Triangle Calculator

Worked example

Sides 5, 6 and 7 close into a triangle with an area of 14.7 by Heron’s formula, and angles of 44.4, 57.1 and 78.5 degrees. They add to 180, which is the check worth doing on any answer of this kind.

How to work it out yourself

1. 1.Enter the three sides in any order. The labels follow: angle A is always opposite side a.
2. 2.Check the triangle inequality first if the answer looks wrong — any two sides must add to more than the third, or no triangle exists at all.
3. 3.Use the height line when you need the altitude for a construction drawing: it is twice the area divided by the side you are measuring onto.

The formula

1. Perimeter5 + 6 + 7
2. Areasqrt(max(0, 9 * (9 - 5) * (9 - 6) * (9 - 7)))
3. Angle A, opposite side aacos((pow(6, 2) + pow(7, 2) - pow(5, 2)) / (2 * 6 * 7)) * 180 / pi
4. Angle B, opposite side bacos((pow(5, 2) + pow(7, 2) - pow(6, 2)) / (2 * 5 * 7)) * 180 / pi
5. Angle C, opposite side c180 - 44.415309 - 57.12165
6. Height onto side a2 * 14.696938 / 5
7. Inradius14.696938 / 9
8. Circumradius5 * 6 * 7 / (4 * 14.696938)

Source: Wolfram MathWorld — triangle, Wolfram MathWorld — law of cosines

Questions people actually ask

Can any three lengths make a triangle?

No. Each side has to be shorter than the other two added together — the triangle inequality. Sides of 2, 3 and 9 cannot close: the short pair cannot reach across the long one. This calculator detects that rather than printing angles from an impossible cosine.

How do you find an angle from three sides?

The law of cosines, rearranged: cos A = (b² + c² − a²) ÷ 2bc. Take the inverse cosine and you have the angle. For a right triangle the term with the longest side vanishes and it reduces to Pythagoras, which is the check that the formula is the same one you already know.

What is Heron’s formula?

Area from the three sides alone, with no angle and no height: take half the perimeter, subtract each side from it in turn, multiply the four numbers together and take the square root. It is what makes a field measurable with a tape and nothing else.

What are the inradius and circumradius for?

The inradius is the radius of the largest circle that fits inside the triangle, touching all three sides; the circumradius is the circle through all three corners. They matter in layout work — the inradius is the biggest round thing that fits in a triangular space.

Related

• Right Triangle CalculatorHypotenuse, area, perimeter and both acute angles of a right triangle from its two legs, with the Pythagorean step shown before the square root.
• Area CalculatorArea of a rectangle, triangle, circle, sector, ellipse, trapezoid, parallelogram or regular polygon — in square feet, square yards, acres and square meters.
• Special Right Triangle CalculatorThe 45-45-90 and 30-60-90 triangles from one known side, with every other side given as an exact multiple rather than a decimal that hides the ratio.
• Pythagorean Theorem CalculatorSolves a² + b² = c² in either direction — the hypotenuse from two legs, or the missing leg from a leg and the hypotenuse — with the squares shown.
• Law of Sines CalculatorSolves a triangle from one side and two angles: the third angle, the remaining two sides and the area, using a/sin A = b/sin B.
• Diagonal & Rafter Length CalculatorHypotenuse of a right triangle in feet and inches — brace lengths, rafter runs, and squaring a layout.

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