Calculators > Geometry > Plane > (Trigonometry) > Law of Sines

Law of Sines Calculator

Using the Law of Sines Calculate:
Enter:
Side a =
Angle B =
Side b =
Angle Units in:
Length Units in:
Results Significant Figures
Answer:
Sides:
a =
b =
c =

Angles:
A =
B =
C =

Other:
P =
s =
K =
r =
R =
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This Calculation Equation & Triangle

law of sines angle A equation in terms of B's

Triangle Diagram with Angles A, B and C and sides opposite those angles a, b and c respectively
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

Law of Sines Calculator

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

Some calculation choices are redundent but included anyway for exact letter designations.

Calculation Methods

To calculate any angle, A, B or C, say B, enter the opposite side b then another angle-side pair such as A and a or C and c. The performed calculations follow the side side angle (SSA) method and only use the law of sines to complete calculations for other unknowns.

To calculate any side, a, b or c, say b, enter the opposite angle B and then another angle-side pair such as A and a or C and c. The performed calculations follow the angle angle side (AAS) method and only use the law of sines to complete calculations for other unknowns.

Base Triangle ABC.

Law of Sines

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 sines states:

equation for law of sines

Equations from Law of Sines solving for angles A, B, and C

law of sines angle A equation in terms of B's      law of sines angle A equation in terms of C's

law of sines angle B equation in terms of A's      law of sines angle B equation in terms of C's

law of sines angle C equation in terms of A's      law of sines angle C equation in terms of B's

Equations from Law of Sines solving for sides a, b, and c

law of sines side a equation in terms of B's      law of sines side a equation in terms of C's

law of sines side b equation in terms of A's      law of sines side b equation in terms of C's

law of sines side c equation in terms of A's      law of sines side c equation in terms of B's

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)

References/ Further Reading

Weisstein, Eric W. "Law of Sines" From MathWorld-- A Wolfram Web Resource. http://mathworld.wolfram.com/LawofSines.html

Zwillinger, Daniel (Editor-in-Chief). CRC Standard Mathematical Tables and Formulae, 31st Edition New York, NY: CRC Press, p. 512, 2003.

http://hyperphysics.phy-astr.gsu.edu/hbase/lcos.html

http://hyperphysics.phy-astr.gsu.edu/hbase/lsin.html