Obtuse Triangle Calculator
Solve obtuse triangles from 3 sides or 2 sides + angle. Detect the obtuse angle, classify, find all properties including altitude through the obtuse vertex, and compare with a right triangle.
Acute Triangle Calculator
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.
Acute Triangle Calculator
Planning notes, formulas, and examples
About the Acute Triangle Calculator
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.
When This Page Helps
Use this page when you want to verify that a triangle is acute and inspect the rest of its geometry at the same time. It keeps the acute-angle test, side and angle measures, and the derived radii, medians, and heights together so the classification is not separated from the triangle itself.
How to Use the Inputs
- Enter the three side lengths a, b, and c.
- Or click a preset to load a common acute triangle.
- Select the measurement unit (mm, cm, in, m, or ft).
- The calculator checks if all angles are under 90° (acute condition).
- View all properties: angles, area, altitudes, medians, inradius, circumradius.
- Compare visual angle bars — all bars should stay under the 90° line.
- Check the comparison table to see how your triangle differs from right/obtuse variants.
Formula used
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)Example Calculation
Result: An acute triangle with all three angles below 90°
The calculator checks the side-length inequalities or the cosine-based angle calculations to confirm that every interior angle is acute, then reports the rest of the triangle measurements from that validated shape.
Tips & Best Practices
- An equilateral triangle (all sides equal) is always acute — all angles are exactly 60°.
- The acute condition a² < b² + c² for all sides is the Pythagorean inequality — the 3D analog determines acute vs obtuse dihedral angles.
- In an acute triangle, the circumcenter, incenter, centroid, and orthocenter are all inside the triangle.
- As any angle approaches 90°, the triangle transitions from acute to right. Going beyond gives obtuse.
- The most "extreme" acute triangle is nearly equilateral. The closer to equilateral, the more balanced the properties.
When To Use This Calculator
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. Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.
How To Check The Result
Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.
Practical Notes
Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.
Sources & Methodology
Last updated:
Frequently Asked Questions
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.
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