How to Find the Area of a Triangle: Base × Height and Heron's Formula

Half base times height

The classic formula is A = ½ × base × height. Any side can serve as the base; the height is the perpendicular distance from that base to the opposite corner. The one-half factor is there because a triangle is exactly half of the rectangle you could draw around its base and height. For a triangle with a base of 10 in and a height of 6 in, the area is ½ × 10 × 6 = 30 square inches.

The catch is that you need the perpendicular height, not the length of a slanted side. In a right triangle the two legs are already perpendicular, so one is the base and the other is the height. In other triangles you may have to drop a perpendicular from a corner to find the height first.

Heron's formula from three sides

When you have measured all three sides but no height, Heron's formula is the tool. First find the semi-perimeter, s = (a + b + c) ÷ 2 — half the way around the triangle. Then the area is the square root of s times the three quantities (s − a), (s − b), and (s − c) multiplied together. It works for any triangle, no right angle required.

A worked example: sides of 3, 4, and 5 units. The semi-perimeter is (3 + 4 + 5) ÷ 2 = 6. Then A = √(6 × (6 − 3) × (6 − 4) × (6 − 5)) = √(6 × 3 × 2 × 1) = √36 = 6 square units. That matches the ½ × base × height answer, since 3-4-5 is a right triangle with legs 3 and 4: ½ × 3 × 4 = 6.

Choosing the right method

Both formulas give the exact same area for the same triangle, so the choice is purely about what you can measure. If you can drop a clean perpendicular and read off a height, half base times height is quicker. If you can only stretch a tape along the three edges — common for an irregular plot of land or a fabric offcut — Heron's formula turns those three numbers straight into an area.

Frequently asked questions

Which side is the base?

Any of the three sides can be the base. Just make sure the height you pair with it is the perpendicular distance from that same side to the opposite vertex.

What if Heron's formula gives a negative under the root?

That means the three lengths cannot form a triangle. If one side is longer than the other two combined, the shape does not close, and the formula signals it with a negative value.

Does Heron's formula work for right triangles?

Yes, for every triangle. For a right triangle it gives the same answer as ½ × leg × leg, which is a handy way to check your work.

How do I find the height if I only have the sides?

Compute the area with Heron's formula first, then rearrange A = ½ × base × height into height = 2A ÷ base. This gives the perpendicular height for whichever side you chose as the base.