Online Calculator Resource

# Law of Sines Calculator

Law of Sines Calculator

Primary Equation:

$$A = \sin^{-1} \left[ \dfrac{a \sin B}{b} \right]$$

Sides:
a =
b =
c =

Angles:
A =
B =
C =

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

## This Calculation Equation & Triangle

$$A = \sin^{-1} \left[ \dfrac{a \sin B}{b} \right]$$

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

Some calculation choices are redundant but are 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.

## 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:

$$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$$

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

$$A = \sin^{-1} \left[ \dfrac{a \sin B}{b} \right]$$
$$A = \sin^{-1} \left[ \dfrac{a \sin C}{c} \right]$$
$$B = \sin^{-1} \left[ \dfrac{b \sin A}{a} \right]$$
$$B = \sin^{-1} \left[ \dfrac{b \sin C}{c} \right]$$
$$C = \sin^{-1} \left[ \dfrac{c \sin A}{a} \right]$$
$$C = \sin^{-1} \left[ \dfrac{c \sin B}{b} \right]$$

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

$$a = \dfrac{b \sin A}{\sin B}$$
$$a = \dfrac{c \sin A}{\sin C}$$
$$b = \dfrac{a \sin B}{\sin A}$$
$$b = \dfrac{c \sin B}{\sin C}$$
$$c = \dfrac{a \sin C}{\sin A}$$
$$c = \dfrac{b \sin C}{\sin B}$$

## 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)

Weisstein, Eric W. "Law of Sines" From MathWorld-- A Wolfram Web Resource. Law of Sines.

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

Cite this content, page or calculator as:

Furey, Edward "Law of Sines Calculator"; from http://www.calculatorsoup.com - Online Calculator Resource.