Heron's Formula Calculator
Enter the three side lengths of any triangle and this calculator instantly finds the area using Heron's formula. No height needed. You also get the semiperimeter, perimeter, all three altitudes, all three interior angles, the inradius, and the circumradius, with a full step-by-step breakdown showing exactly how each result is reached.
Formula
Worked example
For a triangle with sides 5, 6, and 7: s = (5+6+7)/2 = 9. Product = 9 x 4 x 3 x 2 = 216. Area = sqrt(216) approximately 14.6969 square units. The inradius r = 14.6969 / 9 approximately 1.6330, and the circumradius R = (5x6x7)/(4x14.6969) approximately 3.5707.
What is Heron's formula?
Heron's formula calculates the area of a triangle using only its three side lengths, no height measurement required. Named after the first-century Greek mathematician Heron of Alexandria who proved it in his work Metrica, the formula states: A = sqrt(s(s-a)(s-b)(s-c)), where s is the semiperimeter (half the perimeter) and a, b, c are the three side lengths. This makes it uniquely powerful for triangles where the height is inconvenient or impossible to measure directly, such as in land surveying, navigation, and construction.
How to use this calculator
Enter the three side lengths of any triangle in the fields above. The calculator instantly computes the area using Heron's formula, along with the semiperimeter, perimeter, all three altitudes (heights), all three interior angles, the inradius of the inscribed circle, and the circumradius of the circumscribed circle. Switch between metric (metres) and imperial (feet) units using the units selector. The step-by-step panel below the results walks through each calculation with your exact numbers substituted in.
The triangle inequality and valid inputs
Not every set of three lengths can form a triangle. For a valid triangle, each side must be strictly less than the sum of the other two: a + b > c, a + c > b, and b + c > a. This is called the triangle inequality. If you enter side lengths that violate it, the calculator returns no result. For example, sides 1, 2, and 10 cannot form a triangle because 1 + 2 = 3 < 10. The degenerate case where one side equals the sum of the other two produces a straight line with zero area, which the calculator also rejects.
Beyond area: altitudes, angles, inradius, and circumradius
Once you have the area and semiperimeter, many other triangle properties follow directly. Each altitude (height) is twice the area divided by the corresponding base: h_a = 2A/a. Interior angles come from the law of cosines: cos(A) = (b^2 + c^2 - a^2)/(2bc). The inradius, which is the radius of the largest circle that fits inside the triangle, equals the area divided by the semiperimeter: r = A/s. The circumradius, the radius of the circle passing through all three vertices, is R = (abc)/(4A). These relationships are exact and hold for any valid triangle.
Triangle classification by side length
| Type | Side condition | Angle property |
|---|---|---|
| Equilateral | a = b = c | All angles = 60 degrees |
| Isosceles | Two sides equal | Two base angles are equal |
| Scalene | a, b, c all different | All angles are different |
| Right | a² + b² = c² (longest side c) | One angle = 90 degrees |
| Acute | a² + b² > c² for all c | All angles < 90 degrees |
| Obtuse | a² + b² < c² for largest c | One angle > 90 degrees |
Three ways to classify a triangle based on how its side lengths relate to each other.
Frequently asked questions
Why is it called Heron's formula?
The formula is named after Heron of Alexandria (also written Hero), a Greek mathematician and engineer who lived around 60 AD. He proved the formula in his book Metrica, though some historians believe Archimedes may have known it earlier. The formula itself expresses the area of a triangle purely in terms of its three side lengths.
What is the semiperimeter and why is it used?
The semiperimeter is simply half the perimeter: s = (a + b + c) / 2. It appears in Heron's formula because it produces a clean algebraic symmetry: the four factors s, (s-a), (s-b), and (s-c) are all non-negative for any valid triangle (each equals zero only for a degenerate triangle). This symmetry also makes the formula numerically convenient to compute.
Does the formula work for right triangles?
Yes, Heron's formula works for all triangle types including right triangles. For a right triangle with legs 3 and 4 and hypotenuse 5: s = (3+4+5)/2 = 6, and A = sqrt(6 x 3 x 2 x 1) = sqrt(36) = 6 square units. You can verify this with the standard formula: 0.5 x 3 x 4 = 6.
What happens when I enter sides that do not form a triangle?
The calculator returns no result. Three lengths form a valid triangle only if each is strictly less than the sum of the other two (the triangle inequality). If any of those conditions fails, there is no triangle with those side lengths, and the formula has no defined output.
Can I use this for triangles measured in feet or other units?
Yes. Switch the units selector to Imperial to work in feet. Heron's formula works in any consistent unit of length. If you enter sides in metres, the area comes out in square metres. If you enter feet, the area is in square feet.
What is the inradius?
The inradius is the radius of the inscribed circle, the largest circle that fits entirely inside the triangle and touches all three sides. It equals the area divided by the semiperimeter: r = A/s. A larger inradius relative to the side lengths means the triangle is closer to equilateral.
What is the circumradius?
The circumradius is the radius of the circumscribed circle, the unique circle that passes through all three vertices of the triangle. It equals the product of all three sides divided by four times the area: R = (a x b x c) / (4A). For an equilateral triangle with side length a, this simplifies to R = a / sqrt(3).