How to Find the Area of a Triangle When You Only Know the Three Sides

Most area formulas want a height, and a height is the one thing you often cannot measure. A surveyor with three chain lengths, a carpenter with three offcuts, a student with a diagram and no perpendicular drawn on it: all of them know the sides and nothing else. Heron formula exists for exactly that situation, and it has been doing this job for about two thousand years.

What s means and why you halve the perimeter

The letter s stands for the semiperimeter, which is just half the distance around the triangle. Add all three sides, divide by two, and that is s. It is a genuinely odd looking quantity the first time you meet it, because half a perimeter is not a length you can point at anywhere on the shape.

Here is what makes it worth the trouble. The three quantities s minus a, s minus b and s minus c are each guaranteed to be positive for any real triangle, and each one measures how far that side falls short of the other two combined. If any of them comes out negative or zero, the three lengths you were given cannot close into a triangle at all, and the formula tells you so by refusing to produce a real square root. That is a useful side effect rather than an error.

A worked example

Take a triangle with sides of 13, 14 and 15 units. Add them and you get 42, so s is 21. Now the three subtractions: 21 minus 13 is 8, 21 minus 14 is 7, and 21 minus 15 is 6.

Multiply all four numbers together. 21 times 8 is 168, times 7 is 1176, times 6 is 7056. The square root of 7056 is exactly 84, so the area is 84 square units. This particular triangle is a favourite in textbooks because every number stays whole the whole way through, which almost never happens with real measurements.

Worth checking by another route. This triangle turns out to have a height of exactly 12 when you stand it on its base of 14, and the ordinary area formula gives ½ times 14 times 12, which is 84. Two completely different methods landing on the same number is the strongest confirmation you can get.

Checking the three lengths can actually form a triangle

Any two sides of a triangle have to add up to more than the third. Give someone lengths of 3, 4 and 20 and there is no triangle to be had, because a side of 3 and a side of 4 cannot span a gap of 20 no matter how you angle them.

Heron formula catches this on its own. With 3, 4 and 20 the semiperimeter is 13.5, and 13.5 minus 20 is negative 6.5. One negative factor makes the whole product negative, and a negative number has no real square root. If your calculator returns an error or a complex number here, it is not broken. It is telling you the measurements are wrong.

When to use this and when not to

Use Heron when three sides are all you have. If you know a base and a perpendicular height, half base times height is faster and involves no square root. If you know two sides and the angle sitting between them, half ab sin C is faster again. Heron is the method for the case where nobody measured an angle and nobody could get at a height.

One practical warning about precision. The formula multiplies four numbers that can each be small, and small errors in the side lengths compound quickly. On a long thin triangle, where one side is nearly as long as the other two combined, the differences become tiny and the result gets very sensitive to rounding. Carry more decimal places than feels necessary on shapes like that, and round only at the end.

Mistakes that catch people out

• Using the perimeter instead of the semiperimeter. Halve it first. Forgetting doubles every factor and inflates the answer badly.
• Rounding s too early. If the perimeter is odd, s carries a half. Keep it as 10.5 rather than 10 or 11.
• Forgetting the square root. The product is not the area. 7056 is not an area, 84 is.
• Mixing units. All three sides have to be in the same unit before you start, and the answer comes out in that unit squared.
• Treating an error as a failure of the formula. A negative product means the three lengths cannot form a triangle. Check the measurements rather than the arithmetic.

Practice questions

Common questions

What is Heron formula?

Heron formula finds the area of a triangle from its three side lengths alone. Add the sides and halve the total to get s, subtract each side from s, multiply all four numbers, then take the square root.

Do I need to know any angles?

No. That is the whole point of the formula. It works from the three side lengths on their own, with no angle and no height measured.

What does s stand for?

s is the semiperimeter, which is half the distance around the triangle. Add the three sides and divide by two.

What if the answer under the square root is negative?

The three lengths cannot form a triangle. Any two sides must add to more than the third, and a negative product means that rule is broken.

Is Heron formula less accurate than other methods?

It is exact in principle, but on very long thin triangles small measurement errors get amplified. Carry extra decimal places and round only at the end.

Can I use it on a right triangle?

Yes, and it gives the same answer as half the product of the two legs. On a 3, 4, 5 triangle Heron returns 6, and half of 3 times 4 is also 6.

Related lessons and calculators

• Area of a Triangle: Every Formula and When to Use Each for the overview of all four methods and how to choose between them.
• How to Find the Height of a Triangle From the Area and Base for going the other way once you have the area.
• Triangle Solver to get the angles as well as the area from the same three sides.
• Area of 2D Shapes Calculator which runs Heron for you alongside the other plane shapes.
• Distance and Pythagoras Calculator for turning coordinates into the side lengths this formula needs.
• Area Converter if the answer needs to end up in acres or hectares.
• National Council of Teachers of Mathematics for curriculum standards and how this topic is sequenced in school.
• Area formulas for every common shape, a calculator covering fifteen shapes and an index of every geometry lesson here.