Solve obtuse triangles from 3 sides or 2 sides + angle. Detect the obtuse angle, classify, find all properties including altitude through the obtuse vertex, and compare with a right triangle.
An obtuse triangle is a triangle that has exactly one angle greater than 90°. The remaining two angles are both acute (less than 90°), and together with the obtuse angle they sum to 180°. Obtuse triangles appear everywhere — from roof pitches with low slope to the shadow geometry of leaning structures.
Given three side lengths a, b, c, you can determine whether the triangle is obtuse by checking if the square of the longest side exceeds the sum of squares of the other two: if c² > a² + b², the angle opposite c is obtuse. The angles themselves are found via the law of cosines: cos C = (a² + b² − c²)/(2ab). When cos C < 0, angle C is obtuse.
Obtuse triangles have unique geometric properties. The circumcenter (center of the circumscribed circle) lies outside the triangle, on the opposite side of the longest edge from the obtuse vertex. The altitude from the obtuse vertex is internal, but the altitudes from the two acute vertices land outside the triangle (their feet lie on extensions of the opposite sides). The orthocenter is also external.
This calculator accepts either three sides or two sides plus an included angle. It validates the input, classifies the triangle as obtuse (or indicates if it's actually acute or right), computes all six elements (3 sides, 3 angles), finds the area, perimeter, all three altitudes, inradius, circumradius, and the medians. It highlights which altitudes are external, compares with a corresponding right triangle, and includes presets and a reference table for common obtuse triangles.
Obtuse Triangle problems often require several dependent steps, and a small arithmetic slip can propagate through every derived value. This calculator is tailored to that workflow: you enter side a (value), side b (value), side c (value), and it returns angle a, angle b, angle c, area in one consistent pass. It is useful for homework checks, worksheet generation, tutoring walkthroughs, and fast field/design estimates where you need reliable geometry results without rebuilding the full derivation each time.
Law of cosines: c² = a² + b² − 2ab·cos C Angle: C = arccos((a²+b²−c²)/(2ab)) Obtuse test: c² > a² + b² (longest side c) Area (Heron): A = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2 Area (SAS): A = ½ab·sin C Altitude: h_a = 2A/a Inradius: r = A/s Circumradius: R = a/(2·sin A)
Result: Obtuse angle C ≈ 130.54°, Area ≈ 11.40, Perimeter = 21
Sides 5, 6, 10: c² = 100 > 25 + 36 = 61, so obtuse. Angle C = arccos((25+36−100)/(2·5·6)) = arccos(−0.65) ≈ 130.54°. Area via Heron: s = 10.5, A = √(10.5·5.5·4.5·0.5) ≈ 11.40.
This obtuse triangle tool links the entered values (side a (value), side b (value), side c (value), included angle c (°)) to the target geometry relationships used in class and practice problems. Instead of solving each intermediate step manually, you can validate setup and arithmetic quickly while still tracing which measurements drive the final result.
Formula focus: the calculator formula
Obtuse Triangle shows up in school geometry, technical drafting, construction layout checks, and early engineering design estimates. When values are changed repeatedly, the calculator helps you compare scenarios quickly and see how sensitive the shape is to each dimension.
Start with the primary outputs (angle a, angle b, angle c, area) and then use the remaining cards/tables to confirm consistency with your diagram. Keep units consistent across inputs, and round only at the end if your assignment or project specifies a fixed precision.
A triangle is obtuse if exactly one of its angles is greater than 90°. Equivalently, the square of the longest side is greater than the sum of squares of the other two sides.
No. The three angles must sum to 180°, so at most one angle can exceed 90°.
In an obtuse triangle, the circumcenter lies outside the triangle, on the opposite side of the longest edge from the obtuse vertex. Use this as a practical reminder before finalizing the result.
When the foot of an altitude lands on the extension of a side (not between the two vertices), the altitude is external. This happens for altitudes from acute vertices in obtuse triangles.
Use Heron's formula or ½ab sin C — both work perfectly for obtuse triangles. The sine of an obtuse angle is still positive.
Yes. When the longest side satisfies c² = a² + b² exactly, the triangle is right (90°). Greater → obtuse; less → acute.