How to Find the Area of a Triangle

Which formula you need depends on what you have

There are three standard ways to find a triangle's area, and picking between them is just a matter of looking at what the question gave you.

• Base and perpendicular height → A = ½ × b × h
• All three sides → Heron's formula
• Two sides and the angle between them → A = ½ × a × b × sin(C)

Method 1 — half base times height

This is the one most people learn first, and it is the quickest when it applies.

Any side can be the base. The height is the perpendicular distance from that base to the opposite corner — measured at a right angle, not along a slanted edge.

For a triangle with base 10 and height 6: A = ½ × 10 × 6 = 30.

For an obtuse triangle the height may fall outside the triangle, which looks wrong but is fine. Extend the base line and measure to the corner from there.

Method 2 — Heron's formula

When you have the three side lengths and no height, Heron's formula gets you there without needing to find one.

First find the semi-perimeter, which is half the way around: s = (a + b + c) ÷ 2.

Then: A = √(s(s−a)(s−b)(s−c))

For the 3-4-5 triangle loaded into the calculator: s = (3 + 4 + 5) ÷ 2 = 6. Then A = √(6 × 3 × 2 × 1) = √36 = 6.

This is the method the calculator on this page uses, which is why it asks for three sides rather than a base and height — three sides also let it give you the perimeter, which a base and height cannot.

Method 3 — two sides and the included angle

If you know two sides and the angle between them, then A = ½ × a × b × sin(C).

The angle has to be the one between the two sides you are using. An angle somewhere else in the triangle will not do, and using the wrong one is the usual mistake here.

This is really the same formula as ½ base × height in disguise: b × sin(C) is the height, worked out with trigonometry instead of measured.

Checking whether a triangle can exist

Before calculating from three sides, it is worth a quick check: any two sides must add up to more than the third. Sides of 1, 1 and 9 cannot form a triangle — the two short sides cannot reach across the long one.

The calculator on this page checks this for you and will tell you when your three lengths cannot make a triangle, rather than returning a nonsensical answer.

Inside Heron's formula, an impossible triangle makes one of the bracketed terms negative, so you end up taking the square root of a negative number. That is the arithmetic telling you the shape does not exist.

The 3-4-5 triangle

The example loaded above is worth recognising on sight. Because 3² + 4² = 5², it is a right-angled triangle — which means you could also have used ½ × 3 × 4 = 6, treating the two short sides as base and height.

Both methods give 6, as they must. It is a useful pair of numbers to keep in mind when you want to check that a formula you are unsure about is behaving.

The 3-4-5 is also used in building work to get a true right angle: measure 3 units along one edge and 4 along the other, and when the diagonal between them is exactly 5, the corner is square.