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Obtuse Triangle Calculator

Calculate area, all angles, and perimeter of any obtuse triangle from three side lengths using Heron's formula and the Law of Cosines.

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What Is an Obtuse Triangle?

An obtuse triangle is a triangle containing exactly one interior angle greater than 90 degrees. Because all three interior angles of any triangle must sum to 180 degrees, only one angle can be obtuse. The side opposite the obtuse angle is always the longest side of the triangle, satisfying the Pythagorean inequality a² + b² < c², where c is the longest side. According to Wolfram MathWorld, a triangle qualifies as obtuse whenever its largest angle θ falls strictly between 90° and 180°.

Core Formulas Used by This Calculator

Verifying the Obtuse Condition

Before computing any triangle property, confirm the triangle is genuinely obtuse. With sides a, b, and c where c is the longest, apply the test: a² + b² < c². If this inequality holds, the triangle is obtuse. If a² + b² = c², the triangle is right-angled. If a² + b² > c², the triangle is acute. The calculator automatically performs this check and returns an error for invalid side combinations.

Finding the Obtuse Angle with the Law of Cosines

The Law of Cosines, detailed by Texas A&M University Open Resources (Section 8.5), is the standard method for solving oblique (non-right) triangles when all three sides are known: cos C = (a² + b² − c²) / (2ab). Because a² + b² < c² in every obtuse triangle, the numerator is always negative, so cos C < 0 and angle C must exceed 90°. Applying the inverse cosine function (arccos) yields the exact angle measure in degrees.

Computing the Remaining Acute Angles

With angle C established, angles A and B are found using the Law of Cosines: cos A = (b² + c² − a²) / (2bc) and cos B = (a² + c² − b²) / (2ac). Both resulting angles must be positive and less than 90°. As a verification: A + B + C = 180° exactly.

Area via Heron's Formula

Heron's formula, presented in Triangles and Vectors (ScholarWorks@GVSU), computes triangle area from three side lengths alone without requiring any angle measurement. First calculate the semi-perimeter: s = (a + b + c) / 2. Then compute: Area = √(s · (s − a) · (s − b) · (s − c)). This formula applies to any valid triangle regardless of its type.

Perimeter and Altitudes

The perimeter is P = a + b + c. Each altitude (perpendicular height) from a vertex to its opposite base equals h = 2 · Area / base. Note that for obtuse triangles, the altitudes from the two vertices holding acute angles fall outside the triangle—a geometric property that distinguishes obtuse triangles from acute ones.

Worked Example

Consider sides a = 6, b = 8, c = 12:

  • Obtuse check: 6² + 8² = 100 < 144 = 12² → confirmed obtuse.
  • Semi-perimeter: s = (6 + 8 + 12) / 2 = 13
  • Area: √(13 × 7 × 5 × 1) = √455 ≈ 21.33 square units
  • Angle C: cos C = (36 + 64 − 144) / 96 = −44/96 ≈ −0.4583 → C ≈ 117.3°
  • Angle A: cos A = (64 + 144 − 36) / 192 ≈ 0.8958 → A ≈ 26.4°
  • Angle B: 180° − 117.3° − 26.4° = 36.3°
  • Perimeter: 6 + 8 + 12 = 26 units

Real-World Applications

Obtuse triangles appear throughout engineering, architecture, and navigation. Wide, shallow-pitch roof trusses form obtuse triangular profiles. Land surveyors use the Law of Cosines to compute distances across irregular terrain where direct measurement is impossible. In physics, vector resolution produces obtuse triangular configurations when two forces act at angles exceeding 90° to one another. Civil engineers also rely on obtuse triangle geometry when analyzing asymmetric bridge bracing and frame structures.

Reference

Frequently asked questions

How do you determine if a triangle is obtuse from its three side lengths?
Identify the longest side c and test the inequality a squared plus b squared less than c squared. For example, sides 5, 7, and 10 satisfy 25 + 49 = 74, which is less than 100, so the triangle is obtuse. If the sum equals c squared the triangle is right-angled, and if it exceeds c squared the triangle is acute. This single inequality is the definitive obtuse test.
What formula is used to find the obtuse angle in an obtuse triangle?
The Law of Cosines finds the obtuse angle: cos C = (a squared + b squared minus c squared) divided by (2ab), where c is the longest side. Because a squared + b squared is less than c squared in any obtuse triangle, the numerator is always negative, so cos C is negative. Applying the inverse cosine (arccos) to a negative value always returns an angle strictly between 90 and 180 degrees, which is the obtuse angle.
How is the area of an obtuse triangle calculated using Heron's formula?
First compute the semi-perimeter: s = (a + b + c) / 2. Then apply Heron's formula: Area = square root of s times (s minus a) times (s minus b) times (s minus c). For sides 6, 8, and 12, s equals 13 and the area equals the square root of 455, approximately 21.33 square units. The formula requires only side lengths, with no angle measurement needed.
Can an obtuse triangle also be isosceles?
Yes. An isosceles triangle has two equal sides, and if those equal sides are shorter than the third side the triangle can be obtuse. Sides of length 5, 5, and 8 form an obtuse isosceles triangle because 5 squared plus 5 squared equals 50, which is less than 64, confirming obtuse. The obtuse angle opposite the side of length 8 equals arccos of negative 0.28, approximately 106.3 degrees, while each base angle measures about 36.9 degrees.
What is the difference between an obtuse triangle and an acute triangle?
An obtuse triangle has exactly one angle greater than 90 degrees, while an acute triangle has all three angles less than 90 degrees. For sides a, b, and c where c is longest, the obtuse condition is a squared plus b squared less than c squared, and the acute condition is a squared plus b squared greater than c squared. Geometrically, altitudes from the two acute vertices of an obtuse triangle fall outside the triangle, whereas all three altitudes of an acute triangle fall inside it.
Why is the longest side always opposite the obtuse angle?
In any triangle, the longest side is opposite the largest angle. Because the obtuse angle exceeds 90 degrees, it is automatically the largest angle in the triangle; the remaining two angles must each be less than 90 degrees so all three sum to 180 degrees. This side-angle relationship, a fundamental theorem of triangle geometry, guarantees that the side facing the obtuse angle is always longer than both other sides.