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Heron's Formula Calculator

Find the area of a triangle from its three side lengths, with its perimeter and semi-perimeter, using Heron's formula.

What this tool does

Find the area of a triangle from its three side lengths, with its perimeter and semi-perimeter, using Heron's formula. You fill in these fields: Side a, Side b, and Side c. It shows triangle area, semi-perimeter (s), and perimeter.

Inputs

  • Side a: A number. Must be greater than 0. Enter all three sides in the same unit, such as meters or feet.
  • Side b: A number. Must be greater than 0.
  • Side c: A number. Must be greater than 0.

How to use

  1. Enter the side a.
  2. Enter the side b.
  3. Enter the side c.
  4. Read the result: triangle area, semi-perimeter (s), and perimeter. It appears once every required field has a valid value.

Worked examples

Example 1. Inputs:

  • Side a: 3
  • Side b: 4
  • Side c: 5

Results:

  • Triangle area: 6 square units
  • Semi-perimeter (s): 6 units
  • Perimeter: 12 units

How this is worked out: Heron's formula: s = (a + b + c) / 2 = (3 + 4 + 5) / 2 = 6. Area = sqrt(s (s - a) (s - b) (s - c)) = sqrt(6 x 3 x 2 x 1) = sqrt(36) = 6. Perimeter = 3 + 4 + 5 = 12. This is the 3-4-5 right triangle, whose area is 3 x 4 / 2 = 6, which agrees.

Example 2. Inputs:

  • Side a: 7
  • Side b: 10
  • Side c: 5

Results:

  • Triangle area: 16.248077 square units
  • Semi-perimeter (s): 11 units
  • Perimeter: 22 units

How this is worked out: s = (7 + 10 + 5) / 2 = 11. Area = sqrt(11 x (11 - 7) x (11 - 10) x (11 - 5)) = sqrt(11 x 4 x 1 x 6) = sqrt(264). Since 16.248^2 = 263.9975 and 2 x 16.248 x 0.0000769 = 0.0025, sqrt(264) = 16.24808 to 5 decimals. Perimeter = 22.

Example 3. Inputs:

  • Side a: 2
  • Side b: 2
  • Side c: 2

Results:

  • Triangle area: 1.732051 square units
  • Semi-perimeter (s): 3 units
  • Perimeter: 6 units

How this is worked out: Equilateral triangle, side 2. s = 6 / 2 = 3. Area = sqrt(3 x 1 x 1 x 1) = sqrt(3) = 1.7320508. Equilateral area check: (sqrt(3) / 4) x 2^2 = sqrt(3) = 1.7320508. Perimeter = 6.

Limitations

  • Works for flat triangles only, using ordinary Euclidean geometry.
  • Very thin triangles have an area close to zero, and small measuring errors in their sides change the area a lot.
  • The result is only as accurate as the three side lengths you enter.

FAQs

Q: Which side lengths can I enter?
A: Enter three positive lengths in the same unit. Each side must be shorter than the other two added together. If it is not, the three sides cannot make a triangle and the tool says so.
Q: What is the semi-perimeter?
A: It is half of the perimeter, which is the three sides added together and divided by two. Heron's formula uses it, and the tool shows it in the results.
Q: What unit is the area in?
A: The area is in square units of the side length. Sides entered in meters give an area in square meters.
Q: Can I find the area from an angle or a height instead?
A: No. This tool uses only the three side lengths. If you know a base and the height perpendicular to it, the area is base times height divided by two.

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