Area of Triangle Given 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, using Heron’s formula. It works for any triangle, as long as the side lengths satisfy the triangle inequality (each side is less than the sum of the other two).

Formula

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

A = √(s(s − a)(s − b)(s − c))

where s is the semiperimeter:

s = (a + b + c) / 2

• a, b, c – side lengths, in any unit (m, cm, ft, etc.)
• s – semiperimeter, same unit as the sides
• A – area, in square units (m², cm², ft², etc.)

The formula works for any triangle – acute, obtuse, or right – as long as the three lengths actually form a triangle. It does not require knowing any angle or height.

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 × (s − a) × (s − b) × (s − c) = 9 × 4 × 3 × 2 = 216
4. Take the square root: √216 ≈ 14.6969 m²
5. Rounded to two decimals: 14.70 m²

The calculator uses the equivalent expanded form (1/4)√((a+b+c)(−a+b+c)(a−b+c)(a+b−c)) which gives the same result and avoids a separate semiperimeter step. For 5, 6, 7 it returns 14.70 m².

When to Use This Formula

Use Heron’s formula when you know all three side lengths and do not know the height or any angle. Common situations:

• Measuring a plot of land with a tape measure – you get three distances, no angles.
• Finding the area of a triangular face in construction or carpentry when only side lengths are given.
• Solving geometry problems where the triangle is scalene (all sides different) and no height is drawn.

If you do know the base and height, the simpler formula A = ½ × base × height is easier. If you know two sides and the included angle, use A = ½ × a × b × sin(C). Heron’s formula is the only direct method when you have only the three sides.

How to Read the Result

The area is always in square units. If you enter side lengths in meters, the area is in square meters; if in feet, square feet; if in centimeters, square centimeters. The calculator does not convert units – you must use the same unit for all three sides. For example, if you have sides 5 cm, 6 cm, 7 cm, the area is 14.70 cm², not 14.70 m².

The result is always a positive number for a valid triangle. The semiperimeter output is also shown for reference – you can use it to verify the intermediate step or for other calculations.

Common Mistakes and Misconceptions

• Mixing units: Entering one side in meters and another in centimeters gives a meaningless result. Convert all sides to the same unit first.
• Rounding too early: If you round s to 9.0 and then round each subtraction, you might get 14.69 or 14.70. Keep full precision until the final step.
• Assuming the triangle is right: Heron’s formula works for all triangles, but many people incorrectly try to use ½ × base × height without knowing the height.
• Ignoring the triangle inequality: If one side is greater than or equal to the sum of the other two, no triangle exists. For example, sides 1, 2, 5 cannot form a triangle. The calculator will return an error or a non-real result (square root of a negative number).

Limits of the Formula

Heron’s formula is mathematically exact for any valid triangle, but it has practical limits:

• Input precision: The area is very sensitive to small errors in side lengths, especially for nearly degenerate triangles (where one side is close to the sum of the other two). A 1% error in a side can lead to a much larger percentage error in the area.
• Numerical stability: For very large or very small numbers, floating-point rounding can cause tiny errors. The expanded form used here is more stable than the classic form for extreme ratios.
• Units: The formula only works if all three sides are in the same unit. The result is in that unit squared.

For practical measurement, always measure each side to the same precision. If you are using a tape measure, that usually means to the nearest centimeter or inch.

Frequently Asked Questions

Can I use Heron’s formula for a right triangle?

Yes. Heron’s formula works for any triangle, including right triangles. For a 3-4-5 right triangle, it gives an area of 6, which matches ½ × 3 × 4 = 6. It is just a different path to the same result.

What if the sides do not form a triangle?

If the triangle inequality is violated (one side ≥ sum of the other two), the expression under the square root becomes negative, and the area is not a real number. The calculator will show an error or a non-numeric result. No such triangle exists in Euclidean geometry.

Do I need to know the height or an angle?

No. The whole point of Heron’s formula is that you only need the three side lengths. If you have the height, you can use the simpler base-height formula. If you have two sides and the included angle, use the sine formula.

Can I enter sides in different units?

No. All three sides must be in the same unit. If you have sides in feet and inches, convert everything to one unit first (e.g., all to inches or all to feet). The calculator does not perform unit conversions.

Why does my manual calculation differ from the calculator by a small amount?

Likely due to rounding. If you round the semiperimeter or intermediate products, you will lose precision. The calculator keeps full precision internally and only rounds the final display. For most practical purposes, a difference of 0.01 or 0.02 in the last digit is negligible.

Is Heron’s formula the only way to find the area from three sides?

It is the most direct method. You could also find an angle using the law of cosines and then use the sine area formula, but that is more steps. Heron’s formula is the standard single-step solution.

Related

• Pivot Table Terminology: Key Terms Explained
• Volume Equation for Pyramid: Formula & Examples
• Trustworthy AI: Explain, Share Data Safely, Resist Attacks

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 | (1/4)*sqrt((a+b+c)*(−a+b+c)*(a−b+c)*(a+b−c)) | 2
out semiperim | Semiperimeter s | m | (a+b+c)/2 | 2