Find the missing side of a triangle: right triangles from two sides, or any triangle from SSS, SAS, ASA/AAS or SSA, with all angles, area and perimeter.
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Triangle Side Formula
The formula the calculator uses depends on what you already know about the triangle. Each solve-for mode targets a different combination of sides and angles.
For a right triangle, the missing side comes from the Pythagorean theorem:
c^2 = a^2 + b^2
When you know two sides and the angle between them (SAS), the third side comes from the law of cosines:
c = sqrt( a^2 + b^2 - 2*a*b*cos(C) )
When you know one side with two angles (ASA or AAS) or two sides and a non-included angle (SSA), the remaining sides come from the law of sines:
a / sin(A) = b / sin(B) = c / sin(C)
When you know all three sides (SSS), the angles come from the rearranged law of cosines:
cos(A) = (b^2 + c^2 - a^2) / (2*b*c)
- a, b, c: the three side lengths. Each side is opposite the angle named with the matching capital letter.
- A, B, C: the three interior angles, entered in degrees or radians. They always add up to 180 degrees.
- c (right triangle): the hypotenuse, the longest side and the one opposite the 90-degree angle.
The task selector shows the required inputs. Right-triangle mode accepts both legs or a hypotenuse and one leg. SAS finds the side opposite the included angle; ASA/AAS scales the identified known side; SSA reports all valid missing-side lengths; SSS returns all angles plus area, perimeter and type. Each side and angle input has its own unit selector, which converts entered quantities. Choose the length result unit beside the inputs, and angle output units and precision under Result precision and angles.
Which Inputs Solve for a Side
For a general triangle, use a supported three-measurement case including a side. For a known right triangle, two sides suffice because the 90-degree angle is already known. Use the table to choose the matching task.
| You know | Case | Method | Result |
|---|---|---|---|
| 2 sides of a right triangle | Right | Pythagorean theorem | One missing side |
| 2 sides + included angle | SAS | Law of cosines | Third side |
| 1 side + 2 angles | ASA / AAS | Law of sines | Both other sides |
| 2 sides + non-included angle | SSA | Law of sines | 0, 1, or 2 triangles |
| 3 sides | SSS | Law of cosines | All three angles |
Any set of three sides must also satisfy the triangle inequality: the two shorter sides must add up to more than the longest side, or no triangle exists.
| Common right triangle | Sides (a, b, c) | Check |
|---|---|---|
| 3-4-5 | 3, 4, 5 | 9 + 16 = 25 |
| 5-12-13 | 5, 12, 13 | 25 + 144 = 169 |
| 8-15-17 | 8, 15, 17 | 64 + 225 = 289 |
| 7-24-25 | 7, 24, 25 | 49 + 576 = 625 |
Examples
Example 1: missing side of a right triangle. A right triangle has legs a = 6 and b = 8. The hypotenuse is c = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10. If instead you knew the hypotenuse c = 10 and one leg a = 6, the other leg would be b = sqrt(10^2 – 6^2) = sqrt(64) = 8.
Example 2: third side from two sides and the included angle (SAS). A triangle has a = 5, b = 7, and the included angle C = 60 degrees. Then c = sqrt(5^2 + 7^2 – 2 times 5 times 7 times cos(60)) = sqrt(25 + 49 – 70 times 0.5) = sqrt(74 – 35) = sqrt(39), which is about 6.24.
Frequently Asked Questions
How many measurements do I need to find a side? A known right triangle needs two sides. A general triangle needs a supported combination of three measurements including a side: two sides plus an angle, or one side plus two angles. SSS uses three known sides to find angles. Three angles alone fix shape without size.
Why did the SSA mode give me two answers? When you enter two sides and an angle that is not between them, the law of sines can produce two valid triangles, one with an acute angle and one with an obtuse angle. This is the ambiguous case. The calculator reports every triangle that satisfies the triangle inequality so you can choose the one that matches your problem.
Do I have to use degrees? No. Each input angle can use degrees or radians; changing its unit converts the current value. Select the angle result unit and maximum decimal places under Result precision and angles. Length inputs also convert when their units change. Results remain the last calculated snapshot until you select Calculate.
