the other equal to one side of the triangular cross-section.

Step-by-Step Method

1. Sketch or identify the triangular cross-section. Note its base and perpendicular height.
2. Calculate the area of one triangular face: ½ × base × height.
3. Multiply by 2 for both triangular ends.
4. Identify the three sides of the triangle (these are the widths of the rectangular faces).
5. Multiply each triangle side by the length of the prism to get each rectangle's area.
6. Add all five face areas together.

Worked Example 1 — Foundation Level

Question: A triangular prism has a cross-section that is a right-angled triangle with base 6 cm and height 8 cm. The hypotenuse is 10 cm and the prism length is 15 cm. Find the total surface area.

Working:

Step 1 — Area of one triangular face = ½ × 6 × 8 = 24 cm².

Step 2 — Two triangular faces = 2 × 24 = 48 cm².

Step 3 — Three rectangular faces:

• Bottom: 6 × 15 = 90 cm².
• Back: 8 × 15 = 120 cm².
• Sloped face (hypotenuse): 10 × 15 = 150 cm².

Step 4 — Total SA = 48 + 90 + 120 + 150 = 408 cm².

Answer: The total surface area is 408 cm².

Worked Example 2 — Higher Level

Question: A triangular prism has an equilateral triangular cross-section with side 6 cm. The prism is 12 cm long. Find the total surface area. Give your answer to 1 decimal place.

Working:

Step 1 — For an equilateral triangle with side 6 cm, the height = (√3/2) × 6 = 3√3 cm.

Step 2 — Area of one triangular face = ½ × 6 × 3√3 = 9√3 ≈ 15.588 cm².

Step 3 — Two triangular faces = 2 × 9√3 = 18√3 ≈ 31.177 cm².

Step 4 — All three sides of the equilateral triangle are 6 cm, so the three rectangles are identical: 3 × (6 × 12) = 3 × 72 = 216 cm².

Step 5 — Total SA = 31.177 + 216 = 247.2 cm² (1 d.p.).

Answer: The total surface area is 247.2 cm².

Worked Example 3 — Exam Style

Question: A Toblerone-shaped box (triangular prism) has total surface area 840 cm². The triangular cross-section is isosceles with base 8 cm, equal sides 5 cm each, and height 3 cm. Find the length of the box.

Working:

Step 1 — Area of one triangular face = ½ × 8 × 3 = 12 cm².

Step 2 — Two triangular faces = 24 cm².

Step 3 — The three sides of the triangle are 8 cm, 5 cm, and 5 cm. Let the prism length = L.

Step 4 — Three rectangular faces: 8L + 5L + 5L = 18L.

Step 5 — Total SA: 24 + 18L = 840.

Step 6 — 18L = 816. L = 816 ÷ 18 = 45.3 cm (1 d.p.).

Answer: The length of the box is 45.3 cm.

Common Mistakes

• Forgetting one of the rectangular faces. A triangular prism has three rectangular faces, not two. Make sure you include all three sides of the triangle as widths.
• Using the slant height of the triangle instead of the perpendicular height. The area formula ½ × base × height requires the perpendicular height of the triangle, not the length of a slant side.
• Confusing the prism length with the triangle base. The prism length (depth) is the dimension running between the two triangular faces. The triangle base is part of the cross-section.

Exam Tips

• Sketch the net of the prism — it shows all five faces clearly, making it easier to calculate each area.
• If the cross-section is a right-angled triangle, use Pythagoras' theorem to find the hypotenuse (the third side) if it is not given.
• For isosceles or equilateral triangles, you may need to calculate the perpendicular height using Pythagoras.
• Label each face with its dimensions before calculating — this avoids mixing up measurements.

Practice Questions

Q1 (Foundation): A triangular prism has a right-angled triangle cross-section with legs 3 cm and 4 cm. The prism is 10 cm long. Find the total surface area.

Q2 (Foundation): A triangular prism has an isosceles triangle cross-section with base 10 cm, equal sides 13 cm, and height 12 cm. The prism is 20 cm long. Find the total surface area.

Q3 (Higher): The total surface area of a triangular prism is 336 cm². The cross-section is a right-angled triangle with legs 5 cm and 12 cm. Find the length of the prism.

Practise triangular prism surface area questions with instant feedback free on GCSEMathsAI.

Related Topics

• Surface Area — surface area of other 3D shapes.
• Surface Area of a Cuboid — surface area with rectangular faces only.
• Nets of 3D Shapes — visualising faces of prisms.

Summary

The surface area of a triangular prism is found by adding the areas of two triangular ends and three rectangular side faces. Always use the perpendicular height for the triangle area formula, find all three sides of the triangle for the rectangular faces, and multiply each by the prism length. Sketching the net is a reliable way to visualise all five faces. Common errors include missing a rectangular face and confusing slant height with perpendicular height. For reverse problems, set up an equation with the unknown length and solve.