Acute Triangle Calculator
This calculator checks whether a triangle is acute, right, or obtuse, and calculates every other property in one go. Enter the three side lengths a, b, and c in any unit, and the tool first checks the triangle inequality to confirm the values form a valid triangle, then applies the law of cosines to solve for all three angles. If every angle comes out under 90 degrees the triangle is acute, a right triangle has one angle at exactly 90 degrees, and an obtuse triangle has one angle above it. Alongside the classification you get the largest angle called out separately, plus a full breakdown covering perimeter, semi-perimeter, and area calculated using Heron's formula. The calculator also returns the three altitudes, the perpendicular height from each vertex to its opposite side, the inradius, the radius of the circle inscribed inside the triangle, and the circumradius, the radius of the circle passing through all three corners. Every figure updates instantly as you type, so you can adjust a side length and watch the angles, area, and triangle type change in real time. This is handy for checking geometry homework, verifying a triangle you have measured or drawn, or seeing how altering one side shifts a triangle from acute to right to obtuse. All working runs in full precision behind the scenes and is only rounded for the figures shown on screen.
Side Lengths
Triangle Type Check
Enter three side lengths on the left to see the triangle type and a full breakdown of all properties.
Measurements
Circles and Angles
What Is an Acute Triangle?
An acute triangle is one where all three interior angles are strictly less than 90 degrees. Because the angles of any triangle must sum to exactly 180 degrees, having all angles less than 90 degrees means none of the three angles dominates the shape. Acute triangles are sometimes described as "sharp" triangles because all their corners are pointed rather than flat or wide.
The three main types of triangle by angle are:
- Acute triangle: all angles are less than 90 degrees (for example, 60, 70, and 50 degrees)
- Right triangle: exactly one angle is 90 degrees
- Obtuse triangle: exactly one angle is greater than 90 degrees
How to Check if a Triangle Is Acute
Given three side lengths a, b, and c, first check whether they form a valid triangle using the triangle inequality: every pair of sides must sum to more than the third side. If that passes, identify the longest side and call it c. The triangle is:
- Acute if a² + b² > c²
- Right if a² + b² = c² (the Pythagorean theorem)
- Obtuse if a² + b² < c²
This test only needs to be applied to the longest side because the shorter sides always produce a positive difference, and it is only the largest angle that can reach or exceed 90 degrees.
Formulas Used by This Calculator
| Property | Formula |
|---|---|
| Angle A (law of cosines) | A = arccos((b² + c² − a²) / (2bc)) |
| Angle B (law of cosines) | B = arccos((a² + c² − b²) / (2ac)) |
| Angle C | C = 180 − A − B degrees |
| Semi-perimeter | s = (a + b + c) / 2 |
| Area (Heron's formula) | Area = √(s(s−a)(s−b)(s−c)) |
| Altitude ha | ha = 2 × Area / a |
| Inradius | r = Area / s |
| Circumradius | R = (a × b × c) / (4 × Area) |
Worked Example
Using the default values of a = 5, b = 6, c = 7 (all units):
- Triangle inequality: 5 + 6 = 11 > 7, 5 + 7 = 12 > 6, 6 + 7 = 13 > 5. Valid triangle.
- Acuteness check (longest side c = 7): 5² + 6² = 25 + 36 = 61 > 49 = 7². The triangle is acute.
- Angle A = arccos((36 + 49 − 25) / (2 × 6 × 7)) = arccos(60/84) = arccos(0.7143) ≈ 44.42°
- Angle B = arccos((25 + 49 − 36) / (2 × 5 × 7)) = arccos(38/70) = arccos(0.5429) ≈ 57.12°
- Angle C = 180 − 44.42 − 57.12 ≈ 78.46°
- Semi-perimeter: s = (5 + 6 + 7) / 2 = 9
- Area = √(9 × 4 × 3 × 2) = √216 ≈ 14.70 square units
- Inradius: r = 14.70 / 9 ≈ 1.63 units
- Circumradius: R = (5 × 6 × 7) / (4 × 14.70) ≈ 3.57 units
All three angles (44.42°, 57.12°, 78.46°) are less than 90 degrees, confirming the triangle is acute.
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Method: Angles calculated via the law of cosines (ISO 80000-2). Area by Heron's formula. Inradius r = Area / s. Circumradius R = abc / (4 × Area). All intermediate values are computed in full floating-point precision; results are rounded only for display.
This calculator is for educational and general-purpose use. Results depend on the side lengths you enter. For very large or very small values, normal floating-point rounding applies. Always verify critical measurements independently.