Area of a Triangle: Every Formula and When to Use Each

Geometry Lessons

The four formulas, in one place

A = ½ b hbase and perpendicular height
A = √( s(s-a)(s-b)(s-c) )three sides, Heron’s formula, s is half the perimeter
A = ½ a b sin Ctwo sides and the angle between them
A = ½ | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) |three corner coordinates

There is no single formula for the area of a triangle. There are four in common use, and picking between them is not a matter of taste. It comes down to one question: what were you actually given? Every one of these returns the same number for the same triangle, so the only thing that separates them is which measurements you already have in front of you.

Which one you need, based on what you know

1

A base and its perpendicular height

Use A = ½ b h. The default, and the one to reach for whenever you have a flat side and a measurement square to it.

2

All three side lengths, no angles

Use Heron’s formula. Common with land parcels and any shape you measured with a tape rather than drew.

3

Two sides and the angle between them

Use A = ½ a b sin C. The angle has to sit between the two sides. Any other angle and this does not work.

4

Coordinates of the three corners

Use the coordinate method, also called the shoelace formula. Standard for survey points and anything drawn on a grid.

If you have the area already and want to work backwards to a missing measurement, that is a rearrangement rather than a new formula. The most common one is finding a height, covered in full further down.

Where the half comes from

Draw a rectangle around any triangle so the two share a base and a height. The triangle fills exactly half of it. That holds no matter how slanted the triangle is, because the leftover corner pieces always fold in to fill the other half precisely. The half is not a correction factor someone bolted on, it is the fixed ratio between a triangle and the rectangle it sits inside, and it is why every formula on this page carries one.

The same triangle, worked four ways

Take a triangle with sides of 5, 6 and 7 units. Below it is solved by all four routes. The four calculations were run independently against each other and they agree to four decimal places at 14.6969, which is a useful thing to know if you ever get two different answers and need to work out which one to trust.

Four panels showing the same triangle with sides of 5, 6 and 7 units, solved by base times height, Heron formula, half a b sin C, and the coordinate method. Every panel returns an area of 14.70.
The height of 4.20 in the first panel is not a measurement anyone handed over. It had to be derived from the area, which is exactly the rearrangement most people come here looking for.

Base times height, worked through

A triangular bed has a base of 7 meters along a fence and stands 4.2 meters back from it, measured square to the fence. The area is ½ times 7 times 4.2, which is 14.7 square meters. The only thing to be careful about is that 4.2 must be the perpendicular distance, not the length of a slanted side.

Heron’s formula, worked through

With sides of 5, 6 and 7 and no angles at all, first find s, the semiperimeter, which is half the perimeter. Here 5 plus 6 plus 7 is 18, so s is 9. Then subtract each side from s in turn: 9 minus 5 is 4, 9 minus 6 is 3, 9 minus 7 is 2. Multiply all four numbers together, 9 times 4 times 3 times 2 is 216, and take the square root, which is 14.697.

One check worth doing before you start. If the two shorter sides do not add up to more than the longest side, no such triangle exists and Heron’s formula will hand you the square root of a negative number. That is the formula telling you the measurements are wrong, not a mistake in your arithmetic.

Two sides and the included angle, worked through

With sides of 7 and 6 meeting at an angle of 44.42 degrees, the area is ½ times 7 times 6 times sin 44.42, which is 21 times 0.6999, giving 14.697. Make sure your calculator is in degrees rather than radians, because that single setting is responsible for more wrong answers on this formula than anything else.

The angle has to sit between the sides

This formula only works when the angle you use is the one enclosed by the two sides you picked. Reach for a different angle and the sine value is simply wrong for the shape, and nothing in the arithmetic will warn you.

½ x 7 x 6 x sin 44.42°14.70

44.42 degrees is the angle where the sides of 7 and 6 actually meet

½ x 7 x 6 x sin 78.46°20.57

78.46 degrees is a real angle in this triangle, just not the one between those two sides

From coordinates, worked through

With corners at (0, 0), (7, 0) and (4.29, 4.20), multiply each x by the difference of the other two y values, add the three results, take the size of the total and halve it. That gives ½ times the absolute value of 0 times (0 minus 4.20), plus 7 times (4.20 minus 0), plus 4.29 times (0 minus 0), which is ½ times 29.4, or 14.7.

The absolute value matters. Go round the corners clockwise instead of counterclockwise and the total comes out negative. Area is never negative, so take the size and ignore the sign. This method extends to any polygon with any number of corners, which is why it is the one surveyors reach for.

Go deeper on this topic

Working backwards from the area

Once you have the area, the same formulas rearrange to give you a measurement you are missing.

Each method in full

The four routes above each get their own walkthrough, with the mistakes that catch people out and practice questions.

Heights of particular triangles

When the triangle has symmetry, the height falls out of the side lengths alone with no area needed.

Right triangles

A right angle gives you shortcuts the other triangles do not have, and one height that catches people out.

Checking and solving from the sides alone

Before any area, it is worth knowing whether three lengths describe a real triangle, and what angles they force.

Comparing two triangles

Same shape at any size, or identical in both. The tests differ, and two that look convincing do not work.

Centres and circles

Three different centres, each defined by a different property, plus the two circles that go with them.

Out in the field

Where these formulas earn their keep on real measurements rather than textbook numbers.

Mistakes that catch people out

  • Using a slanted side as the height. The height is perpendicular to the base. A slanted side is always longer, so the area comes out too big.
  • Picking an angle that is not between the two sides. Half a b sin C needs the enclosed angle. Any other angle gives a wrong answer with no warning.
  • Leaving the calculator in radians. Sin 44.42 in radians is a different number entirely, and the result will still look plausible.
  • Forgetting the square root in Heron’s formula. The product under the root is not the area. Here it is 216, while the area is 14.7.
  • Dropping the absolute value in the coordinate method. Going round the corners clockwise gives a negative total. Take the size of it.
  • Mixing units. A base in centimeters with a height in meters gives an answer that is wrong by a factor of a hundred.

Common questions

What is the formula for the area of a triangle?

The most common one is A = ½bh, half the base times the perpendicular height. Three others exist for when you do not have a height: Heron’s formula for three sides, half a b sin C for two sides and the angle between them, and the coordinate method for three corner points.

Which formula should I use if I only know the three sides?

Heron’s formula. Add the three sides and halve the total to get s, then take the square root of s times each of the three differences between s and a side.

Do I need the height to find the area of a triangle?

No. Three of the four formulas never mention a height. If all you have is three sides, or two sides and an angle, or three coordinates, you can get the area without ever finding a perpendicular distance.

Why is there a half in every area of a triangle formula?

Because a triangle fills exactly half of the rectangle that shares its base and height. That ratio holds for every triangle regardless of shape, so the half appears in all four formulas.

Does the area change if I pick a different side as the base?

No. Each of the three sides has its own matching perpendicular height, and every base and height pair gives the same area. A longer base simply comes with a shorter height.

Can I find the area of a triangle from coordinates?

Yes, using the coordinate or shoelace method. Multiply each x by the difference of the other two y values, add the results, then halve the size of the total. It works for any polygon, not just triangles.

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