About

An obtuse triangle contains exactly one interior angle exceeding 90°. This geometric constraint means standard right-triangle shortcuts fail. Miscalculating the area of such a triangle propagates errors into structural load estimates, land surveys, and material cost projections. This calculator applies validated formulas - Heron's, SAS (12 ⋅ a ⋅ b ⋅ sinC), and base-height - while enforcing the obtuse constraint and the triangle inequality theorem. Results assume Euclidean plane geometry. The tool approximates to 6 decimal places; for geodetic-scale triangles on curved surfaces, corrections for Earth's curvature apply.

Formulas

The primary formula for computing triangle area from three known sides uses Heron's Formula. First compute the semi-perimeter s:

s = a + b + c2

Then the area A is:

A = √s⋅(s − a)⋅(s − b)⋅(s − c)

When two sides and the included angle are known (SAS method):

A = 12 ⋅ a ⋅ b ⋅ sin(C)

The simplest base-height method:

A = 12 ⋅ b ⋅ h

To verify a triangle is obtuse, check via the cosine rule. If the largest side is c:

c² > a² + b² &implies; obtuse

Where a, b, c are side lengths, C is the included angle in degrees, h is the perpendicular height to base b, and s is the semi-perimeter.

Reference Data

Triangle Type	Angle Condition	Side Relationship	Area Formula	Example Area
Right	One angle = 90°	c² = a² + b²	12 ⋅ a ⋅ b	6 u² (3,4,5)
Equilateral	All angles = 60°	a = b = c	√34 ⋅ a²	43.30 u² (a=10)
Isosceles Acute	All < 90°, two equal	a = b ≠ c	Heron's or SAS	24 u² (5,5,6)
Isosceles Obtuse	One > 90°, two equal sides	a = b, c² > a² + b²	Heron's or SAS	11.98 u² (5,5,9)
Scalene Acute	All < 90°	All sides different	Heron's	29.33 u² (7,8,9)
Scalene Obtuse	One > 90°	c² > a² + b²	Heron's	19.90 u² (4,7,10)
Obtuse (120°)	One angle = 120°	c² = a² + b² + a⋅b	SAS preferred	21.65 u² (5,10,120°)
Nearly Degenerate	One angle → 179°	Longest side ≈ sum of others	Area → 0	≈0 u²
Common Obtuse	100°	Sides 6, 8, 12	Heron's	23.62 u²
Common Obtuse	110°	Sides 5, 7, 11	Heron's	16.50 u²
Common Obtuse	95°	Sides 9, 12, 16	SAS	53.82 u²
Common Obtuse	135°	Sides 4, 6	SAS	8.49 u²
Surveying Example	105°	Sides 50m, 80m	SAS	1931.85 m²

Frequently Asked Questions

For side-based methods, it applies the cosine rule: if the square of the longest side exceeds the sum of squares of the other two sides (c² > a² + b²), the triangle is obtuse. For angle-based methods, it checks that exactly one angle exceeds 90° and the sum of all angles equals 180°. A warning is displayed if the inputs form a valid triangle that is not obtuse.

Heron's formula yields a real result only when the three sides satisfy the triangle inequality: every side must be strictly less than the sum of the other two. If a + b ≤ c, the expression under the radical becomes zero or negative, producing NaN. This calculator validates the triangle inequality before computation and reports the specific violated constraint.

No. In an obtuse triangle, the altitude drawn from a vertex of an acute angle falls outside the triangle. The foot of the height lands on the extension of the opposite side beyond the triangle boundary. This is why the base-height method requires the perpendicular distance to the base line (extended if necessary), not just the distance to the visible side segment.

When angles approach 180°, Heron's formula suffers from catastrophic cancellation in floating-point arithmetic because s − c approaches 0. The SAS formula (12absinC) is more numerically stable for near-degenerate cases, as sin of small supplementary angles remains well-conditioned.

No. The sum of interior angles in a Euclidean triangle is exactly 180°. If two angles each exceeded 90°, their sum alone would exceed 180°, violating this constraint. An obtuse triangle has exactly one angle in the open interval (90°, 180°) and two acute angles.

Thermal expansion alters physical side lengths. Steel expands at approximately 12 μm/m/°C. For a 10m beam at 40°C above reference, the length increases by 4.8mm. For precision surveying or structural work, measure at reference temperature or apply correction factors before entering values into the calculator.