Obtuse Triangle Calculator

What Is an Obtuse Triangle?

An obtuse triangle (also known as an obtuse-angled triangle) is a triangle in which exactly one interior angle measures greater than 90 degrees (> 90° and < 180°). Because the sum of all three interior angles in any Euclidean triangle always equals 180 degrees, the remaining two angles must be strictly acute (< 90°).

Together with acute triangles, obtuse triangles make up the family of oblique triangles (triangles that contain no 90-degree right angle).

How to Identify an Obtuse Triangle

You can determine whether a triangle is obtuse using either its angles or its side lengths:

• By Angles: If any single interior angle $\theta$ satisfies $90^\circ < \theta < 180^\circ$, the triangle is obtuse.
• By Sides (Converse of the Pythagorean Theorem): Let $c$ be the longest side, and let $a$ and $b$ be the two shorter sides. If $c^2 > a^2 + b^2$, the triangle is obtuse. If $c^2 = a^2 + b^2$, it is a right triangle, and if $c^2 < a^2 + b^2$, it is an acute triangle.

Key Formulas for Obtuse Triangles

Obtuse triangles follow standard trigonometric and geometric relations:

1. Law of Cosines

To solve for an unknown side or angle when given two sides and the included angle (SAS) or three sides (SSS):

$$c^2 = a^2 + b^2 - 2ab \cos(\gamma)$$

$$\cos(\gamma) = \frac{a^2 + b^2 - c^2}{2ab}$$

When $\gamma > 90^\circ$, $\cos(\gamma)$ is negative, which means $-2ab\cos(\gamma)$ is positive, explaining why $c^2 > a^2 + b^2$.

2. Law of Sines

Relates side lengths to the sines of their opposite angles:

$$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} = 2R$$

where $R$ is the circumradius of the triangle.

3. Area of an Obtuse Triangle

You can calculate area using several methods depending on available measurements:

• Trigonometric Area: $\text{Area} = \frac{1}{2}ab \sin(\gamma)$
• Heron's Formula: $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$ is the semi-perimeter.
• Base and Altitude: $\text{Area} = \frac{1}{2} \times \text{base} \times h$. Note that for the two shorter sides of an obtuse triangle, the altitude falls outside the triangle on the extended base line.

4. Inradius and Circumradius

The inscribed circle radius $r$ and circumscribed circle radius $R$ are given by:

$$r = \frac{\text{Area}}{s} = \frac{\sqrt{s(s-a)(s-b)(s-c)}}{s}$$

$$R = \frac{abc}{4 \times \text{Area}}$$

In an obtuse triangle, the circumcenter and the orthocenter always lie outside the triangle perimeter.

Step-by-Step Example

Suppose a triangle has side lengths $a = 5\text{ cm}$, $b = 7\text{ cm}$, and $c = 10\text{ cm}$.

1. Verify Triangle Inequality: $5 + 7 = 12 > 10$, so it forms a valid triangle.
2. Check Obtuse Condition: $a^2 + b^2 = 5^2 + 7^2 = 25 + 49 = 74$. Since $c^2 = 10^2 = 100 > 74$, the triangle is obtuse.
3. Calculate Angle $\gamma$: $\cos(\gamma) = \frac{5^2 + 7^2 - 10^2}{2(5)(7)} = \frac{74 - 100}{70} = \frac{-26}{70} \approx -0.3714 \implies \gamma \approx 111.80^\circ$.
4. Calculate Semi-Perimeter: $s = \frac{5 + 7 + 10}{2} = 11\text{ cm}$.
5. Calculate Area: $\text{Area} = \sqrt{11(11-5)(11-7)(11-10)} = \sqrt{11 \times 6 \times 4 \times 1} = \sqrt{264} \approx 16.248\text{ cm}^2$.