How to find the area of a triangle: the formula and why it works, with a right triangle, with the height outside the triangle, with 3 sides (Heron's formula), with 2 sides and an angle, on a coordinate plane, and the mistakes that lose the marks

"How to find the area of a triangle" is one search with five different students behind it: a Grade 6 child meeting ½ × base × height for the first time, a Grade 8 child whose triangle is on a grid, a Geometry student given three sides and no height, a Grade 10 or 11 student given two sides and an angle, and a CBSE Class 9 student who has just been told about a man called Heron. This page does all five, worked in full with numbers, says where each one sits in the curriculum, and lists the mistakes that lose most of the marks, which are the same in every country.

The formula, and why it is a half

Area = ½ × base × height.

Draw any triangle inside a rectangle with the same base and the same height. The triangle is exactly half the rectangle: cut along the height and the two off-cuts fit into the two spare corners. A rectangle's area is base × height, so the triangle's is half of that. A child who has cut one out of paper once never forgets the ½; a child who only memorised the formula drops it under exam pressure.

Example 1. Base 10 cm, height 6 cm. Area = ½ × 10 × 6 = 30 cm².

The height is the perpendicular distance from the base to the opposite corner, drawn at a right angle to the base. It is not the slanted side, unless the triangle has a right angle there.

Right triangles: the two legs are the base and height

In a right triangle, the two sides that meet at the right angle are already perpendicular, so one is the base and the other is the height. The longest side (the hypotenuse) is never used in the area.

Example 2. Legs 5 and 12 (hypotenuse 13). Area = ½ × 5 × 12 = 30. The 13 plays no part.

When the height falls outside the triangle

In an obtuse triangle (one angle more than 90°), the perpendicular from the top corner may land outside the base. Extend the base with a dotted line and measure the height to that line; the formula is unchanged.

Example 3. Base 8, and the perpendicular height to the extended base is 5. Area = ½ × 8 × 5 = 20. Students who "measure to the corner along the side" get a larger, wrong number here.

Three sides and no height: Heron's formula

Given the three sides a, b, c and nothing else:

1. Semi-perimeter: s = (a + b + c) ÷ 2
2. Area = √[s(s − a)(s − b)(s − c)]

Example 4. Sides 5, 6 and 7. s = (5 + 6 + 7) ÷ 2 = 9. Area = √(9 × (9 − 5) × (9 − 6) × (9 − 7)) = √(9 × 4 × 3 × 2) = √216 ≈ 14.70 square units.

Check with a known triangle. Sides 3, 4, 5 (a right triangle, so the area should be ½ × 3 × 4 = 6). s = 6, area = √(6 × 3 × 2 × 1) = √36 = 6. Correct. Doing this check once convinces students that the formula is not magic.

Heron's formula also gives the height if you need it afterwards: height = 2 × area ÷ base. For the 5-6-7 triangle with base 7, height = 2 × 14.7 ÷ 7 = 4.2.

Two sides and the angle between them: ½ab sin C

Area = ½ × a × b × sin C, where C is the angle between sides a and b.

Example 5. Sides 8 and 10 with a 30° angle between them. Area = ½ × 8 × 10 × sin 30° = ½ × 80 × 0.5 = 20 square units.

Why it works: b × sin C is the height of the triangle when a is the base, so this is ½ × base × height in disguise. The angle must be the included one (between the two sides you know). If you are given two sides and an angle that is not between them, use the sine rule to find another angle first, and be aware there may be two possible triangles.

Three points on a coordinate plane

If a side is horizontal or vertical, count. For A(1, 1), B(5, 1), C(3, 6): AB is horizontal with length 5 − 1 = 4; the height is the vertical distance from C to the line y = 1, which is 6 − 1 = 5. Area = ½ × 4 × 5 = 10.

If no side is horizontal or vertical, use the coordinates directly (the "shoelace" formula):

Area = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

Same points: ½ |1(1 − 6) + 5(6 − 1) + 3(1 − 1)| = ½ |−5 + 25 + 0| = ½ × 20 = 10. It agrees. The vertical bars mean "take the positive value"; if your working gives a negative, drop the sign. If it gives zero, the three points are on one straight line and there is no triangle.

Equilateral triangles

All sides equal to s: Area = (√3 ÷ 4) × s² ≈ 0.433 × s². For s = 6: (√3 ÷ 4) × 36 = 9√3 ≈ 15.59. It comes from ½ × base × height with the height found by Pythagoras (height = s × √3 ÷ 2).

Which method, when

The five mistakes that lose most marks

1. Using the slanted side as the height. The height must make a right angle with the base. In a right triangle only, a side can be the height.
2. Forgetting the half. Base × height is the rectangle. Write the ½ before anything else.
3. Wrong units. Area is in square units: cm², m², square units on a grid. "30 cm" loses the mark even with 30 correct.
4. Heron's formula with the wrong s. s is half the perimeter, not the perimeter. If any bracket (s − a) comes out negative, either s is wrong or the three lengths cannot make a triangle.
5. Using ½ab sin C with the wrong angle. The angle must be the one between the two sides you multiply. Label the triangle before writing anything.

A short practice set (answers below)

1. Base 12 cm, height 7 cm.
2. Right triangle, legs 9 and 40.
3. Sides 7, 8, 9 (Heron).
4. Sides 6 and 9 with a 60° angle between them (sin 60° ≈ 0.866).
5. Points (0, 0), (6, 0), (2, 5).
6. Equilateral triangle, side 10.

Answers: 1) 42 cm² 2) 180 3) s = 12, √(12 × 5 × 4 × 3) = √720 ≈ 26.8 4) ½ × 54 × 0.866 ≈ 23.4 5) ½ × 6 × 5 = 15 6) 25√3 ≈ 43.3

Where to get help

A 1-on-1 online maths class with a tutor who holds a master's in mathematics and teaches Grade 6 to 12 geometry and trigonometry, GCSE and A-level, IB and CBSE Class 7 to 11: the area of a triangle taught from the half-rectangle up, Heron's formula checked on a 3-4-5 triangle, ½ab sin C connected to the height rather than memorised. The geometry page lists the topics by grade; the free 30-minute first class is a real lesson on this week's homework. The quadratics lesson is the next algebra topic most of these students meet.

Questions parents ask

See how we teach this, 1-on-1 online →