How to Find the Area of a Triangle (4 Methods)

The area of a triangle is one of the first formulas everyone learns, but "½ × base × height" only helps when you actually know the height. In real problems you're often given sides, or two sides and an angle, instead. Here are four methods that cover almost every case — and our triangle calculator can apply them for you.

Method 1: Base and height

The classic formula is:

area = ½ × base × height

The height must be the perpendicular distance from the base to the opposite vertex — not the slanted side. For a triangle with a base of 10 and a perpendicular height of 6:

area = ½ × 10 × 6 = 30

This is the go-to method whenever you can read off a base and its matching height.

Method 2: Heron's formula (three sides)

When you know all three side lengths but no height, use Heron's formula. First find the semi-perimeter s:

s = (a + b + c) ÷ 2

Then:

area = √( s(s − a)(s − b)(s − c) )

For a triangle with sides 3, 4 and 5:

1. s = (3 + 4 + 5) ÷ 2 = 6
2. area = √(6 × (6−3) × (6−4) × (6−5)) = √(6 × 3 × 2 × 1) = √36 = 6

Heron's formula is wonderfully general — it never needs an angle or a height. Our triangle calculator uses exactly this method and shows each step.

Method 3: Two sides and the included angle

If you know two sides and the angle between them, use the trigonometric formula:

area = ½ × a × b × sin(C)

where C is the angle between sides a and b. For a = 8, b = 5 and C = 30°:

area = ½ × 8 × 5 × sin(30°) = ½ × 40 × 0.5 = 10

You can evaluate the sin(30°) part on our scientific calculator — just make sure it's in degree mode.

Method 4: Coordinates (the shoelace formula)

If your triangle is defined by three points (x₁,y₁), (x₂,y₂), (x₃,y₃), the shoelace formula gives the area directly:

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

This is the method computers use, because it works straight from coordinates with no need to find side lengths first.

Check the triangle is valid first

Before any of this, three given lengths must satisfy the triangle inequality: the sum of any two sides has to exceed the third. The lengths 2, 3, 10 fail (2 + 3 < 10), so no triangle exists and no area can be found. The triangle calculator checks this automatically and warns you if the sides can't form a triangle.

Classifying your triangle

Once you have the sides, you can also classify the triangle:

• By sides: equilateral (all equal), isosceles (two equal) or scalene (all different).
• By angles: acute, right (one 90° angle) or obtuse. A quick test: if a² + b² = c² for the longest side c, it's a right triangle.

Pick the method that fits your data

There's no single "best" formula — the right one depends on what you're given:

• Base and matching height → ½ × base × height
• Three sides → Heron's formula
• Two sides and the angle between → ½ ab sin C
• Three coordinates → shoelace formula

Match the method to your information and finding a triangle's area becomes routine.