Area of Triangle with 3 Sides (Heron’s Formula)

This calculator works out the area of a triangle when you know the lengths of all three sides. Enter the three side lengths, and you get the area in square units, plus the semiperimeter and the height to the longest side. The calculation uses Heron’s formula, which is the standard method for this situation.

Formula

Heron’s formula gives the area A of a triangle from its three side lengths a, b, and c:

A = √(s(s − a)(s − b)(s − c))

where s is the semiperimeter:

s = (a + b + c) / 2

All lengths are in the same unit (meters, feet, cm, etc.). The area comes out in the square of that unit (m², ft², cm²). The semiperimeter is simply half the perimeter.

Worked Example

Using the default values: a = 5 m, b = 6 m, c = 7 m.

1. Calculate the semiperimeter: s = (5 + 6 + 7) / 2 = 18 / 2 = 9 m.
2. Subtract each side from s: (s − a) = 9 − 5 = 4, (s − b) = 9 − 6 = 3, (s − c) = 9 − 7 = 2.
3. Multiply: s × 4 × 3 × 2 = 9 × 4 × 3 × 2 = 216.
4. Take the square root: √216 ≈ 14.6969 m².

Rounded to two decimals, the area is 14.70 m². The calculator shows the same result. The height to side a is 2 × 14.6969 / 5 ≈ 5.88 m.

When to Use Heron’s Formula

Use Heron’s formula when you know all three side lengths and you do not have the height or any angle. This is common in land surveying, construction, and geometry problems where you measure the three edges of a triangular plot or object directly. It works for any triangle – scalene, isosceles, or equilateral – as long as the three sides satisfy the triangle inequality (each side is less than the sum of the other two). For a right triangle, you can also use base × height / 2 if you know the legs, but Heron’s formula works without knowing which side is the hypotenuse.

How to Choose the Right Sides

Any order of sides gives the same area, so you do not need to worry about which side is “base� or “height.� The formula is symmetric in a, b, and c. Just enter the three measured lengths in the same unit. If your sides are in different units (e.g., one in feet and one in inches), convert them all to one unit first. The calculator assumes all inputs are in the same unit.

Common Mistakes and Pitfalls

• Unit mix-ups: Mixing meters and centimeters will give a wrong area. Convert everything to one unit first.
• Triangle inequality violation: If one side is greater than or equal to the sum of the other two, the triangle cannot exist, and the square root becomes negative or zero. The calculator will show an error or zero – check your measurements.
• Rounding too early: If you round the semiperimeter before subtracting, you lose precision. Keep full precision until the final step.
• Forgetting the square root: The product s(s−a)(s−b)(s−c) is not the area; you must take the square root.

Limitations and Alternatives

Heron’s formula is exact for any triangle, but it is not always the most convenient. If you know two sides and the included angle, use Area = ½ × a × b × sin(C). If you know the base and height, use Area = ½ × base × height. For a triangle on a coordinate plane, the shoelace formula is easier. Heron’s formula is best when you only have side lengths – no angles or height.

For very large or very small triangles, numerical stability can be an issue with the standard formula. A more stable variant sorts the sides so that a ≥ b ≥ c and uses Area = ¼ × √((a+(b+c)) × (c−(a−b)) × (c+(a−b)) × (a+(b−c))). This avoids subtracting nearly equal numbers, which can cause rounding errors in floating-point arithmetic. For most everyday measurements, the standard formula is fine.

Frequently Asked Questions

Can I use Heron’s formula for any triangle?

Yes, as long as the three sides satisfy the triangle inequality (each side is less than the sum of the other two). It works for scalene, isosceles, and equilateral triangles.

What if the sides are in different units?

Convert all sides to the same unit before entering them. The area will be in the square of that unit. For example, if you convert everything to meters, the area is in square meters.

Why do I get an error or a negative number?

That happens when the three lengths cannot form a triangle – one side is too long relative to the others. Check your measurements; the longest side must be less than the sum of the other two.

Is there a simpler formula for a right triangle?

Yes, for a right triangle you can use Area = ½ × (leg1) × (leg2). But Heron’s formula still works if you only know the three side lengths.

Does the order of sides matter?

No. The formula is symmetric, so any order gives the same area. You can enter the sides in any order.

How do I find the height from the area?

Once you have the area A and a chosen base side b, the height to that side is h = 2A / b. The calculator shows the height to side a as an example.

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in a | Side a | m | 5 | 0
in b | Side b | m | 6 | 0
in c | Side c | m | 7 | 0
out area | Area | m² | sqrt((a+b+c)/2*(((a+b+c)/2)-a)*(((a+b+c)/2)-b)*(((a+b+c)/2)-c)) | 2
out semi | Semiperimeter s | m | (a+b+c)/2 | 2
out h | Height to side a | m | 2*sqrt((a+b+c)/2*(((a+b+c)/2)-a)*(((a+b+c)/2)-b)*(((a+b+c)/2)-c))/a | 2