What Defines an Obtuse Triangle?
An obtuse triangle is characterised by having one interior angle greater than 90° (the obtuse angle) and two interior angles that each measure less than 90° (the acute angles). All three angles must sum to 180°, which is true for every triangle.
Obtuse triangles belong to the broader category of oblique triangles, alongside their counterpart: acute triangles, where all three angles fall below 90°. Right triangles form a separate classification with one 90° angle. Understanding these distinctions matters when solving geometry problems or designing structures.
The obtuse angle is always the largest interior angle. If you know any two angles, the third is trivial to find using the angle sum property.
Area Formulas for Obtuse Triangles
Four primary approaches exist for calculating the area, depending on which measurements you know. Select the formula that matches your available data.
Area = ½ × base × height
Area = ½ × a × b × sin(C)
Area = ¼ × √[(a + b + c) × (−a + b + c) × (a − b + c) × (a + b − c)]
Area = ½ × a × b × sin(β) × sin(γ) / sin(β + γ)
a, b, c— The three side lengths of the trianglebase— The side chosen as the reference baseheight— The perpendicular distance from the base to the opposite vertexC, β, γ— Interior angles in degrees or radianssin— The sine trigonometric function
Determining If Your Triangle Is Obtuse
Inspection of the angles is the most direct method. If any single interior angle exceeds 90°, the triangle is obtuse—verification complete.
When only two angles are known, compute the missing angle:
Third angle = 180° − Angle₁ − Angle₂
If the result exceeds 90°, your triangle is obtuse. If all three computed angles fall below 90°, it is acute.
Side lengths alone can also reveal the triangle type. For an obtuse triangle with sides a, b, and c (where c is the longest side), the relationship c² > a² + b² holds true. Conversely, acute triangles satisfy c² < a² + b².
Common Pitfalls When Working With Obtuse Triangles
Pay attention to these practical considerations to avoid errors.
- Height measurement can be tricky — In an obtuse triangle, the altitude from the obtuse angle vertex to the opposite side may extend outside the triangle's boundary (the foot of the perpendicular lies beyond the base segment). Always measure the perpendicular distance carefully, not along a side.
- Angle sum verification saves time — Before using area formulas, confirm your three angles sum to exactly 180°. Measurement or rounding errors compound quickly; catching this early prevents cascading calculation mistakes.
- Trigonometric functions require correct angle format — Ensure angles are in the same units (degrees or radians) throughout your calculation. Many calculator errors stem from mixing units without conversion. The sine function behaves identically, but numerical output will be nonsensical if units are inconsistent.
- Heron's formula applies universally — The square-root formula works for any non-degenerate triangle, including obtuse ones, and requires only the three side lengths. It avoids trigonometry altogether and is especially useful when angles are unknown or difficult to measure.
Using the Obtuse Triangle Calculator
Input whichever combination of measurements suits your situation. The tool adapts to:
- Three angles (to identify triangle type)
- Base and height (simplest area calculation)
- Two sides and the included angle
- All three side lengths (Heron's method)
- Two angles, one side, and computed area
The calculator instantly classifies your triangle, confirms whether it is obtuse, and delivers the area. No manual angle arithmetic or formula lookup is necessary—focus on your problem while the tool handles the algebra.