Obtuse Triangle Calculator - Sides, Angles, and Area

The obtuse triangle calculator decides whether a triangle is obtuse from its three side lengths, and returns the three interior angles, area, perimeter, and longest side in a single pass. The verdict compares the square of the longest side to the sum of the squares of the two shorter sides.

• • Classify a measured triangle: Take three measured side lengths from a sketch, a survey, or a worksheet and read off whether the result is obtuse, right, or acute without doing the law of cosines by hand.
• • Solve all three interior angles: Recover every interior angle in degrees from the three side inputs, so a downstream calculation that needs a specific angle does not have to run the law of cosines separately.
• • Get area and perimeter together: Pull the Heron's formula area and the side sum in the same pass, so coverage, fencing, and material takeoffs share one source of truth with the obtuse verdict.

An obtuse triangle has exactly one interior angle greater than 90 degrees, and the side opposite that obtuse angle is the longest of the three. The other two angles are acute. The verdict uses a strict greater-than rule against 90.

For the matching angle-based family where all three interior angles are strictly less than 90 degrees, the Acute Triangle Calculator takes the same three side inputs and returns the acute verdict alongside the angles, area, and perimeter.

The calculator reads the three side lengths, validates the triangle inequality, sorts the sides so the longest is in a known slot, and compares c squared with a squared plus b squared. The same three side lengths drive the law of cosines for the angles and Heron's formula for the area, so a single pass returns the verdict, the three angles, the longest side, the area, and the perimeter.

• a, b, c: the three side lengths, all in the same length unit, with the side labeled opposite its matching interior angle
• longest side c: the longest of the three sides; the interior angle opposite this side is the largest angle and the one the calculator uses for the obtuse verdict
• A, B, C: the three interior angles, in degrees, with A opposite side a, B opposite side b, and C opposite side c; they always sum to 180 degrees
• s: the semi-perimeter, equal to (a + b + c) / 2; used in Heron's formula to compute the area
• area: the area of the triangle, in square units that match the length unit used for the sides

The law of cosines is the workhorse for the angles. Each interior angle is the inverse cosine of a ratio built from the squares of the sides, with the cosine clamped before the inverse cosine to keep floating-point rounding inside the valid range.

According to Wolfram MathWorld, the law of cosines gives every interior angle from the three side lengths using a^2 = b^2 + c^2 - 2bc cos(A)

When the squares test lands on equality, c^2 = a^2 + b^2, the verdict is right, and the Right Triangle Calculator can be used to solve the same three sides using the Pythagorean theorem and the SOH-CAH-TOA ratios.

These four ideas sit behind the verdict and the area.

The squares test is the verdict, the longest side is what makes the test fail, the law of cosines turns three sides into three angles, and Heron's formula turns the same three sides into the area.

When the input set is two sides and the obtuse included angle rather than three full sides, the Area Obtuse Triangle Calculator covers the SAS area workflow on the same kind of triangle without needing the law of cosines first.

Enter the three side lengths in the same length unit, and read the result panel from the top down. The verdict, the angles, the longest side, the area, and the perimeter all come from the same three inputs.

1. 1 Pick a single length unit: Use meters, feet, inches, or any one unit for all three side inputs. Mixing units is the most common cause of a wrong area and a wrong verdict.
2. 2 Enter sides a, b, and c: Type the three side lengths into the matching fields. Each side must be shorter than the sum of the other two, otherwise the inputs do not describe a triangle.
3. 3 Read the verdict first: The Triangle type row at the top of the result panel says Obtuse, Right, or Acute. A Right or Acute verdict means the triangle is not obtuse, even though the angles, area, and perimeter are still returned.
4. 4 Use the angles and longest side: Copy the largest, middle, and smallest angles into a worksheet, and use the longest side as the side opposite the largest angle in a follow-up calculation.

A surveyor measures a lot corner with sides 4 m, 5 m, and 7 m. Enter a = 4, b = 5, c = 7. The calculator returns verdict Obtuse, largest angle 101.54°, middle 44.42°, smallest 34.05°, longest side 7, area 9.80 square meters, and perimeter 16 m.

When the three sides need to be cross-checked against the SSS, SAS, or ASA solution of an oblique triangle, the Oblique Triangle Calculator handles the same input cases in a single form using the law of sines and the law of cosines.

Putting the verdict, the three angles, the longest side, the area, and the perimeter in one panel turns a three-side measurement into a complete answer.

• • Verdict from the squares test: The Triangle type row reads Obtuse, Right, or Acute, so the user does not compute three cosines by hand and compare each to 90 degrees.
• • All three angles from three sides: The law of cosines runs once per angle, so the full angle set appears together with the largest angle labeled as the obtuse one when the verdict is Obtuse.
• • Longest side identified: The longest side is named explicitly, and it is always the side opposite the largest angle, which makes follow-up roof, fence, or deck calculations straightforward.
• • Area and perimeter included: Heron's formula and the side sum run in the same pass, so coverage, fencing, and trim work share the same source of truth as the obtuse verdict.

If the triangle is part of a larger design, the area feeds material coverage and the perimeter feeds trim or fencing.

When the three sides are all different, the Scalene Triangle Area Calculator is the matching scalene-area page that uses the same Heron's formula on unequal sides and is a quick way to recheck the area output.

A few input choices and assumptions decide whether the verdict and the area match the triangle you are trying to solve.

• • The calculator accepts side lengths only. If the input is one side and two angles, or two sides and a non-included angle, an extra conversion step is needed that this tool does not perform.
• • The verdict uses the strict mathematical definition of obtuse. A triangle that lands at 90.01 is labelled obtuse, and one that lands at 89.99 is labelled acute. Treat 90 plus or minus a small tolerance as the boundary.

If the sides come from a drawing, double-check whether the values are edge-to-edge or include a wall or material thickness before entering the sides.

According to Wolfram MathWorld, an obtuse triangle has exactly one interior angle greater than 90 degrees and the side opposite that angle is the longest side

According to Wolfram MathWorld, the area of a triangle with sides a, b, c is sqrt(s(s-a)(s-b)(s-c)) where s is the semi-perimeter (a + b + c) / 2

When the same worksheet also needs an area for a triangle where one side and the height are known rather than all three sides, the Triangle Area Calculator covers that base-and-height path in the same math-conversion cluster.