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# Law of Cosines Calculator

Law of Cosines Calculator

Primary Equation:

$A = \cos^{-1} \left[ \dfrac{b^2+c^2-a^2}{2bc} \right]$
Sides:
a =
b =
c =

Angles:
A =
B =
C =

Other:
P =
s =
K =
r =
R =

## This Calculation Equation & Triangle

$$A = \cos^{-1} \left[ \dfrac{b^2+c^2-a^2}{2bc} \right]$$ A = angle A
B = angle B
C = angle C
a = side a
b = side b
c = side c
P = perimeter
s = semi-perimeter
K = area
r = radius of inscribed circle
R = radius of circumscribed circle

*Length units are for your reference-only since the value of the resulting lengths will always be the same no matter what the units are.

## Calculator Use

Uses the law of cosines to calculate unknown angles or sides of a triangle. In order to calculate the unknown values you must enter 3 known values.

To calculate any angle, A, B or C, enter 3 side lengths a, b and c.  This is the same calculation as Side-Side-Side (SSS) Theorem. To calculate side a for example, enter the opposite angle A and the two other adjacent sides b and c. Using different forms of the law of cosines we can calculate all of the other unknown angles or sides. This is the same calculation as Side-Angle-Side (SAS) Theorem. ## Law of Cosines

If a, b and c are the lengths of the legs of a triangle opposite to the angles A, B and C respectively; then the law of cosines states:

$$a^2 = b^2 + c^2 - 2bc \cos A$$
$$b^2 = a^2 + c^2 - 2ac \cos B$$
$$c^2 = a^2 + b^2 - 2ab \cos C$$

### Law of Cosines solving for sides a, b, and c

$$a = \sqrt{b^2 + c^2 - 2bc \cos A }$$
$$b = \sqrt{a^2 + c^2 - 2ac \cos B }$$
$$c = \sqrt{a^2 + b^2 - 2ab \cos C }$$

### Law of Cosines solving for angles A, B, and C

$$A = \cos^{-1} \left[ \dfrac{b^2+c^2-a^2}{2bc} \right]$$
$$B = \cos^{-1} \left[ \dfrac{a^2+c^2-b^2}{2ac} \right]$$
$$C = \cos^{-1} \left[ \dfrac{a^2+b^2-c^2}{2ab} \right]$$

## Triangle Characteristics

Triangle perimeter, P = a + b + c

Triangle semi-perimeter, s = 0.5 * (a + b + c)

Triangle area, K = √[ s*(s-a)*(s-b)*(s-c)]

Radius of inscribed circle in the triangle, r = √[ (s-a)*(s-b)*(s-c) / s ]

Radius of circumscribed circle around triangle, R = (abc) / (4K)