Area of an Obtuse Triangle Calculator

What Is an Obtuse Triangle?

An obtuse triangle is a triangle that has one interior angle greater than 90 degrees. Because the three interior angles of any triangle must sum to exactly 180 degrees, a triangle can have at most one obtuse angle. The other two angles are always acute (less than 90 degrees). For example, a triangle with angles of 120, 40, and 20 degrees is obtuse because 120 degrees is greater than 90 degrees.

Obtuse triangles look "flattened" or tilted compared to acute or right-angled triangles. A key practical point: if you draw the perpendicular height from the vertex opposite a side adjacent to the obtuse angle, that height falls outside the triangle rather than inside it. The area formula still works correctly regardless.

How to Calculate the Area of an Obtuse Triangle

Method 1: Base and Height

The simplest formula for any triangle's area is:

Area = 0.5 x base x height

The height must be the perpendicular distance from the chosen base to the opposite vertex. For an obtuse triangle, if you pick a side adjacent to the obtuse angle as your base, the height foot falls outside the triangle. It is generally easier to choose the side opposite the obtuse angle as the base, in which case the height falls inside the triangle.

Worked example: base = 8 cm, perpendicular height = 5 cm.
Area = 0.5 x 8 x 5 = 20 cm²

Method 2: Two Sides and the Included Angle (SAS)

If you know two sides and the angle between them, use:

Area = 0.5 x a x b x sin(C)

where a and b are the two side lengths and C is the angle between them. For an obtuse triangle, C may be the obtuse angle itself (greater than 90 degrees). The sine function handles angles above 90 degrees correctly; sin(120°) = sin(60°) = approximately 0.866.

Worked example: a = 7 cm, b = 9 cm, C = 120°.
Area = 0.5 x 7 x 9 x sin(120°) = 0.5 x 7 x 9 x 0.8660 = 27.28 cm²

Method 3: Heron's Formula (Three Sides)

When you know all three side lengths but no angle or height, use Heron's formula:

1. Calculate the semi-perimeter: s = (a + b + c) / 2
2. Calculate the area: Area = √(s × (s - a) × (s - b) × (s - c))

The formula works for any triangle. To confirm the triangle is obtuse, check whether the square of the longest side is greater than the sum of squares of the other two sides.

Worked example: a = 6 cm, b = 8 cm, c = 11 cm.
s = (6 + 8 + 11) / 2 = 12.5
Area = √(12.5 × 6.5 × 4.5 × 1.5) = √548.4375 = 23.42 cm²
(Check: 11² = 121 > 6² + 8² = 100, so the triangle is indeed obtuse.)

When Is a Triangle Obtuse? The Side-Length Test

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• General Triangle Calculator: sides, angles, area, and perimeter.

Sources and method: Standard Euclidean geometry. Area formula A = 0.5bh is established in Euclid's Elements. SAS area formula A = 0.5ab sin(C) follows from the definition of the sine ratio. Heron's formula is attributed to Hero of Alexandria (circa 60 AD). All three formulas are consistent with each other and produce identical results for the same triangle.

This calculator is for educational purposes. Results depend on the accuracy of the measurements you enter. For very small or very large triangles, precision in your input values will significantly affect the area result. The obtuse classification shown is based on the inputs provided.