Team
math

Area of Obtuse Triangle Calculator

An obtuse triangle has one interior angle exceeding 90°, making its longest side opposite that wide angle. Whether you have two sides and the included angle, all three side lengths, or just base and height, this calculator finds the area instantly. Engineers, architects, and geometry students rely on multiple formulas depending on available measurements.

Last updated:

Creators Anna Szczepanek, PhD Mathematics
5,301 people find this calculator helpful

Understanding Obtuse Triangles

An obtuse triangle is defined by a single interior angle greater than 90°. This distinguishes it fundamentally from acute triangles, where all angles remain below 90°.

  • One angle exceeds 90° — the obtuse angle — while the other two angles are acute.
  • The longest side always lies opposite the obtuse angle.
  • Can be scalene or isosceles — with either all different sides or two equal sides. An equilateral triangle (all equal sides) cannot be obtuse.
  • Real-world examples include roof trusses, land plots, and structural reinforcements where angles exceed right angles.

The key distinction for calculation is knowing which measurements you possess. Different data sets require different area formulas.

Area Formulas for Obtuse Triangles

Four primary methods exist for finding area, each suited to different input scenarios:

Base × Height Method:

Area = 0.5 × b × h

Three Sides (Heron's Formula):

Area = 0.25 × √[(a + b + c)(−a + b + c)(a − b + c)(a + b − c)]

Two Sides and Included Angle (SAS):

Area = 0.5 × a × b × sin(γ)

Two Angles and Included Side (ASA):

Area = 0.5 × a² × sin(β) × sin(γ) ÷ sin(β + γ)

  • b — Base length
  • h — Perpendicular height from base to opposite vertex
  • a, b, c — All three side lengths
  • γ (gamma) — Angle between sides a and b
  • β (beta), γ (gamma) — Two known angles
  • a — Side between the two known angles

Common Pitfalls When Calculating Obtuse Triangle Area

Avoid these frequent mistakes when applying obtuse triangle formulas.

  1. Height must be perpendicular — When using base × height, the height must be a perpendicular line from the base to the opposite vertex. In obtuse triangles, this perpendicular often falls outside the triangle itself. Measure or calculate it carefully, as slanting heights give incorrect results.
  2. Angle measurement affects SAS calculations — The SAS formula (0.5 × a × b × sin(γ)) requires the angle γ between your two known sides. Using an angle not between those sides will produce a completely different and wrong area. Always verify which angle sits between your measured sides.
  3. Heron's formula demands all three sides — The three-side method requires exact side lengths. Even small measurement errors propagate through the square root operation. When sides are 10, 17, and 25 inches, a measurement error of 1 inch can shift the calculated area by several square inches.
  4. ASA formula requires the included side — For two angles and a side, the side must be between your two angles, not opposite either one. Confusing which side lies between your angles is a common source of error in this less-intuitive formula.

Step-by-Step Calculation Examples

Example 1: Three sides of 10, 17, and 25 inches

Using Heron's formula:

  • Sum: 10 + 17 + 25 = 52
  • Calculate: (52)(42)(28)(27) = 1,668,312
  • Square root: √1,668,312 ≈ 1,291.64
  • Multiply by 0.25: 0.25 × 1,291.64 = 61.19 in²

Example 2: Two angles 97° and 34°, with 13 inches between them

  • Apply ASA formula: area = 0.5 × 13² × sin(97°) × sin(34°) ÷ sin(131°)
  • Calculate: 0.5 × 169 × 0.9903 × 0.5592 ÷ 0.7547
  • Result: 62.14 in²

Both examples show how different inputs lead to accurate areas using the appropriate formula.

Frequently Asked Questions

What makes a triangle obtuse rather than acute or right-angled?

A triangle becomes obtuse when one of its three interior angles exceeds 90 degrees. The other two angles must sum to less than 90° to keep the total at 180°. A right triangle has exactly one 90° angle, while an acute triangle has all angles below 90°. The obtuse angle determines which side is longest — always the one opposite that wide angle. This distinction matters for real-world applications like roof design, where knowing whether a structure is obtuse or acute affects load distribution.

Why does the perpendicular height fall outside an obtuse triangle?

In an obtuse triangle, when you extend the base as a line, the perpendicular from the opposite vertex drops onto that extended line rather than the base segment itself. This occurs because the vertex is positioned such that a true perpendicular cannot land within the base. The perpendicular height is still valid mathematically — you measure it as the straight-line distance from the vertex to the extended base line. Always measure or mark this perpendicular carefully, as it's a different value than any side length.

Can an equilateral or right triangle ever be obtuse?

No. An equilateral triangle has all angles at 60°, making it acute. A right triangle has one angle at exactly 90°, leaving no room for an angle exceeding 90°. Only scalene triangles (all sides different) or isosceles triangles (two sides equal) can be obtuse. This constraint comes from the 180° angle sum rule — if one angle exceeds 90°, the remaining two must total less than 90°, preventing equal angles.

Which formula should I use if I have two sides but no angle between them?

If you have two sides but the angle between them is unknown, you'll need the third side to use Heron's formula. The SAS method (two sides and included angle) specifically requires the angle between those two sides. If you have two sides and a non-included angle, you may be able to solve for the third side using the Law of Cosines first, then apply Heron's formula. Otherwise, insufficient data exists for a unique area calculation.

How accurate is Heron's formula for obtuse triangles?

Heron's formula is mathematically exact for all triangle types, including obtuse. The accuracy depends entirely on your side measurements. If you measure sides to the nearest 0.1 inch, your calculated area is reliable to several decimal places. However, measurement errors compound through the formula — small errors in side lengths amplify when summed and multiplied. Always double-check your side measurements, especially for applications where precision matters, like land surveying or structural engineering.

What if I only know the base and height of an obtuse triangle?

The simplest formula applies: multiply base by height and divide by two. This works regardless of where the height falls relative to the base. Even though the perpendicular may extend beyond the triangle visually, the calculation 0.5 × base × height remains valid. This method bypasses angle measurements entirely, making it ideal when you have direct measurements of the perpendicular distance from base to opposite vertex.

More math calculators (see all)

Interval Notation Calculator Percentage Increase Calculator Gamma Function Calculator Lowest Common Denominator Calculator Simplify Cube Root Calculator Slope Percentage Calculator Power Set Calculator Right Triangle Side and Angle Calculator