Classifying Triangles Calculator
Our free triangle calculator solves classifying triangles problems. Get worked examples, visual aids, and downloadable results.
Formula
Classification by sides and angles using the law of cosines
Triangles are classified by sides (equilateral = 3 equal, isosceles = 2 equal, scalene = 0 equal) and by angles using the Pythagorean relationship: if a^2 + b^2 = c^2 it is right, if > then acute, if < then obtuse. Angles are calculated using the law of cosines.
Worked Examples
Example 1: Classifying a Scalene Acute Triangle
Problem: Classify the triangle with sides 5, 6, and 7. Determine its type by sides and angles.
Solution: By sides: All sides different (5, 6, 7) = Scalene\nPythagorean test: 5^2 + 6^2 = 25 + 36 = 61 > 49 = 7^2 = Acute\nAngle A = arccos((36 + 49 - 25) / (2 x 6 x 7)) = arccos(60/84) = 44.42 deg\nAngle B = arccos((25 + 49 - 36) / (2 x 5 x 7)) = arccos(38/70) = 57.12 deg\nAngle C = 180 - 44.42 - 57.12 = 78.46 deg\nAll angles < 90 = Acute confirmed
Result: Classification: Acute Scalene | Angles: 44.42, 57.12, 78.46 deg
Example 2: Identifying a Right Isosceles Triangle
Problem: Classify a triangle with sides 5, 5, and 7.071 (5 x sqrt(2)).
Solution: By sides: Two sides equal (5, 5) = Isosceles\nPythagorean test: 5^2 + 5^2 = 25 + 25 = 50 = 7.071^2 = Right\nAngle A = arccos((25 + 50 - 25) / (2 x 5 x 7.071)) = 45 deg\nAngle B = 45 deg\nAngle C = 90 deg\nClassification: Right Isosceles (45-45-90 triangle)
Result: Classification: Right Isosceles | Angles: 45, 45, 90 deg
Frequently Asked Questions
How are triangles classified by their sides?
Triangles are classified into three categories based on their side lengths. An equilateral triangle has all three sides equal, which also means all three angles are 60 degrees. An isosceles triangle has exactly two sides of equal length, and the angles opposite those equal sides are also equal. A scalene triangle has all three sides of different lengths, meaning all three angles are also different. This classification is fundamental in geometry because the side relationships determine many other properties, including symmetry, angle measures, and the positions of triangle centers.
How do you use the Pythagorean relationship to classify triangles?
The Pythagorean relationship provides a quick way to classify triangles by angles using only side lengths, without computing angles directly. Sort the three sides so that a is the smallest and c is the largest. If a squared plus b squared equals c squared, the triangle is a right triangle. If a squared plus b squared is greater than c squared, the triangle is acute (all angles less than 90 degrees). If a squared plus b squared is less than c squared, the triangle is obtuse (the angle opposite the longest side exceeds 90 degrees). This test extends the Pythagorean theorem beyond right triangles.
What are special right triangles?
Special right triangles have fixed side ratios that make calculations exact without a calculator. In a 45-45-90 triangle (an isosceles right triangle), the two legs are equal and the hypotenuse is leg × √2. If each leg is 1, the hypotenuse is √2 ≈ 1.414. In a 30-60-90 triangle, the sides are in ratio 1 : √3 : 2, where the shortest side is opposite the 30° angle and the hypotenuse is twice the shortest side. These triangles appear constantly in engineering, architecture, and physics — for instance, a roof pitch of 45° forms a 45-45-90 triangle, and an equilateral triangle bisected diagonally creates two 30-60-90 triangles. Knowing these ratios eliminates the need for trigonometric tables in common scenarios.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
Can I use Classifying Triangles Calculator on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.