How to use the Triangle Area Calculator (Heron’s Formula)
- Enter the three side lengths in the same unit.
- The calculator checks that they can form a triangle.
- Read the area, perimeter, height above side a and the triangle type.
- If you get an error, one side is too long for the other two.
Formula used
With s = (a + b + c) ÷ 2, Area = √(s × (s − a) × (s − b) × (s − c)). Height above side a = 2 × Area ÷ a.
Worked example
Here is the Triangle Area Calculator (Heron’s Formula) worked through with the example values shown above, so you can check each step by hand.
| Inputs | |
|---|---|
| Side a | 5 |
| Side b | 6 |
| Side c | 7 |
| Result | |
| Area | 14.696938 |
| Perimeter | 18 |
| Type of triangle | Scalene, acute |
| Height above side a | 5.8787754 |
Heron’s formula
First find the semi-perimeter s, half the sum of the sides. Then Area = √(s(s − a)(s − b)(s − c)). For sides 5, 6 and 7, s = 9, so the area is √(9 × 4 × 3 × 2) = √216 = 14.6969.
The height above side a follows from area = ½ × base × height, so height = 2 × area ÷ a. For the example that is 5.8788.
Checking it is a triangle
Three lengths form a triangle only if the two shorter ones add up to more than the longest (the triangle inequality). Sides 2, 3 and 6 fail, because 2 + 3 is less than 6, and the calculator will say so. If they add up to exactly the longest side, the triangle is flat and has no area.
Heron’s formula can lose accuracy for very thin triangles, so results for nearly flat triangles should be read with some care.
Types of triangle
By sides: an equilateral triangle has three equal sides, such as 6-6-6 with area 15.5885. An isosceles triangle has two equal sides. A scalene triangle has none equal.
By angle: compare the squares of the sides. If the two smaller squares add up to the largest, it is right, like 3-4-5 with area 6. If they add up to more, it is acute, and to less, obtuse, like 2-3-4 (area 2.9047).
Other ways to find the area
If you know the base and height, area = ½ × base × height. If you know two sides and the angle between them, area = ½ × a × b × sin(angle). Heron’s formula is the choice when only the three sides are known, for example when measuring a plot of land.
For a right triangle you can also use the Pythagorean theorem calculator, which gives the area from the two legs.
Frequently asked questions
What is Heron’s formula?
Area = √(s(s − a)(s − b)(s − c)), where s is half the perimeter. It needs only the three sides.
Why does the calculator say the sides cannot form a triangle?
The two shorter sides must add up to more than the longest side. Otherwise they cannot meet.
Which unit is the area in?
Square units of what you entered. Sides in metres give area in square metres.
How do I find the area from base and height?
Multiply them and halve the result: area = ½ × base × height.
What is the height shown?
The perpendicular height measured from side a to the opposite corner.
Does it work for right triangles?
Yes. For 3, 4 and 5 it gives an area of 6, and labels the triangle as right.
This calculator gives estimates for information only. Actual figures from a bank, employer, tax authority or doctor may differ. It is not financial, tax, legal or medical advice.