Acute Triangle Calculator - Sides, Angles, and Area

An acute triangle calculator solves a triangle whose three interior angles are all strictly less than 90 degrees. Enter the three side lengths a, b, and c and the tool returns the three interior angles, the area by Heron's formula, the perimeter, and a verdict stating whether the triangle is acute, right, or obtuse.

• • Classify a measured triangle: Use three side lengths from a survey, a sketch, or a classroom problem and read off whether the result is acute, right, or obtuse.
• • Solve all three angles at once: Skip the manual law of cosines work and read all three interior angles in degrees, ready to copy into a worksheet.
• • Get area and perimeter together: Heron's formula and the perimeter land in the same panel so coverage, fencing, or material estimates use the same source of truth as the angle check.
• • Check homework and exam answers: Verify a geometry problem with side lengths 5, 6, and 7 is in fact acute, and confirm the worked area and angles against the answer key.

An acute triangle is one of the three angle-based families in triangle geometry, sitting beside right triangles, where one angle is exactly 90 degrees, and obtuse triangles, where one angle is more than 90. A triangle with one angle of 90.0001 is obtuse, not acute, so the verdict uses a strict less-than rule.

For the general-purpose triangle page that also accepts base and height or two sides with an included angle, the Triangle Calculator covers the same problem in a more flexible form.

The calculator reads the three side lengths, validates that they form a triangle, and runs the law of cosines once for each interior angle. The same semi-perimeter that drives the area is used in the verdict step, so a single pass through the math returns the angles, area, perimeter, and the acute-or-not verdict.

• a, b, c: the three side lengths of the triangle, all in the same length unit, with the side labelled opposite its matching interior angle
• 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. Once the three sides are known, each angle is the inverse cosine of a single ratio built from the squares of the sides. Heron's formula is the area counterpart: it takes the same three side lengths, builds the semi-perimeter, and returns the area without needing the height or any angle. Floating-point rounding can push the cosine argument outside the valid range, so the cosine is clamped before the inverse cosine is taken.

According to Wolfram MathWorld, the law of cosines states that for a triangle with sides a, b, c and angle A opposite a, a^2 = b^2 + c^2 - 2bc cos(A), so cos(A) = (b^2 + c^2 - a^2) / (2bc)

When the verdict lands on Right instead of Acute, the Right Triangle Calculator solves the same three sides using the Pythagorean theorem and the SOH-CAH-TOA ratios.

These four ideas are the math behind the calculator. Once they are clear, the verdict and the area fall out.

These four ideas are connected. The law of cosines returns the angles from the sides, the angle sum rule guards against bad input, and Heron's formula turns the same sides into the area. The acute rule sits on top of the angles and tells the user whether the triangle really is acute.

If the input triangle ends up obtuse, the Area Obtuse Triangle Calculator focuses on the same Heron's formula area calculation for triangles that fail the strict acute rule.

Measure or read the three side lengths, then enter them in the form. The result panel fills in the angles, area, perimeter, and the verdict in a single pass.

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.
3. 3 Read the verdict first: The Triangle type row at the top of the result panel says Acute, Right, or Obtuse. A Right or Obtuse verdict means the inputs do not describe an acute triangle.
4. 4 Use the angles, area, and perimeter: Copy the three interior angles into a worksheet, use the area for material coverage, and use the perimeter for trim work. All three outputs come from the same three side inputs.

Suppose a wooden support brace is cut with sides 5 cm, 6 cm, and 7 cm. Enter a = 5, b = 6, c = 7. The acute triangle calculator returns verdict Acute, angles 44.42, 57.12, and 78.46 degrees, area 14.70 square cm, and perimeter 18 cm. The verdict is the answer to the original question: yes, this brace is an acute triangle.

When the three measured sides happen to include two equal sides, the Isosceles Triangle Area Calculator returns the area, altitude, perimeter, base angles, and apex angle from just the base and the equal side length.

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

• • Answers the acute question directly: The Triangle type row labels the triangle Acute, Right, or Obtuse, so the user does not compare three angles to 90 degrees by hand.
• • All three angles from three sides: The law of cosines runs once per angle, so the full angle set appears together without rerunning the calculator.
• • Area and perimeter included: Heron's formula and the side sum run in the same pass, so coverage, fencing, and trim work all use the same source of truth.
• • Decimal-friendly input: Field-measured lengths such as 5.27, 6.05, and 7.18 are accepted without rounding. The math runs at full precision and only the final display rounds.
• • Rejects impossible inputs: Zero or negative sides, and sides that violate the triangle inequality, are caught before any angle math runs.

If the triangle is part of a larger design, the area feeds material coverage like sheathing or flooring, while the perimeter feeds trim work. The angles also feed a follow-up slope or pitch calculation, especially for right or obtuse follow-up cases.

If the three input sides are equal and the verdict is the special 60-60-60 equilateral case, the Equilateral Triangle Area Calculator returns the area from a single side without the general law of cosines.

A few input choices and assumptions decide whether the result matches the triangle you are trying to solve.

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

If the sides come from a drawing, double-check whether the values are edge-to-edge or include a wall or material thickness. Stripping that thickness before entering the sides keeps the result aligned with the measured triangle.

According to Wikipedia, an acute triangle is a triangle in which all three interior angles are strictly less than 90 degrees

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 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.