Triangle Theorems Calculator
Classify a triangle as equilateral, isosceles or scalene, and find its angles with the law of cosines.
Angles are found with the law of cosines; the triangle inequality a + b > c must hold for a valid triangle.
Classification
scalene
right triangle
Angle at a
36.87°
Angle at b
53.13°
Angle at c
90°
Sum = 180°
Perimeter
12 units
Angles are rounded to 2 decimal places; they sum to 180° for any valid triangle.
Last updated: January 2026
How this calculator is verified
Checked by True Calculator automated test suite on
- Formula verified against a published worked example in the automated test suite
- Edge cases (zero, negative, boundary and unit-mismatch inputs) covered by unit tests
The full verification method is on our how we verify page. Found an error? Tell us and we will re-check it.
When to Use This Calculator
Use this calculator whenever a geometry problem hands you three side lengths and asks what shape they make. Students revising the law of cosines, engineers checking the triangle formed by truss members, and woodworkers or tilers laying out triangular sections all start from the same three numbers. The tool decides validity, names the classification, and computes every interior angle — useful for drawing accurate plans, checking roof rafters, or setting out land boundaries. Because the angles come from the sides themselves, no protractor is needed, and the 180° sum provides a built-in check that the numbers are consistent. It is equally at home in a classroom exercise or a real measuring-tape problem.
How to Use This Calculator
- Step 1: Enter the lengths of all three sides of the triangle.
- Step 2: Check the validity message first — sides must satisfy a + b > c.
- Step 3: Read the classification: equilateral, isosceles or scalene, and acute, right or obtuse.
- Step 4: Note the three angles, which always sum to 180°, and the perimeter.
Worked Example
Enter sides 3, 4 and 5. The triangle inequality holds and the calculator returns a scalene, right-angled triangle. The angle opposite the side of 5 is exactly 90°, and the other two angles are 36.87° and 53.13°, found with the law of cosines. The perimeter is 12 units. Since 3² + 4² = 9 + 16 = 25 = 5², the right angle makes sense. Equal sides would instead produce an isosceles or equilateral triangle with matching angles.
Tips and Common Mistakes
- •Tip 1: Sides of 5, 5, 5 produce an equilateral triangle with three 60° angles — check the sum of angles afterwards.
- •Tip 2: A triangle with sides 2, 2, 3 is isosceles but obtuse, with the largest angle at 97.18°.
- •Tip 3: Use the perimeter to sanity-check real-world problems like fencing or border lengths.
- ✗Mistake 1: Avoid entering sides that violate a + b > c — no triangle can exist with them.
- ✗Mistake 2: Don't assume a triangle is right-angled just because it looks like one; verify 3-4-5 style ratios first.
Frequently Asked Questions
Which triangle theorems does this use?
The triangle inequality (a + b > c) checks validity, the law of cosines finds each angle, and the sum of angles (always 180°) confirms the result. Side comparisons classify the shape.
What makes a triangle equilateral, isosceles or scalene?
Equilateral means all three sides equal, isosceles means exactly two equal, and scalene means all different. A 3-4-5 triangle is scalene because every side has a distinct length.
How do I know a triangle is right-angled?
If the squares of the two shorter sides add up to the square of the longest side, the angle opposite the longest side is 90°. For 3-4-5, 9 + 16 = 25, so it is right-angled.
What is the difference between acute and obtuse?
An acute triangle has all angles under 90°, while an obtuse triangle has one angle over 90°. With sides 2, 2, 3, the largest angle is about 97.18°, making it obtuse.
Why do my sides sometimes return 'cannot form a triangle'?
The triangle inequality fails: the sum of any two sides must exceed the third. Sides of 1, 1 and 3 cannot close into a triangle because 1 + 1 is less than 3.
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