Area of Obtuse Triangle Calculator — Three Sides
Find the area of an obtuse triangle (one angle greater than 90°) from its three side lengths. The calculator uses Heron's formula and the law of cosines to compute the area, all three angles, and the heights — and it flags whether the triangle is truly obtuse.
Computed with Heron's formula from three side lengths
- 1
Semi-perimeter
(5 + 6 + 9) ÷ 2 = 10Half the perimeter — the key intermediate in Heron's formula. - 2
Product inside the root
10 × 5 × 4 × 1 = 200 - 3
Area
√200 = 14.1421
How does this calculator work?
Area of an obtuse triangle from three sides: s = (a+b+c)/2, Area = √[s(s−a)(s−b)(s−c)] (Heron's formula). The obtuse angle is opposite the longest side — confirmed when a²+b² < c². Angles follow from the law of cosines: cos(C) = (a²+b²−c²)/(2ab).
Formula
How this is calculated
An obtuse triangle is a triangle in which one interior angle is greater than 90°. The obtuse angle is always opposite the longest side — so if c is the longest side, angle C is obtuse and the condition a² + b² < c² holds. All other triangle properties (area, perimeter, heights) are computed exactly the same way as for any triangle.
Given three side lengths, the area is found with Heron's formula: first compute the semi-perimeter s = (a+b+c)/2, then Area = √[s(s−a)(s−b)(s−c)]. This formula requires no angles and works for any valid triangle. The three angles are recovered separately via the law of cosines: cos(A) = (b²+c²−a²)/(2bc), and similarly for B and C. The height from each side then follows from Area = ½ × side × height.
The only restriction is the triangle inequality: each side must be strictly less than the sum of the other two. The calculator checks this and warns if the inputs do not form a valid triangle. If all three computed angles turn out to be below 90°, the triangle is acute rather than obtuse — the area result is still correct, and the calculator notes this.
Frequently asked questions
A triangle is obtuse when one angle exceeds 90°. Given side lengths, check whether a²+b² < c² for the longest side c — if so, angle C is obtuse. This calculator computes all three angles and tells you which, if any, is obtuse.
In an obtuse triangle the altitude from the vertex of the obtuse angle falls outside the triangle (the foot of the perpendicular lands on an extension of the base), making it awkward to measure the height directly. Heron's formula uses only the three known side lengths and avoids this problem entirely.
Yes — Heron's formula and the law of cosines work for any valid triangle. The calculator flags when the triangle is not actually obtuse, but the area and angle results are correct regardless.
Also known as
TG we-Calculate Editorial Team. (2026). Area of Obtuse Triangle Calculator — Three Sides [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/area-obtuse-triangle-calculator
TG we-Calculate Editorial Team. "Area of Obtuse Triangle Calculator — Three Sides." TG we-Calculate. 2026. https://we-calculate.com/calculator/area-obtuse-triangle-calculator.
TG we-Calculate Editorial Team, "Area of Obtuse Triangle Calculator — Three Sides," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/area-obtuse-triangle-calculator
@misc{wecalculate_area_obtuse_triangle_calculator, title = {Area of Obtuse Triangle Calculator — Three Sides}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/area-obtuse-triangle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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