03 // Bidirectional STEM Calculators // MATHa / sin(A) = b / sin(B)

Law of Sines Triangle Proportion Calculator

The Law of Sines states that in any planar triangle, the ratio of the length of a side to the sine of its opposite interior angle is constant and equal to the circumdiameter 2R: a / sin(A) = b / sin(B).

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

Pick which value you want to calculate (or type into any field):Solving for: b — Side b

Side bCalculated

Side length opposite angle B

0.110,000.00

Side a

Side length opposite angle A

0.110,000.00

Opposite Angle A

Interior angle opposite side a in degrees

1178

deg

Opposite Angle B

Interior angle opposite side b in degrees

1178

deg

Calculated Result · Side b (b)

14.1421m

Converted units:

Step-by-Step Calculation

01Canonical Law: a / sin(A) = b / sin(B)

02Given Inputs (Base Units): a = 10 m, A (°) = 30, B (°) = 45

03Solve for b (Side b): b = 14.1421 m

Real-World Reference Values

Circumcircle Diameter (2R)2R = a / sin(A) = 20.0

Diameter of the circle passing through all 3 vertices

30°-45° Special Ratiosin(30°)=0.5, sin(45°)=√2/2

b = a × √2 ≈ 1.41421 × a

Triangle Angle SumC = 180° - (A + B)

Third interior angle in Euclidean plane

20-Row Comparison Table

Side b (b) · Side a (a)

Compare across:

Law of Sines Triangle Proportion Calculator — Side a → Side b
#	Side a (m)	Side b (m)	Δ %
01	10 mCurrent	14.1421 m	0.00%
02	72.9888 m	103.2217 m	+629.89%
03	145.8775 m	206.3019 m	+1358.78%
04	218.7663 m	309.3822 m	+2087.66%
05	291.655 m	412.4625 m	+2816.55%
06	364.5438 m	515.5427 m	+3545.44%
07	437.4325 m	618.623 m	+4274.33%
08	510.3213 m	721.7032 m	+5003.21%
09	583.21 m	824.7835 m	+5732.10%
10	656.0988 m	927.8638 m	+6460.99%
11	728.9875 m	1,030.94 m	+7189.88%
12	801.8763 m	1,134.02 m	+7918.76%
13	874.765 m	1,237.10 m	+8647.65%
14	947.6538 m	1,340.18 m	+9376.54%
15	1,020.54 m	1,443.27 m	+10105.43%
16	1,093.43 m	1,546.35 m	+10834.31%
17	1,166.32 m	1,649.43 m	+11563.20%
18	1,239.21 m	1,752.51 m	+12292.09%
19	1,312.10 m	1,855.59 m	+13020.98%
20	1,384.99 m	1,958.67 m	+13749.86%

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Frequently Asked Questions

What is the geometric meaning of the constant ratio a / sin(A)?+

The ratio a / sin(A) equals 2R, the exact diameter of the circumscribed circle passing through the triangle's three vertices.

What is the SSA ambiguous case?+

When solving for an angle B using arcsin, because sin(180° - B) = sin(B), an obtuse supplement 180° - B may also be valid if A + (180° - B) < 180°.

How do I find the third side c?+

First compute the third angle C = 180° - (A + B), then set B = C in the solver to obtain side c.