Calculate a triangle’s missing sides, angles, area, and perimeter from the few measurements you already have.
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Triangle Formula
Choose the known input case: SSS, SAS, ASA/AAS or SSA. Enter only its labeled measurements, relabeling the triangle consistently if needed. The calculator solves the other sides and angles and reports area, perimeter, radii, medians, perpendicular heights and internal angle bisectors. SSA can have zero, one or two valid triangles.
When all three sides are known (SSS) or you have two sides and the included angle (SAS), it uses the law of cosines:
c^2 = a^2 + b^2 - 2*a*b*cos(C)
When you have a side with its opposite angle plus one more angle or side (ASA, AAS, or SSA), it uses the law of sines:
a / sin(A) = b / sin(B) = c / sin(C)
Area is found from whichever inputs are available, using Heron’s formula for three sides, the included-angle formula for two sides and an angle, or base times height:
Area = sqrt( s*(s-a)*(s-b)*(s-c) ), s = (a+b+c)/2
Area = (1/2)*a*b*sin(C) = (1/2)*base*height
The remaining properties come from the solved side lengths and the area:
Perimeter = a + b + c
Circumradius R = a / (2*sin(A)); Inradius r = Area / s
Median m_a = sqrt(2*b^2 + 2*c^2 - a^2) / 2
- a, b, c: the three side lengths, where each side is opposite the angle of the matching capital letter.
- A, B, C: the three interior angles, entered in degrees or radians, which always add to 180 degrees.
- s: the semi-perimeter, half of the perimeter.
- Area: the surface enclosed by the three sides.
- R: the circumradius, the radius of the circle passing through all three vertices.
- r: the inradius, the radius of the largest circle that fits inside the triangle.
- m_a: the median from vertex A to the midpoint of side a (medians m_b and m_c follow the same pattern).
The task selector shows only the required inputs. Choose the missing area, base or height explicitly; base and height alone do not determine a complete triangle. Right-triangle mode lets you choose the two known sides, with c as hypotenuse. Equilateral mode accepts a side, area or perimeter. Expand Result precision and angles for the maximum decimal places and angle result unit. Each input angle has its own degree/radian unit control, which converts an entered value when changed.
What You Need to Solve a Triangle
A triangle is fully determined only by certain combinations of three measurements. Use this to know what to enter and which method the calculator applies.
| You know | Case | Method used | Solvable? |
|---|---|---|---|
| 3 sides | SSS | Law of cosines | Yes, one triangle |
| 2 sides + included angle | SAS | Law of cosines | Yes, one triangle |
| 2 angles + any side | ASA / AAS | Law of sines | Yes, one triangle |
| 2 sides + non-included angle | SSA | Law of sines | Ambiguous, 0, 1, or 2 triangles |
| 3 angles only | AAA | None | No, shape only, size unknown |
The triangle inequality must also hold for any set of three sides: the two shorter sides must add up to more than the longest side, or no triangle exists.
| Triangle type | Defining property | Area shortcut |
|---|---|---|
| Equilateral | All sides equal, all angles 60 degrees | (sqrt(3)/4) * side^2 |
| Right | One angle is 90 degrees | (1/2) * leg1 * leg2 |
| Isosceles | Two sides and two angles equal | (1/2) * base * height |
| Scalene | All sides and angles different | Heron’s formula |
Examples
Example 1: three sides (SSS). A triangle has sides a = 6, b = 8, and c = 10. Find angle C and the area. Using the law of cosines, cos(C) = (6^2 + 8^2 – 10^2) / (2 times 6 times 8) = (36 + 64 – 100) / 96 = 0, so C = 90 degrees. The semi-perimeter is s = (6 + 8 + 10) / 2 = 12, and Heron’s formula gives Area = sqrt(12 times 6 times 4 times 2) = sqrt(576) = 24. The perimeter is 24.
Example 2: two angles and a side (AAS). A triangle has A = 40 degrees, B = 60 degrees, and side a = 8. Then C = 180 – 40 – 60 = 80 degrees. By the law of sines, b = a times sin(B) / sin(A) = 8 times sin(60) / sin(40) = about 10.78, and c = 8 times sin(80) / sin(40) = about 12.26. The perimeter is about 31.04.
Frequently Asked Questions
How many values do I need to enter? For a general triangle, choose a valid three-measurement case that includes a side. Three angles alone give shape without size. The separate right-triangle, equilateral and area tasks require only their displayed inputs; redundant or conflicting extra inputs are not used.
Why did I get two different triangles? When you enter two sides and an angle that is not between them (the SSA case), the law of sines can have two valid answers, one with an acute angle and one with an obtuse angle. This is the ambiguous case. The calculator reports each triangle that satisfies the triangle inequality so you can pick the one that matches your problem.
Can I switch between degrees and radians? Yes. Each angle input has a degree/radian selector; changing it converts the entered quantity. Choose the angle result unit separately under Result precision and angles. Results stay unchanged until you select Calculate, and very small or large nonzero values use scientific notation.
