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Triangle Calculator

Enter any three valid values and get every side, every angle, the area and a drawing of the triangle.

Known values

Three sides.

Try:

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

Area

6 cm²
a = 3 cmb = 4 cmc = 5 cmA36.87°B53.13°C90°

Sides and angles

Side a
3 cm
Side b
4 cm
Side c
5 cm
Angle A
36.8699°
Angle B
53.1301°
Angle C
90°

Measures

Area (Heron's formula)
6 cm²
Area (½ ab sin C)
6 cm²
Perimeter
12 cm
Heights (on a, b, c)
4, 3, 2.4 cm
Medians (to a, b, c)
4.272, 3.6056, 2.5 cm
Inradius r
1 cm
Circumradius R
2.5 cm
Type
right, scalene

Solving the triangle: step by step

  1. 1Check the triangle inequality
    a+b>c,  a+c>b,  b+c>aa+b>c,\; a+c>b,\; b+c>aa+b>c,a+c>b,b+c>a
    ↓
    3+4>5,  3+5>4,  4+5>33+4>5,\; 3+5>4,\; 4+5>33+4>5,3+5>4,4+5>3

    All three hold, so a triangle exists.

  2. 2Law of cosines for A
    cos⁡A=b2+c2−a22bc\cos A=\dfrac{b^2+c^2-a^2}{2bc}cosA=2bcb2+c2−a2​
    ↓
    A=arccos⁡ ⁣(42+52−322⋅4⋅5)=36.8699∘A=\arccos\!\left(\dfrac{4^2+5^2-3^2}{2\cdot 4\cdot 5}\right)=36.8699^\circA=arccos(2⋅4⋅542+52−32​)=36.8699∘
  3. 3Law of cosines for B
    cos⁡B=a2+c2−b22ac\cos B=\dfrac{a^2+c^2-b^2}{2ac}cosB=2aca2+c2−b2​
    ↓
    B=arccos⁡ ⁣(32+52−422⋅3⋅5)=53.1301∘B=\arccos\!\left(\dfrac{3^2+5^2-4^2}{2\cdot 3\cdot 5}\right)=53.1301^\circB=arccos(2⋅3⋅532+52−42​)=53.1301∘
  4. 4Angles add to a straight angle
    C=180∘−A−BC=180^\circ-A-BC=180∘−A−B
    ↓
    C=180∘−36.8699∘−53.1301∘=90∘C=180^\circ-36.8699^\circ-53.1301^\circ=90^\circC=180∘−36.8699∘−53.1301∘=90∘

Area, heights and circles: step by step

  1. 1Semi-perimeter
    s=a+b+c2s=\dfrac{a+b+c}{2}s=2a+b+c​
    ↓
    s=3+4+52=6s=\dfrac{3+4+5}{2}=6s=23+4+5​=6
  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=6(3)(2)(1)=6K=\sqrt{6(3)(2)(1)}=6K=6(3)(2)(1)​=6
  3. 3Area from two sides and the angle between them
    K=12absin⁡CK=\tfrac12 ab\sin CK=21​absinC
    ↓
    K=12⋅3⋅4⋅sin⁡(90∘)=6K=\tfrac12\cdot 3\cdot 4\cdot \sin(90^\circ)=6K=21​⋅3⋅4⋅sin(90∘)=6

    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=4,  hb=3,  hc=2.4h_a=4,\; h_b=3,\; h_c=2.4ha​=4,hb​=3,hc​=2.4
  5. 5Medians
    ma=122b2+2c2−a2m_a=\tfrac12\sqrt{2b^2+2c^2-a^2}ma​=21​2b2+2c2−a2​
    ↓
    ma=4.272,  mb=3.6056,  mc=2.5m_a=4.272,\; m_b=3.6056,\; m_c=2.5ma​=4.272,mb​=3.6056,mc​=2.5
  6. 6Inscribed circle
    r=Ksr=\dfrac{K}{s}r=sK​
    ↓
    r=66=1r=\dfrac{6}{6}=1r=66​=1
  7. 7Circumscribed circle
    R=abc4KR=\dfrac{abc}{4K}R=4Kabc​
    ↓
    R=3⋅4⋅54⋅6=2.5R=\dfrac{3\cdot 4\cdot 5}{4\cdot 6}=2.5R=4⋅63⋅4⋅5​=2.5
  1. Tools
  2. Mathematics
  3. Triangle Calculator

About Triangle Calculator

This triangle calculator is free, needs no sign-up and runs in your browser, so the numbers you type are never uploaded. Choose which three values you know (three sides, two sides and the angle between them, two angles and a side, or two sides and a non-included angle) and it solves the rest with the law of cosines and the law of sines, showing each step.

Besides the missing sides and angles it reports the area two ways (Heron’s formula and one half a b sin C), the perimeter, the three heights, the three medians, the inscribed and circumscribed circle radii, and whether the triangle is acute, right or obtuse and scalene, isosceles or equilateral. A scaled drawing shows the result with every side and angle labelled.

The SSA case is handled honestly: when the data fits two different triangles, both are shown, each with its own steps and diagram. A separate right triangle mode solves from any two values, such as two legs or a hypotenuse and an acute angle.

How to use Triangle 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 formulas the calculator uses

Sides are written a, b, c and the angle opposite each is A, B, C. The law of cosines finds a side from two sides and the included angle, or an angle from three sides. The law of sines links each side to the sine of its opposite angle.

Heron’s formula gives the area from the three sides alone, with s as the semi-perimeter.

c² = a² + b² − 2ab cos C
a / sin A = b / sin B = c / sin C
s = (a + b + c) / 2
Area = √(s(s − a)(s − b)(s − c)) = ½ ab sin C
r = Area / s     R = abc / (4 · Area)

Worked examples

  • SSS: a = 3, b = 4, c = 5 → gives A = 36.87°, B = 53.13°, C = 90°

    A right triangle with area 6, perimeter 12, inradius 1 and circumradius 2.5.

  • SSS: a = 13, b = 14, c = 15 → gives Area = 84

    Heron’s formula: s = 21, so the area is √(21 · 8 · 7 · 6) = 84.

  • SAS: a = 5, C = 40°, b = 7 → gives c = 4.5141, A = 45.4°, B = 94.6°

    The law of cosines gives the third side, then the angles.

  • SSA: a = 6, b = 8, A = 30° → gives Two triangles: B = 41.81° or 138.19°

    Side a is shorter than b but longer than b sin A = 4, so c = 11.4003 or c = 2.4561.

  • Right triangle: hypotenuse 10, A = 30° → gives a = 5, b = 8.6603

    Opposite leg = c sin A and adjacent leg = c cos A.

Which mode for which data

ModeYou knowSolutions
SSSThree sidesOne, if the triangle inequality holds
SASTwo sides and the angle between themAlways one
ASATwo angles and the side between themOne, if the angles add to less than 180°
AASTwo angles and a side opposite one of themOne, if the angles add to less than 180°
SSATwo sides and an angle opposite one of themNone, one or two
RightAny two values with a 90° angleOne

Good to know

The ambiguous case (SSA)

With two sides and an angle that is not between them, the side opposite the angle can sometimes swing to two positions. If side a is shorter than side b, compare a with the height h = b sin A. When a is less than h no triangle exists, when a equals h the triangle is a right triangle, and when a is between h and b there are two triangles. If a is at least as long as b, only one triangle fits.

The calculator lists both triangles in the two-solution case so you can pick the one that matches your drawing or problem.

Limits and precision

Results are computed in double-precision floating point and shown to four decimal places, so a displayed value can differ from an exact textbook answer in the last digit. Triangles with a very small angle are sensitive to rounding in the inputs.

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.

How do I solve a triangle with three sides?

Use the law of cosines to find each angle: cos A = (b² + c² − a²) / (2bc). Do the same for B, then C = 180° − A − B. The SSS mode shows these steps.

What is the ambiguous case of the law of sines?

It is the SSA situation, where two sides and a non-included angle can fit two different triangles. Because sin B = sin(180° − B), the angle B can be acute or obtuse. The calculator shows both when both are valid.

Can a triangle have two obtuse angles?

No. The three angles add to 180°, so at most one angle can be 90° or more. In SSA the second solution is rejected when it would break this rule.

Which area formula should I use?

Use Heron’s formula when you know all three sides, and one half a b sin C when you know two sides and the angle between them. The calculator shows both, and they agree.

What are the medians and heights of a triangle?

A height is the perpendicular distance from a vertex to the opposite side. A median joins a vertex to the midpoint of the opposite side. The three medians meet at the centroid.

Does it work in radians?

Yes. Switch the angle unit to radians. Inputs and outputs, including the steps, then use radians, and the numbers you already typed are converted.

Why does it say no triangle exists?

Either the sides break the triangle inequality (one side is at least the sum of the other two), the angles add to 180° or more, or in SSA the side opposite the angle is shorter than the height it must reach.

Popular Triangle Calculator pages

  • 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 Sines CalculatorEnter two angles and a side to find the others with the law of sines.
  • 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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