Triangle Area from Three Sides

What This Heron's Formula Calculator Does

This calculator finds the area of any triangle when you know the lengths of all three sides. You enter the three side lengths (a, b, and c) in the same unit, and it returns the area using Heron's formula.

It is useful when you cannot easily measure the height of a triangle but you can measure its edges. That makes it handy for land surveying, carpentry, roofing, fabric and material cutting, garden bed planning, and geometry homework. Because it only needs side lengths, you do not need to know any angles.

How Heron's Formula Works

Heron's formula computes triangle area directly from the three sides. First find the semi-perimeter s, which is half the perimeter:

s = (a + b + c) / 2

Then plug s and the three sides into the area equation:

area = sqrt( s (s - a) (s - b) (s - c) )

Before any area exists, the three sides must satisfy the triangle inequality: each side must be shorter than the sum of the other two (a + b > c, a + c > b, and b + c > a). If a side equals or exceeds the sum of the other two, the sides cannot form a real triangle, and (s - a), (s - b), or (s - c) becomes zero or negative, so the formula has no valid (positive) area.

Worked Example With Real Numbers

Suppose a triangle has sides a = 6, b = 8, and c = 10 (all in centimeters).

Step 1 - Semi-perimeter: s = (6 + 8 + 10) / 2 = 24 / 2 = 12.

Step 2 - Side differences: s - a = 12 - 6 = 6, s - b = 12 - 8 = 4, s - c = 12 - 10 = 2.

Step 3 - Multiply: 12 x 6 x 4 x 2 = 576.

Step 4 - Square root: area = sqrt(576) = 24 square centimeters.

As a check, 6-8-10 is a right triangle (6 squared plus 8 squared equals 10 squared), so its area should be (1/2) x 6 x 8 = 24. Heron's formula agrees.

Tips and Common Mistakes

A few small errors cause most wrong answers. Keep these in mind:

• Use one consistent unit. Mixing centimeters and inches gives a meaningless result. Convert everything first.
• The answer is in square units. If sides are in meters, the area is in square meters.
• Take the square root last. A frequent slip is computing s(s-a)(s-b)(s-c) and forgetting the final square root.
• Watch for impossible triangles. Sides like 2, 3, and 9 fail the triangle inequality (2 + 3 < 9) and have no area.
• Very thin (nearly degenerate) triangles, where one side is almost the sum of the other two, produce a tiny area that is sensitive to rounding, so keep extra decimal places in your inputs.

Factors That Affect the Result

The area depends only on the three side lengths, not on orientation or which side you call a, b, or c. Swapping the labels gives the same answer because multiplication is commutative.

Accuracy depends on your measurements. Because the side values are squared and multiplied inside the formula, small measurement errors are magnified in the area, especially for long, narrow triangles. For real-world work, measure each side carefully and consider averaging repeated measurements.

Heron's formula is exact for flat (planar) triangles. For a triangle drawn on a sphere or a steeply curved surface, it is only an approximation, so use it for everyday flat layouts and standard geometry problems.