Obtuse Triangle Calculator

Enter three side lengths to check if the triangle is obtuse and calculate all properties with step-by-step solutions.

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Step-by-Step Solution

Understanding Obtuse Triangles

An obtuse triangle is a triangle in which one of the angles measures more than 90 degrees. This is in contrast to acute triangles (all angles less than 90 degrees) and right triangles (one angle exactly 90 degrees). Every triangle has exactly 180 degrees in total, so only one angle can be obtuse.

How to Identify an Obtuse Triangle

Given three sides a, b, and c (where c is the longest side), the triangle is obtuse if and only if:

Obtuse Triangle Test

If the square of the longest side is greater than the sum of squares of the other two sides:

c² > a² + b²

Triangle Inequality

Three sides form a valid triangle only if the sum of any two sides exceeds the third.

a + b > c, a + c > b, b + c > a

Area (Heron's Formula)

Area using semi-perimeter s = (a + b + c) / 2.

A = sqrt(s(s-a)(s-b)(s-c))

Key Formulas

Angles Using the Law of Cosines

Each angle can be found using the law of cosines. For angle A opposite side a:

cos(A) = (b² + c² - a²) / (2bc)

Similarly for angles B and C. The angle whose cosine is negative is the obtuse angle.

Height of the Triangle

The height relative to a given base can be calculated from the area: h = 2A / base. For an obtuse triangle, the foot of the altitude from the obtuse-angled vertex falls inside the triangle, while altitudes from the other vertices may fall outside.

Properties of Obtuse Triangles

  • Exactly one angle is greater than 90 degrees and less than 180 degrees.
  • The longest side is always opposite the obtuse angle.
  • The circumcenter lies outside the triangle.
  • The orthocenter also lies outside the triangle.
  • The sum of the two acute angles is less than 90 degrees.

Real-World Applications

Obtuse triangles appear frequently in architecture, engineering, and navigation. Roof trusses, sail designs, and land survey plots often contain obtuse triangles. Understanding their properties is essential for structural analysis, area computation, and trigonometric problem-solving.