Area of an Obtuse Triangle Calculator
This calculator finds the area of an obtuse triangle using whichever measurements you have. Pick the method that matches your data: base and height, two sides and the included angle (SAS), all three sides (Heron's formula), or two angles and the side between them (AAS). The result updates instantly, with a step-by-step breakdown and a labelled triangle diagram.
Formula
Worked example
Triangle with sides 10, 17 and 25 units: s = (10 + 17 + 25) / 2 = 26. Area = sqrt(26 x 16 x 9 x 1) = sqrt(3744) = 61.19 sq units. Because 25^2 = 625 > 10^2 + 17^2 = 389, the longest side confirms the triangle is obtuse.
What makes a triangle obtuse?
A triangle is obtuse when one of its interior angles is greater than 90 degrees. Because the three angles of any triangle must sum to exactly 180 degrees, at most one angle can exceed 90 degrees. The side opposite the obtuse angle is always the longest side of the triangle. A key visual feature is that the perpendicular height drawn from the vertex of the obtuse angle to the base falls outside the triangle rather than inside it, which is why the base-height formula can feel counterintuitive at first glance. Despite this, the area formula (Area = 1/2 x base x height) is identical to the one used for acute and right triangles.
Four ways to find the area of an obtuse triangle
The method you use depends on which measurements you already have. If you know the base and the perpendicular height, Area = 1/2 x base x height is the simplest route. If you have two sides and the angle between them, the SAS formula Area = 1/2 x a x b x sin(gamma) avoids needing the height. If you have all three side lengths but no angles, Heron's formula computes the area from the semi-perimeter alone. If you know two angles and the side between them, the AAS formula Area = c^2 x sin(alpha) x sin(beta) / (2 x sin(gamma)) handles the rest, where gamma = 180 - alpha - beta.
How to verify a triangle is obtuse
Given three sides a, b and c, label the longest side c. The triangle is obtuse when c^2 > a^2 + b^2 (by the converse of the Pythagorean theorem). If c^2 = a^2 + b^2, it is a right triangle, and if c^2 < a^2 + b^2, it is acute. Given angles, simply check whether any angle is larger than 90 degrees. This calculator performs that check automatically and reports the triangle type alongside the area.
Understanding the perpendicular height in an obtuse triangle
The perpendicular height (altitude) is the shortest distance from a vertex to the line containing the opposite side. For an acute triangle, all three altitudes fall inside the figure. For an obtuse triangle, the two altitudes drawn from the acute vertices land on extensions of the sides, not on the sides themselves, so they appear to fall outside the triangle when drawn. The altitude from the obtuse vertex, however, always lands inside the triangle. The base-height formula uses whichever base-altitude pair is convenient, regardless of whether the foot of the altitude is inside or outside the boundary.
Which formula to use
| Known measurements | Formula | Notes |
|---|---|---|
| Base (b) and height (h) | Area = 1/2 x b x h | Height must be perpendicular to the base |
| Two sides (a, b) and included angle (gamma) | Area = 1/2 x a x b x sin(gamma) | gamma is the angle between a and b |
| Three sides (a, b, c) | Area = sqrt( s(s-a)(s-b)(s-c) ), s = (a+b+c)/2 | Heron's formula, no angle needed |
| Two angles (alpha, beta) and included side (c) | Area = c^2 x sin(alpha) x sin(beta) / (2 x sin(gamma)) | gamma = 180 - alpha - beta |
Choose the row that matches the measurements you have available.
Frequently asked questions
Is the area formula different for an obtuse triangle?
No. The formula Area = 1/2 x base x height works for every triangle regardless of whether it is acute, right, or obtuse. The only subtlety is that for an obtuse triangle the perpendicular height may fall outside the triangle when drawn, but its length is still measured perpendicularly from the base line, and the calculation is identical.
What is Heron's formula and when should I use it?
Heron's formula calculates the area of a triangle from its three side lengths without needing any angle measurement. You compute the semi-perimeter s = (a + b + c) / 2, then Area = sqrt(s x (s - a) x (s - b) x (s - c)). Use it when you know all three sides but not the height or any angle. It works equally well for acute, right, and obtuse triangles.
How do I find the area if I have two sides and the angle between them?
Use the SAS formula: Area = 1/2 x a x b x sin(gamma), where a and b are the two known sides and gamma is the angle between them. For an obtuse triangle the included angle will be greater than 90 degrees at least at one vertex, and sin(gamma) handles this correctly because the sine function returns a positive value for all angles between 0 and 180 degrees.
Can I calculate the area if I only know the angles?
Not without at least one side length. Angles alone determine the shape of the triangle (all similar triangles share the same angles) but not its size. You need at least one side to anchor the scale. The AAS method uses two angles plus the side between them to recover the area.
How can I tell if a triangle is obtuse from its sides alone?
Label the longest side c. The triangle is obtuse when c^2 > a^2 + b^2, right when c^2 = a^2 + b^2, and acute when c^2 < a^2 + b^2. For example, sides 10, 17 and 25: 25^2 = 625 and 10^2 + 17^2 = 389, so 625 > 389 confirms the triangle is obtuse.
What units does this calculator use?
The calculator works with any consistent unit. If you enter side lengths in centimetres, the area will be in square centimetres. If you use metres, you get square metres. Just make sure all inputs are in the same unit and the output area will match.