Triangle Calculator

Name: Triangle - Geometry Calculator
Author: Jürgen Kummer

Calculations at a general triangle. A triangle or trigon has three corners and three straight sides; the sum of the three angles is 180 degrees. It is the simplest polygon. Every polygon can be made of triangles, which can then be calculated individually.
Enter exactly three values, including at least one side length. When entering three sides, any two sides together must be longer than the third. Please enter angles in degrees, here you can convert angle units. Depending on the combination of sides and angles, different formulas are used for the calculation. A calculation is not always possible and sometimes not unique.

Triangle shape (longest side at the bottom):

Formulas:

SSS: Law of cosines

α = \arccos (\frac{b^{2} + c^{2} - a^{2}}{2 b c})

β = \arccos (\frac{a^{2} + c^{2} - b^{2}}{2 a c})

γ = \arccos (\frac{a^{2} + b^{2} - c^{2}}{2 a b})

SAS:

a = \sqrt{b^{2} + c^{2} - 2 b c \cdot \cos (α)}

b = \sqrt{a^{2} + c^{2} - 2 a c \cdot \cos (β)}

c = \sqrt{a^{2} + b^{2} - 2 a b \cdot \cos (γ)}

SSA: Law of sines

a / \sin (α) = b / \sin (β) = c / \sin (γ)

The law of sines is unique, if the known angular is opposite to the larger of the two given sides, otherwise there are two different solutions.

ASA and AAS: Third angle = 180° - other two angles, then law of sines

p = a + b + c

A = \sqrt{p / 2 \cdot (p / 2 - a) \cdot (p / 2 - b) \cdot (p / 2 - c)}

h_{a} = c \cdot \sin (β)

h_{b} = a \cdot \sin (γ)

h_{c} = b \cdot \sin (α)

r_{c} = a / (2 \cdot \sin (α))

r_{i} = 4 r \cdot \sin (α / 2) \cdot \sin (β / 2) \cdot \sin (γ / 2)

m_{a} = \sqrt{2 \cdot (b^{2} + c^{2}) - a^{2}} / 2

m_{b} = \sqrt{2 \cdot (c^{2} + a^{2}) - b^{2}} / 2

m_{c} = \sqrt{2 \cdot (a^{2} + b^{2}) - c^{2}} / 2

Side length, perimeter, radius and heights have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the angles are in degrees.

The centroid is at the intersection of the median lines, the center of the circumcircle is at the intersection of the perpendicular bisectors, the center of the incircle is at the intersection of the bisecting lines.

perimeter p, area A

sides and angles

heights

median lines and centroid

perpendicular bisectors and circumcircle

bisecting lines and incircle

There are three important special cases of triangles. The most regular is the equilateral triangle with equal angles and equal sides. The right triangle is important for trigonometry and the Pythagorean theorem. The isosceles triangle has two sides of equal length and two equal angles. While all equilateral triangles are equivalent, i.e. they only differ in size, there are an infinite number of different non-equivalent variants of right- and isosceles triangles. Of course, this also applies to the general triangle, which can be calculated here.

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