Area of a Triangle –

The area of a triangle is one of the most fundamental skills in GCSE Maths. At Foundation level you need the classic half-base-times-height formula, while Higher tier students must also learn A = ½ab sin C for triangles where the perpendicular height is not given.

What Is the Area of a Triangle?

The area of a triangle measures the space enclosed within its three sides. Every triangle — whether right-angled, scalene, isosceles, or equilateral — can have its area calculated using the same base-and-height approach.

The formula A = ½ × base × height works because any triangle is exactly half of a rectangle (or parallelogram) with the same base and height. At Higher tier, when you know two sides and the included angle but not the perpendicular height, you use the sine formula instead.

The key to applying these formulas correctly is identifying the correct pair: the base and the perpendicular height that is at right angles to it.

Key Formulas

Step-by-Step Method

1. Identify the base — any side of the triangle can be chosen as the base.
2. Find the perpendicular height — the vertical distance from the base to the opposite vertex, at 90° to the base.
3. Substitute into A = ½ × base × height and calculate. If the perpendicular height is not available but two sides and the included angle are given, use A = ½ab sin C instead.

Worked Example 1 — Foundation Level

Question: A triangle has a base of 10 cm and a perpendicular height of 6 cm. Find its area.

Working: A = ½ × base × height A = ½ × 10 × 6 A = ½ × 60 A = 30

Answer: 30 cm²

Worked Example 2 — Higher Level

Question: Triangle PQR has PQ = 9 cm, PR = 12 cm, and the included angle P = 40°. Find the area correct to 1 decimal place.

Working: A = ½ab sin C A = ½ × 9 × 12 × sin 40° A = 54 × 0.6428… A = 34.7 (1 d.p.)

Answer: 34.7 cm²

Worked Example 3 — Exam Style

Question: The area of a triangle is 48 cm². The base is 16 cm. Work out the perpendicular height.

Working: A = ½ × base × height 48 = ½ × 16 × h 48 = 8h h = 48 ÷ 8 h = 6

Answer: 6 cm

Common Mistakes

• Using the slant side instead of the perpendicular height. The height must be at right angles to the base, not along a side of the triangle. Look for the right-angle marker on the diagram.
• Forgetting to halve. The ½ is essential — without it you calculate the area of a full rectangle, which is double the triangle.
• Using the wrong angle in ½ab sin C. The angle must be the one enclosed between the two sides you are using, not any angle in the triangle.
• Calculator in wrong mode. When using sin, ensure your calculator is set to degrees, not radians.
• Not including square units. Area is always in cm², m², etc. — never cm or m.

Exam Tips

• Always write the formula first — this secures a method mark even if your arithmetic slips.
• If a triangle is part of a compound shape, calculate its area separately then add or subtract.
• For ½ab sin C questions, clearly state the two sides and the included angle before substituting.

Practice Questions

Q1 (Foundation): A triangle has a base of 14 cm and a perpendicular height of 9 cm. Find its area.

Q2 (Foundation): The area of a triangle is 36 cm². The perpendicular height is 8 cm. Find the base.

Q3 (Higher): Triangle ABC has AB = 11 cm, AC = 7 cm, and angle A = 54°. Find the area to 1 decimal place.

Practise area of a triangle questions with instant feedback — completely free on GCSEMathsAI.

Related Topics

• Area of 2D Shapes
• Trigonometry – SOHCAHTOA
• Trigonometry – Sine and Cosine Rules

Summary

• The area of any triangle is A = ½ × base × perpendicular height.
• The base can be any side, but the height must be perpendicular to it.
• At Higher tier, use A = ½ab sin C when you know two sides and the included angle.
• Always check that you have halved your answer — a triangle is half the corresponding rectangle.
• Write the formula explicitly in your working to earn method marks in exams.
• Triangles frequently appear within compound shapes — calculate each triangle separately and combine.