How do you find the area of a triangle from three sides?

Use Heron's formula. First calculate the semi-perimeter: s = (a + b + c) / 2. Then compute the area: Area = sqrt(s x (s - a) x (s - b) x (s - c)). For example, a triangle with sides 5, 6, and 7 has s = 9 and area = sqrt(9 x 4 x 3 x 2) = sqrt(216) = 14.70 square units. The formula works for any valid triangle regardless of whether the angles are known.

What is a valid triangle? Why might the calculator show an error?

Three lengths a, b, and c form a valid triangle only if the sum of any two sides is greater than the third side. This is called the triangle inequality: a + b > c, a + c > b, and b + c > a. If any condition fails, no triangle can be formed. For example, sides 1, 2, and 10 are not a valid triangle because 1 + 2 = 3 is less than 10. If you enter such values the calculator will display an error rather than a result.

What is the difference between an equilateral, isosceles, and scalene triangle?

An equilateral triangle has all three sides equal in length. An isosceles triangle has exactly two sides equal. A scalene triangle has all three sides of different lengths. The triangle area calculator identifies which type your triangle is based on the side lengths you enter. All three types can be solved using Heron's formula.

Triangle Area Calculator (3 Sides)

Calculate the area of any triangle using only its three side lengths. Enter sides a, b, and c and the calculator applies Heron's formula to give you the exact area, perimeter, and triangle type instantly.

No angles required. Works for scalene, isosceles, and equilateral triangles.

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Standard formula Heron's formula (Hero of Alexandria, c. 60 AD). Exact for any valid triangle.

Enter the Three Side Lengths

Side a (any unit: cm, m, mm)

units

Side b

units

Side c

units

These three lengths do not form a valid triangle. The sum of any two sides must be greater than the third side.

Heron's Formula

Step 1: Semi-perimeter s = (a + b + c) / 2

Step 2: Area = sqrt(s × (s−a) × (s−b) × (s−c))

Enter your side lengths to see the working.

The result is in square units matching your input (e.g. if sides are in centimetres, area is in cm²).

Triangle Results

Area
14.70
Square units

Perimeter

Units

Semi-perimeter (s)

Units

Triangle Type

Scalene

Based on sides

Step-by-Step Calculation

Side a5

Side b6

Side c7

Perimeter (a + b + c)18

Semi-perimeter s9

s − a4

s − b3

s − c2

Product inside sqrt216

Area14.6969 sq units

Triangle Properties

Triangle typeScalene

Longest side7

Triangle valid?Yes

Ratio longest:shortest1.40 : 1

Area (4 d.p.)14.6969 sq units

Area (2 d.p.)14.70 sq units

Height from longest side4.20 units

Result: A scalene triangle with sides 5, 6, and 7 has an area of 14.70 square units and a perimeter of 18 units.

How to Calculate Triangle Area from Three Sides

When you know all three sides of a triangle but not the height or any angles, Heron's formula is the standard method. It was described by Hero of Alexandria around 60 AD and works for any triangle, regardless of whether it is acute, right-angled, or obtuse.

The process has two steps:

Calculate the semi-perimeter: s = (a + b + c) / 2
Calculate the area: Area = sqrt(s x (s - a) x (s - b) x (s - c))

Worked Example

For a triangle with sides a = 5, b = 6, and c = 7:

Step	Calculation	Result
Semi-perimeter	s = (5 + 6 + 7) / 2 = 18 / 2	9
s - a	9 - 5	4
s - b	9 - 6	3
s - c	9 - 7	2
Product	9 x 4 x 3 x 2	216
Area	sqrt(216)	14.6969 sq units

These are the default values shown in the calculator above, confirming that the result is 14.70 square units (to 2 decimal places).

Triangle Types by Side Length

Type	Side Lengths	Example
Equilateral	All three sides equal	a = b = c = 5
Isosceles	Exactly two sides equal	a = b = 5, c = 7
Scalene	All three sides different	a = 5, b = 6, c = 7

The Triangle Inequality

Not every set of three lengths forms a valid triangle. For sides a, b, and c to form a triangle, all three of these conditions must hold:

a + b > c
a + c > b
b + c > a

If any condition is not met, the calculator will display an error. For example, sides 1, 2, and 10 fail because 1 + 2 = 3 which is not greater than 10.

Height from the Longest Side

Once you have the area, you can find the height (altitude) from any base using the formula: height = (2 x Area) / base. The calculator shows the height measured from the longest side. This is useful for checking your answer or for practical tasks like cutting timber or marking out shapes on the ground.

Related Calculators

Maths and Stats Calculators: all maths tools on this site.
Area of a Triangle Calculator: base and height method.
Area of a Triangle (Heron) Calculator: extended Heron's formula tool.
Triangle Calculator (general): solve any triangle from mixed inputs.
Pythagorean Theorem Calculator: right-triangle side lengths.

Method: Heron's formula. Hero of Alexandria, Metrica, c. 60 AD. Standard formula reproduced in any geometry textbook (e.g. Coxeter, Introduction to Geometry, Wiley). The triangle inequality is a fundamental theorem in Euclidean geometry.

This calculator uses exact arithmetic (JavaScript's IEEE 754 double precision). Results are rounded for display. For very small or very large triangles, verify with an independent source if precision is critical.

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