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Calcimator

Triangle Calculator

Calculate area, angles, heights, perimeter, inradius, and circumradius of any triangle from three sides.

Inputs

Results

Area

17.32

Perimeter20
Triangle TypeScalene Acute
Angle A38.21°
Angle B60°
Angle C81.79°
Height from A6.93
Inradius1.73
Circumradius4.04

Heron's formula for the area of a triangle

  1. 1.Find the semi-perimeter
    s=5+7+82=10

    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. 2.Multiply the four terms and take the square root
    A=10×(10−5)×(10−7)×(10−8)=17.32

    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.
Heron of Alexandria, Metrica, Book I, Proposition 8·c. 60 CE
How to Use This Calculator
  1. 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).
  2. 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.
  3. All three interior angles are derived from the law of cosines: cos(C) = (a²+b²−c²)/(2ab).
  4. The triangle is automatically classified as equilateral, isosceles, scalene, right, acute, or obtuse.
  5. Inradius (r = Area/s) and Circumradius (R = abc/(4×Area)) are also shown for advanced geometry.
  6. The height from side a (Height from A) is computed as h = 2×Area / a.

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)/2

Worked example, using the default values

  1. Identify Input Parameters
    Side a = 5, Side b = 7, Side c = 8 = 3 input(s) provided
  2. Calculate Area
    Area
    17.3205 = 17.3205
  3. Calculate Perimeter
    Perimeter
    20 = 20
  4. Calculate Triangle Type
    Scalene Acute = Scalene Acute

Engine last updated .

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