Triangle Calculator
Calculate area, angles, heights, perimeter, inradius, and circumradius of any triangle from three sides.
About this calculator
Given only three side lengths, this calculator recovers everything else about a triangle without ever needing a height measurement directly. Area comes from Heron's formula, which multiplies the semi-perimeter against its difference from each side and takes the square root of the result — a purely side-based route to area that works for any triangle, not just right triangles. The three interior angles come from a separate route, the law of cosines, applied once per angle by rearranging c squared equals a squared plus b squared minus 2ab cosine C; Angle C is then found by subtracting the other two from 180 degrees rather than running the law of cosines a third time, since the three angles are guaranteed to sum to a straight line.
At the default sides of 5, 7, and 8, raising Side a while the other two stay fixed increases Area, and among the three sides, Side a has the largest measured effect on Area for an equivalent proportional change — a genuine asymmetry that follows from how the other two sides happen to be set, not a property of triangles in general. Before any of this runs, the triangle inequality is checked — each side must be shorter than the sum of the other two — and a set of lengths that fails this check is reported as an Invalid triangle rather than silently producing an area or angle set that would not correspond to any real shape.
Inputs
Results
Area
17.32
Heron's formula for the area of a triangle
- 1.Find the semi-perimeter
Heron's formula is built around half the perimeter rather than the whole thing, so this is the one quantity every later term is measured against.
- 2.Multiply the four terms and take the square root
Each bracket measures how far one side falls short of the semi-perimeter. Their product grows when the sides are balanced and shrinks toward zero as the triangle flattens, which is why the square root of it is an area.
Heron of Alexandria set out this formula with a worked proof in his Metrica, deriving a triangle's area from its three sides alone — no angle and no height required.
How to Use This Calculator
- Enter all three side lengths (a, b, c) of your triangle; the sides must satisfy the triangle inequality (each side < sum of the other two).
- Area is computed using Heron's formula: A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2, so no height is needed.
- All three interior angles are derived from the law of cosines: cos(C) = (a²+b²−c²)/(2ab).
- The triangle is automatically classified as equilateral, isosceles, scalene, right, acute, or obtuse.
- Inradius (r = Area/s) and Circumradius (R = abc/(4×Area)) are also shown for advanced geometry.
- The height from side a (Height from A) is computed as h = 2×Area / a.
How the result changes with Side a
| Side a | Area |
|---|---|
| 2.5 | 8.47 |
| 3.75 | 13.12 |
| 7.5 | 24.14 |
| 13 | 24.25 |
What each input means
- Side a
- Length of side A.
- Side b
- Length of side B.
- Side c
- Length of side C.
How this is calculated
Formula
A = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2Worked example, using the default values
- Identify Input ParametersSide a = 5, Side b = 7, Side c = 8 = 3 input(s) provided
- Calculate AreaArea17.3205 = 17.3205
- Calculate PerimeterPerimeter20 = 20
- Calculate Triangle TypeScalene Acute = Scalene Acute
Engine last updated . Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Which side has the biggest effect on Area at the default lengths?
Side a. Starting from the default 5, 7, 8 triangle, raising Side a by a given percentage while Sides b and c stay fixed moves Area more than raising either of the other two sides by the same percentage — a fact about this specific triangle's proportions, not a general rule about all triangles.
Why does the calculator need the law of cosines instead of just using Heron's formula for the angles too?
Heron's formula only produces area from the three sides; it does not directly yield an angle. The law of cosines rearranges the relationship between a triangle's sides and one of its angles algebraically, which is what lets the calculator solve for each angle in degrees.
How is Angle C found without running the law of cosines a third time?
The three interior angles of any triangle always add up to 180 degrees, so once Angle A and Angle B are known, Angle C is simply 180 minus the other two. This is faster than a third law-of-cosines calculation and gives the identical result.
What happens if I enter three side lengths that can't form a real triangle?
The calculator checks the triangle inequality — that each side is shorter than the sum of the other two — before computing anything else, and reports Invalid triangle if that check fails. It will not produce a misleading area or angle set for a combination of lengths that cannot close into a shape.
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