Law of Sines Calculator

How does this law of sines calculator work?

Together with the law of cosines, the law of sines can help when dealing with simple or complex math problems by simply using the formulas explained here, which are also used in the algorithm of this law of sines calculator.

A = sin^-1[(a*sin(b))/b]

Assuming that a, b and c are the 3 sides of the triangle opposite to the angles A, B and C as shown in the figure below, the law of sines states that:

■ For the calculation of the three sides (a, b and c) these formulas are applicable:

a = b*sin(A)/sin(B)

a = c*sin(A)/sin(C)

b = a*sin(B)/sin(A)

b = c*sin(B)/sin(C)

c = a*sin(C)/sin(A)

c = b*sin(C)/sin(B)

■ In order to compute the three angles (A, B and C) the equations described here are suitable:

A = sin^-1 [(a*sin(B))/b]

A = sin^-1 [(a*sin(C))/c]

B = sin^-1 [(b*sin(A))/a]

B = sin^-1 [(b*sin(C))/c]

C = sin^-1 [(c*sin(A))/a]

C = sin^-1 [(c*sin(B))/b]

Triangle formulas for area, perimeter and radius

The standard triangle formulas that are used in trigonometry to solve different problems are:

Triangle perimeter (P) = a + b + c

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

Triangle area by Heron equation (A_S) = √[ 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) / (4A_S)

Where:

a = Side a

b = Side b

c = Side c

A = Angle A

B = Angle B

C = Angle C

P = Perimeter

s = Semi-perimeter

S_A = Area

r = radius of inscribed circle

R = radius of circumscribed circle