Scalene Triangle Calculator
Enter any combination of sides and angles to solve a scalene triangle completely. Choose your input mode - three sides (SSS), two sides and the included angle (SAS), two angles and the side between them (ASA), two angles and a non-included side (AAS), or two sides and a non-included angle (SSA). The calculator returns all three sides, all three angles, the area, perimeter, three altitudes, inradius, circumradius and medians, with a step-by-step worked solution.
What is a scalene triangle?
A scalene triangle is a triangle in which all three sides have different lengths and, as a direct consequence, all three interior angles are also different. This distinguishes it from an equilateral triangle (all three sides equal) and an isosceles triangle (exactly two sides equal). Scalene triangles are the most general type of triangle and can be acute (all angles below 90 degrees), right (one angle exactly 90 degrees) or obtuse (one angle above 90 degrees). The classic 30-60-90 right triangle is a well-known example of a right scalene triangle, and the 3-4-5 Pythagorean triple is another.
How to solve a scalene triangle
You need exactly three independent pieces of information to uniquely determine a triangle. The five standard input combinations are: SSS (all three sides), SAS (two sides and the angle between them), ASA (two angles and the side between them), AAS (two angles and a side not between them), and SSA (two sides and an angle not between them - the ambiguous case that can produce zero, one or two solutions). The Law of Cosines handles SSS and SAS directly. The Law of Sines is used for ASA, AAS and SSA. Once all sides and angles are known, Heron's formula gives the area, and simple ratios give the altitudes, inradius, circumradius and medians.
Key formulas for a scalene triangle
Law of Cosines: a^2 = b^2 + c^2 - 2bc cos(A). Law of Sines: a / sin(A) = b / sin(B) = c / sin(C). Heron's area formula: first compute the semiperimeter s = (a + b + c) / 2, then Area = sqrt(s(s-a)(s-b)(s-c)). Altitude to side a: ha = 2 x Area / a. Inradius: r = Area / s. Circumradius: R = (a x b x c) / (4 x Area). Median to side a: ma = 0.5 x sqrt(2b^2 + 2c^2 - a^2). These formulas apply to any scalene triangle regardless of whether it is acute, right or obtuse.
The SSA ambiguous case
SSA (two sides and a non-included angle) is called the ambiguous case because knowing sides a and b and angle A can sometimes yield two distinct triangles, one triangle, or no triangle at all. If sin(B) = b sin(A) / a is greater than 1, no triangle exists. If it equals exactly 1, there is one right triangle. If it is less than 1, there may be one or two solutions: the first uses angle B = arcsin(sin(B)), and the second uses the supplement B' = 180 - B. This calculator returns the principal (smaller-B) solution. If the geometry of your problem requires the second solution, subtract the returned angle B from 180 degrees and re-run using AAS mode.
Triangle types by angles
| Type | Largest angle | All sides | Example angles |
|---|---|---|---|
| Acute scalene | Less than 90 deg | All different | 50 deg, 60 deg, 70 deg |
| Right scalene | Exactly 90 deg | All different | 30 deg, 60 deg, 90 deg |
| Obtuse scalene | Greater than 90 deg | All different | 20 deg, 45 deg, 115 deg |
Every scalene triangle falls into one of these three angle-based sub-types.
Frequently asked questions
What makes a triangle scalene?
A triangle is scalene when all three sides have different lengths. Because the longest side is always opposite the largest angle, and the shortest side is opposite the smallest angle, having all sides different automatically means all three interior angles are different too. No two sides or angles can be equal in a scalene triangle.
Can a scalene triangle have a right angle?
Yes. A right scalene triangle has one 90-degree angle and two acute angles that are different from each other, which means all three angles are different and all three sides (including the hypotenuse as the longest) are different. The classic 3-4-5 triangle and the 30-60-90 triangle are both right scalene triangles.
Why does the SSA input mode have an ambiguous case?
When you know two sides (a and b) and the angle opposite the shorter side (A), swinging side a like a compass arc can sometimes intersect the opposite base line at two different points, giving two valid triangles. This only happens when side b is longer than side a and A is acute. The calculator detects whether zero, one or two solutions exist and returns the principal solution with a status message.
What is the inradius and how is it useful?
The inradius r is the radius of the largest circle that fits entirely inside the triangle, touching all three sides. It equals Area divided by the semiperimeter: r = Area / s. If you are designing a physical object - a triangular tile, bracket or frame - the inradius tells you the size of the largest circular hole or fitting that will sit inside without touching the sides.
How is the circumradius different from the inradius?
The circumradius R is the radius of the circle that passes through all three vertices of the triangle. It equals (a x b x c) / (4 x Area). The circumradius is always larger than the inradius (R is at least twice r for any triangle). It is useful in navigation, surveying and engineering when you need a circle that encloses a triangular region.
How do I find the area without knowing the height?
If you know all three side lengths you can use Heron's formula: compute the semiperimeter s = (a + b + c) / 2, then Area = sqrt(s(s-a)(s-b)(s-c)). If you know two sides and the included angle you can use Area = 0.5 x a x b x sin(C). Both methods give the same result for a given triangle.