4
18
9
216
14.696938
5.878775
4.898979
4.199125
1.632993
4
18
9
216
14.696938
5.878775
4.898979
4.199125
1.632993
Heron's Formula Calculator computes the area of any triangle directly from its three side lengths, without requiring any angle measurement. Named after Heron of Alexandria (c. 10-70 CE), this elegant formula is one of the most celebrated results in classical geometry. Given sides $$a$$, $$b$$, and $$c$$, the formula first calculates the semi-perimeter $$s = \frac{a+b+c}{2}$$, then determines the area as $$A = \sqrt{s(s-a)(s-b)(s-c)}$$.
The beauty of Heron's formula lies in its simplicity and universality. Unlike the standard area formula $$A = \frac{1}{2}bh$$, which requires knowing a height, or the trigonometric formula $$A = \frac{1}{2}ab\sin(C)$$, which requires an angle, Heron's formula works with side lengths alone. This makes it invaluable in situations where only distances can be measured, such as land surveying, satellite positioning, and structural analysis.
This calculator provides a comprehensive set of derived quantities beyond just the area. It computes all three altitudes (heights) using $$h_a = \frac{2A}{a}$$, and the inradius using $$r = \frac{A}{s}$$. The inradius is the radius of the largest circle that fits inside the triangle (the inscribed circle or incircle), touching all three sides. These additional outputs transform a simple area calculation into a complete triangle analysis tool.
In practical applications, surveyors use Heron's formula to calculate land areas when they can measure all boundary lengths but cannot easily determine heights or angles. Structural engineers apply it to compute cross-sectional areas of triangular members. Computer graphics programmers use it extensively for mesh processing, collision detection, and determining whether a point lies inside a triangle.
The formula also has elegant mathematical properties. The expression under the square root, $$s(s-a)(s-b)(s-c)$$, is a symmetric polynomial in the side lengths. It is non-negative exactly when the three sides satisfy the triangle inequality, providing a built-in validity check. When the sides violate the triangle inequality, the expression becomes negative and the square root is undefined, signaling that no valid triangle exists.
Heron's formula connects to many other areas of mathematics. It can be derived from the Law of Cosines, from Brahmagupta's formula for cyclic quadrilaterals (by setting one side to zero), and from the Cayley-Menger determinant in distance geometry. It also generalizes to higher dimensions through the theory of simplices.
Step 1: Compute the semi-perimeter:
$$s = \frac{a + b + c}{2}$$
Step 2: Apply Heron's formula:
$$A = \sqrt{s(s-a)(s-b)(s-c)}$$
Derived quantities:
Heights (altitudes) from $$A = \frac{1}{2} \cdot \text{base} \cdot \text{height}$$:
$$h_a = \frac{2A}{a}, \quad h_b = \frac{2A}{b}, \quad h_c = \frac{2A}{c}$$
Inradius (radius of the inscribed circle):
$$r = \frac{A}{s}$$
The semi-perimeter is half the total boundary length. The area is measured in square units corresponding to the input units. The three heights represent the perpendicular distance from each vertex to the opposite side. The inradius is the radius of the incircle, the circle tangent to all three sides. If the area is 0, the triangle is degenerate (collinear points). If the output shows NaN, the sides violate the triangle inequality.
Inputs
Results
s = 9. Area = sqrt(9·4·3·2) = sqrt(216) ≈ 14.70. Inradius r = 14.70/9 ≈ 1.633.
Inputs
Results
s = 6. Area = sqrt(6·3·2·1) = sqrt(36) = 6. For a 3-4-5 right triangle, the inradius is exactly 1.
Heron's formula calculates the area of a triangle from its three side lengths: A = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2 is the semi-perimeter. It requires no angle or height measurements, only the three sides.
Heron of Alexandria (c. 10-70 CE) was a Greek mathematician and engineer. He described this formula in his work Metrica, though some historians believe the formula was known to Archimedes (c. 287-212 BCE) even earlier.
The inradius (r) is the radius of the incircle — the largest circle that fits inside the triangle, tangent to all three sides. It is calculated as r = Area/s, where s is the semi-perimeter. The incircle's center is called the incenter, equidistant from all three sides.
NaN occurs when the expression under the square root is negative, meaning s(s-a)(s-b)(s-c) < 0. This happens when the three side lengths violate the triangle inequality (the sum of any two sides must exceed the third), making it impossible to form a valid triangle.
Yes. For an equilateral triangle with side length a, s = 3a/2, and the formula gives A = (sqrt(3)/4)·a², which matches the standard equilateral triangle area formula perfectly.
For a right triangle with legs a, b and hypotenuse c, Heron's formula always yields A = ab/2, matching the simple right-triangle area formula. This can be verified algebraically: with c² = a² + b², the Heron expression simplifies to (ab/2)².
Roboculator Team
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