Verify and solve acute triangles from 3 sides. Check that all angles are under 90°, compute all properties including circumcenter inside the triangle, and compare with right and obtuse triangles.
An acute triangle is a triangle in which every angle measures less than 90°. This is the "standard" triangle shape — the kind most people picture when they think of a triangle. The equilateral triangle (all angles 60°) is the most symmetric acute triangle, and any isosceles triangle with apex angle under 90° and base angles under 90° is also acute.
To verify that a triangle is acute from its side lengths, check that the square of every side is less than the sum of squares of the other two: a² < b² + c², b² < a² + c², c² < a² + b². If any of these fails, the triangle is right (equality) or obtuse (inequality reversed for the longest side).
Acute triangles have special geometric properties. The circumcenter (center of the circumscribed circle) lies inside the triangle — unlike obtuse triangles where it falls outside. All three altitudes are internal (their feet land on the actual sides, not extensions). The orthocenter is also located inside the triangle. This makes acute triangles geometrically "well-behaved."
This calculator takes three side lengths, validates the triangle inequality, checks the acute condition, and computes all properties: three angles, area, perimeter, semi-perimeter, all three altitudes, all three medians, inradius, circumradius, and the triangle's classification. Visual angle bars and comparison with right/obtuse variants help you understand what makes a triangle acute. Presets include equilateral, isosceles, and scalene acute triangles, plus a reference table of common acute triangles.
This acute triangle calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Unit and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Acute test: a² < b²+c², b² < a²+c², c² < a²+b² Angles: A = arccos((b²+c²−a²)/(2bc)) Area (Heron's): A = √(s(s−a)(s−b)(s−c)) Altitude: h_a = 2·Area/a Median: m_a = ½√(2b²+2c²−a²) Inradius: r = Area/s Circumradius: R = a/(2·sin A)
Result: For unit=5, the tool returns the solved acute triangle outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in acute triangle formulas and reports derived values, checks, and classifications automatically.
This page is tailored to acute triangle, with outputs tied directly to the form fields (Unit). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
All three angles must be strictly less than 90°. Equivalently, the square of each side must be less than the sum of squares of the other two.
Yes. All angles in an equilateral triangle are 60°, which is less than 90°, so it is always acute.
Inside the triangle. This is unique to acute triangles — for right triangles it's on the hypotenuse, and for obtuse triangles it's outside.
No. An isosceles triangle can be acute, right, or obtuse depending on the apex angle. It's acute only if the apex angle is less than 90°.
Check all three inequalities: a² < b²+c², b² < a²+c², c² < a²+b². If all hold, the triangle is acute.
In an acute triangle all angles < 90° so circumcenter and orthocenter are inside. In an obtuse triangle one angle > 90° so both move outside.