Triangle Area Calculator - 3 Sides
Enter all three side lengths to calculate the area of any triangle using Heron's formula. You also get the perimeter, semiperimeter, all three altitudes, the inradius (inscribed circle radius), the circumradius (circumscribed circle radius), and the triangle type. Switch between metric and imperial units - results update as you type.
Formula
Worked example
Sides 5, 6 and 7: semiperimeter s = (5+6+7)/2 = 9. Radicand = 9 x 4 x 3 x 2 = 216. Area = sqrt(216) approximately 14.6969. Altitudes: h_a = 29.3938/5 = 5.8788, h_b = 4.8990, h_c = 4.1991. Inradius = 14.6969/9 = 1.6330. Circumradius = 210/(4 x 14.6969) = 3.5707.
Heron's formula: calculate triangle area from three sides
Heron's formula (also called Hero's formula) lets you find the area of any triangle when you know the length of all three sides, with no need to measure an angle or an altitude. The formula was described by Hero of Alexandria around 60 AD. You first compute the semiperimeter s = (a + b + c) / 2, then substitute into Area = sqrt(s(s - a)(s - b)(s - c)). Because it only needs the three side lengths it works for scalene, isosceles, equilateral, right and obtuse triangles alike. This calculator also derives the perimeter, all three altitudes, the inradius and the circumradius from the same three inputs, so you get a complete geometric picture in one step.
What else does this calculator compute?
Beyond the area, this tool computes several related properties: Altitudes: each altitude (height) is the perpendicular distance from a vertex to the opposite side. Given the area, the altitude on side a is h_a = 2A / a, and similarly for b and c. These are useful for construction, carpentry and land surveying. Inradius: the radius of the largest circle that fits inside the triangle, r = A / s. A larger inradius means the triangle is closer to equilateral, the shape that maximises area for a given perimeter. Circumradius: the radius of the circle that passes through all three vertices, R = abc / (4A). For a right triangle, the circumradius equals exactly half the hypotenuse. Triangle type: the calculator classifies the triangle by both side lengths (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). An obtuse triangle has one angle above 90 degrees; the largest side squared exceeds the sum of the squares of the other two.
How to check whether three lengths form a valid triangle
Not every set of three positive numbers can be the sides of a triangle. The triangle inequality requires that each side must be strictly less than the sum of the other two: a < b + c, b < a + c and c < a + b. If any of these fails, no triangle exists and the calculator shows no result. For example, sides 1, 2 and 10 cannot form a triangle because 1 + 2 = 3 < 10. A degenerate case where one side exactly equals the sum of the others collapses to a straight line with area zero.
Common uses and worked example
Heron's formula is used in architecture and construction to lay out triangular floor plans, in land surveying when you can measure boundary lengths but not angles, in navigation for triangulation problems, and in geometry classrooms. Worked example: a triangular garden plot has sides of 5 m, 6 m and 7 m. Semiperimeter s = (5 + 6 + 7) / 2 = 9 m. Radicand = 9 x 4 x 3 x 2 = 216. Area = sqrt(216) approximately 14.70 m squared. The altitude onto the 7 m side is 2 x 14.70 / 7 approximately 4.20 m. The inscribed circle has radius 14.70 / 9 approximately 1.63 m.
Triangle types at a glance
| Type | Sides | Largest angle | Circumcenter location |
|---|---|---|---|
| Equilateral | All equal | 60 degrees | Centre (centroid) |
| Isosceles acute | 2 equal | Below 90 degrees | Inside |
| Isosceles right | 2 equal | Exactly 90 degrees | On hypotenuse midpoint |
| Isosceles obtuse | 2 equal | Above 90 degrees | Outside |
| Scalene acute | All different | Below 90 degrees | Inside |
| Scalene right | All different | Exactly 90 degrees | On hypotenuse midpoint |
| Scalene obtuse | All different | Above 90 degrees | Outside |
Classification by sides and angles, with properties that each type guarantees.
Frequently asked questions
What is Heron's formula?
Heron's formula calculates the area of a triangle from its three side lengths without needing an angle or altitude. Given sides a, b and c, you first find the semiperimeter s = (a + b + c) / 2, then compute Area = sqrt(s(s - a)(s - b)(s - c)). It works for any triangle - scalene, isosceles, equilateral, right or obtuse.
Can I use this calculator for a right triangle?
Yes. Enter all three side lengths including the hypotenuse and the formula gives the correct area. For a right triangle you could also compute Area = 0.5 x leg1 x leg2 directly, but Heron's formula produces the same result. The circumradius of a right triangle always equals half the hypotenuse, which this calculator confirms.
Why does the calculator show no result for my inputs?
The three sides may violate the triangle inequality: each side must be strictly less than the sum of the other two (a < b + c, b < a + c, c < a + b). If any side is zero, negative, or greater than or equal to the sum of the other two, no real triangle exists and the calculator returns blank. Check that all three values are positive and that no single side is too long.
What is the inradius of a triangle?
The inradius (r) is the radius of the largest circle that fits entirely inside the triangle, tangent to all three sides. It equals Area divided by the semiperimeter: r = A / s. An equilateral triangle has the largest inradius relative to its side length of any triangle.
What is the circumradius of a triangle?
The circumradius (R) is the radius of the circle that passes through all three vertices of the triangle. It equals (a x b x c) / (4 x Area). For a right triangle, R = hypotenuse / 2. For an obtuse triangle the circumcenter lies outside the triangle; for an acute triangle it lies inside.
How are the three altitudes calculated?
Each altitude is the perpendicular distance from one vertex to the opposite side. Once you have the area, the altitude on side a is h_a = 2 x Area / a, on side b it is h_b = 2 x Area / b, and on side c it is h_c = 2 x Area / c. The shortest altitude always corresponds to the longest side.