Obtuse Triangle Calculator
Enter any combination of sides and angles to solve a complete obtuse triangle. This calculator supports five solve modes - SSS, SAS, ASA, AAS and SSA - and returns all three sides, all three angles, the area, perimeter, three altitudes, three medians, the inradius and the circumradius. The "Show your work" panel walks through each formula step by step. Choose from metric or imperial length units; all outputs update instantly.
What is an obtuse triangle?
An obtuse triangle is a triangle in which exactly one interior angle exceeds 90 degrees. The other two angles must each be acute (less than 90 degrees) because all three angles must sum to 180 degrees. The side opposite the obtuse angle is always the longest side of the triangle. Common everyday examples include the wide triangular gable of a low-pitched roof, many obtuse slices of pie, and the shape formed by an open compass set past 90 degrees.
How to solve an obtuse triangle
You need at least three pieces of information - including at least one side length - to fully determine a triangle. The five standard 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 non-adjacent side), and SSA (two sides and an angle not between them). SSS and SAS use the Law of Cosines to find the first unknown angle or side, then the Law of Sines completes the rest. ASA and AAS use the Law of Sines throughout after deriving the third angle from the 180-degree sum. SSA is the ambiguous case: it can yield zero, one, or two valid triangles depending on the relative lengths.
Key formulas: Law of Cosines and Law of Sines
The Law of Cosines - c² = a² + b² - 2ab cos(C) - generalises the Pythagorean theorem and works for any triangle type. It is the primary tool for SSS and SAS problems. The Law of Sines states that a / sin(A) = b / sin(B) = c / sin(C), linking each side to the sine of its opposite angle. Once two angles are known, the third follows from the 180-degree sum, and any remaining side can be found with a single Law of Sines ratio.
Area, altitudes, medians, inradius and circumradius
Area is most efficiently computed with Heron's formula when all three sides are known: Area = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2 is the semi-perimeter. Each altitude (height) equals twice the area divided by the corresponding base: ha = 2A / a. The three medians each connect a vertex to the midpoint of the opposite side; the median to side a is ma = 0.5 × sqrt(2b² + 2c² - a²). The inradius - radius of the inscribed circle - is r = Area / s. The circumradius - radius of the circumscribed circle - is R = abc / (4 × Area). For an obtuse triangle, the circumcenter (center of the circumscribed circle) lies outside the triangle, on the same side as the longest side.
Triangle types by angles
| Type | Largest angle | All angles condition | Key property |
|---|---|---|---|
| Acute | Below 90° | All three angles < 90° | Orthocenter inside triangle |
| Right | Exactly 90° | One angle = 90° | Pythagorean theorem applies |
| Obtuse | Above 90° | One angle > 90°, two angles < 90° | Circumcenter outside triangle |
How any triangle is classified based on its largest interior angle.
Frequently asked questions
What makes a triangle obtuse?
A triangle is obtuse when exactly one of its interior angles is greater than 90 degrees. Because the three angles must sum to 180 degrees, having one angle above 90 degrees forces the other two to be acute. You cannot have two obtuse angles in the same triangle.
How do I find the missing side of an obtuse triangle?
Use the Law of Cosines if you know two sides and the angle between them (SAS): c² = a² + b² - 2ab cos(C). Use the Law of Sines if you know two angles and one side (ASA or AAS): divide each side by the sine of its opposite angle and set the ratios equal. This calculator handles all five solve modes automatically.
Can I use the Pythagorean theorem on an obtuse triangle?
No, not directly. The Pythagorean theorem (a² + b² = c²) applies only to right triangles. For obtuse triangles, the equivalent relationship is a² + b² < c² (where c is the longest side, opposite the obtuse angle). The generalised version - the Law of Cosines - applies to any triangle including obtuse ones.
What is the SSA ambiguous case and does it affect obtuse triangles?
SSA means you know two sides and an angle that is not between those sides. This configuration can have zero, one, or two valid triangle solutions depending on the values. For an obtuse triangle, if the angle given is obtuse, there can be at most one solution because a triangle cannot have two obtuse angles. This calculator returns the principal solution; if a second solution exists for acute input angles, consider checking both.
Where is the circumcenter of an obtuse triangle?
The circumcenter - the center of the circle passing through all three vertices - lies outside the triangle for any obtuse triangle. It falls on the opposite side of the longest side from the opposite vertex. This is in contrast to acute triangles, where the circumcenter lies inside, and right triangles, where it lies exactly at the midpoint of the hypotenuse.
How is altitude calculated for an obtuse triangle?
For an obtuse triangle, two of the three altitudes fall outside the triangle itself because their foot points lie on extensions of the sides rather than on the sides directly. Despite this, all three altitudes are still real and positive lengths. The formula is always the same: altitude to side a equals twice the area divided by the length of a.