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Law of Sines Calculator

Enter two angles and a side to find the others with the law of sines.

Known values

Two angles and a side opposite one of them.

Try:

Sides a, b, c sit opposite angles A, B, C. In right mode the right angle is at C.

Area

68.3013 cm²
a = 10 cmb = 14.142 cmc = 19.319 cmA30°B45°C105°

Sides and angles

Side a
10 cm
Side b
14.1421 cm
Side c
19.3185 cm
Angle A
30°
Angle B
45°
Angle C
105°

Measures

Area (Heron's formula)
68.3013 cm²
Area (½ ab sin C)
68.3013 cm²
Perimeter
43.4607 cm
Heights (on a, b, c)
13.6603, 9.6593, 7.0711 cm
Medians (to a, b, c)
16.1741, 13.6603, 7.5299 cm
Inradius r
3.1431 cm
Circumradius R
10 cm
Type
obtuse, scalene

Solving the triangle: step by step

  1. 1Angles add to a straight angle
    C=180∘−A−BC=180^\circ-A-BC=180∘−A−B
    ↓
    C=180∘−30∘−45∘=105∘C=180^\circ-30^\circ-45^\circ=105^\circC=180∘−30∘−45∘=105∘
  2. 2Law of sines for b
    bsin⁡B=asin⁡A\dfrac{b}{\sin B}=\dfrac{a}{\sin A}sinBb​=sinAa​
    ↓
    b=10sin⁡45∘sin⁡30∘=14.1421b=\dfrac{10\sin 45^\circ}{\sin 30^\circ}=14.1421b=sin30∘10sin45∘​=14.1421
  3. 3Law of sines for c
    csin⁡C=asin⁡A\dfrac{c}{\sin C}=\dfrac{a}{\sin A}sinCc​=sinAa​
    ↓
    c=10sin⁡105∘sin⁡30∘=19.3185c=\dfrac{10\sin 105^\circ}{\sin 30^\circ}=19.3185c=sin30∘10sin105∘​=19.3185

Area, heights and circles: step by step

  1. 1Semi-perimeter
    s=a+b+c2s=\dfrac{a+b+c}{2}s=2a+b+c​
    ↓
    s=10+14.1421+19.31852=21.7303s=\dfrac{10+14.1421+19.3185}{2}=21.7303s=210+14.1421+19.3185​=21.7303
  2. 2Heron's formula
    K=s(s−a)(s−b)(s−c)K=\sqrt{s(s-a)(s-b)(s-c)}K=s(s−a)(s−b)(s−c)​
    ↓
    K=21.7303(11.7303)(7.5882)(2.4118)=68.3013K=\sqrt{21.7303(11.7303)(7.5882)(2.4118)}=68.3013K=21.7303(11.7303)(7.5882)(2.4118)​=68.3013
  3. 3Area from two sides and the angle between them
    K=12absin⁡CK=\tfrac12 ab\sin CK=21​absinC
    ↓
    K=12⋅10⋅14.1421⋅sin⁡(105∘)=68.3013K=\tfrac12\cdot 10\cdot 14.1421\cdot \sin(105^\circ)=68.3013K=21​⋅10⋅14.1421⋅sin(105∘)=68.3013

    Both methods give the same area.

  4. 4Heights
    ha=2Ka,  hb=2Kb,  hc=2Kch_a=\dfrac{2K}{a},\; h_b=\dfrac{2K}{b},\; h_c=\dfrac{2K}{c}ha​=a2K​,hb​=b2K​,hc​=c2K​
    ↓
    ha=13.6603,  hb=9.6593,  hc=7.0711h_a=13.6603,\; h_b=9.6593,\; h_c=7.0711ha​=13.6603,hb​=9.6593,hc​=7.0711
  5. 5Medians
    ma=122b2+2c2−a2m_a=\tfrac12\sqrt{2b^2+2c^2-a^2}ma​=21​2b2+2c2−a2​
    ↓
    ma=16.1741,  mb=13.6603,  mc=7.5299m_a=16.1741,\; m_b=13.6603,\; m_c=7.5299ma​=16.1741,mb​=13.6603,mc​=7.5299
  6. 6Inscribed circle
    r=Ksr=\dfrac{K}{s}r=sK​
    ↓
    r=68.301321.7303=3.1431r=\dfrac{68.3013}{21.7303}=3.1431r=21.730368.3013​=3.1431
  7. 7Circumscribed circle
    R=abc4KR=\dfrac{abc}{4K}R=4Kabc​
    ↓
    R=10⋅14.1421⋅19.31854⋅68.3013=10R=\dfrac{10\cdot 14.1421\cdot 19.3185}{4\cdot 68.3013}=10R=4⋅68.301310⋅14.1421⋅19.3185​=10
  1. Tools
  2. Mathematics
  3. Triangle Calculator
  4. Law of Sines Calculator

About Law of Sines Calculator

This law of sines calculator solves triangles where you know two angles and a side opposite one of them. It applies a / sin A = b / sin B = c / sin C and shows each substitution. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

The law of sines is the right tool for AAS and ASA. For SSA it can give two answers, so switch to the SSA mode of this calculator to see both.

How to use Law of Sines Calculator

  1. 1Pick the mode that matches what you know: SSS, SAS, ASA, AAS, SSA or Right triangle.
  2. 2Type the three values in the boxes. Sides a, b, c sit opposite angles A, B, C. In right mode fill exactly two boxes.
  3. 3Choose a length unit and whether angles are in degrees or radians. Changing the angle unit converts the numbers you already typed.
  4. 4Read the sides, angles, area and other measures, and compare them with the scaled diagram.
  5. 5Open the step-by-step working to see each law of cosines or law of sines line, then Heron’s formula for the area.
  6. 6In SSA mode, check whether one solution, two solutions or none appear, and read the note that explains why.

Features

  • Six modes: SSS, SAS, ASA, AAS, SSA and right triangle.
  • Both solutions shown in the ambiguous SSA case, each with a diagram and its own working.
  • Area by Heron’s formula and by one half a b sin C, which agree.
  • Perimeter, three heights, three medians, inradius and circumradius.
  • Triangle type by angles (acute, right, obtuse) and by sides (scalene, isosceles, equilateral).
  • A scaled, labelled SVG diagram that follows light and dark mode.
  • Degrees or radians, and eight length units from millimetres to miles.
  • Clear messages when the values cannot form a triangle, instead of a wrong answer.
  • Worked examples and KaTeX formulas for every step.

The law of sines

Each side divided by the sine of its opposite angle is the same for all three sides.

a / sin A = b / sin B = c / sin C = 2R

Worked examples

  • A = 30°, B = 45°, a = 10 → gives b = 14.1421, c = 19.3185

    C = 105°.

Privacy: how it works

  • Everything is calculated by code running in your browser tab. The values you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.

Tips

  • Check the labelling: side a is opposite angle A. Swapping a side with its neighbour gives a different triangle.
  • In SAS the angle must be the one between the two sides. If it is not, use the SSA mode.
  • If two angles already add to 180° or more, no triangle exists. The calculator says so.
  • For a right triangle you can also use SSS with the sides 3, 4 and 5 and read C = 90°.

Frequently asked questions

Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Does it run in my browser?

Yes. Your values are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Your values are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

When do I use the law of sines?

When you know a side and its opposite angle plus one more angle or side. For SAS and SSS use the law of cosines.

What is the 2R in the law of sines?

The common ratio equals twice the circumradius, the radius of the circle through the three vertices.

More Triangle Calculator pages

  • Triangle CalculatorEnter any three valid values and get every side, every angle, the area and a drawing of the triangle.
  • SSS Triangle CalculatorEnter three sides and get all three angles, the area and a scaled drawing.
  • SAS Triangle CalculatorEnter two sides and the angle between them to find the third side and the other angles.
  • ASA Triangle CalculatorEnter two angles and the side between them to get the other two sides and the third angle.
  • AAS Triangle CalculatorEnter two angles and a non-included side to get every other side and angle.
  • SSA Triangle CalculatorEnter two sides and a non-included angle. The calculator shows every triangle that fits.
  • Right Triangle CalculatorEnter any two values of a right triangle and get the legs, the hypotenuse, both acute angles and the area.
  • Area of a Triangle CalculatorEnter three sides to get the area by Heron’s formula, with the heights and the steps.
  • Heron’s Formula CalculatorEnter the three sides and see the semi-perimeter, the product and the area worked out.
  • Law of Cosines CalculatorEnter two sides and the angle between them to find the third side and the other angles.
  • Isosceles Triangle CalculatorEnter the two equal sides and the base to get the angles, the height and the area.
  • Equilateral Triangle CalculatorEnter the side length to get the height, the area and both circle radii.

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