Find the hypotenuse, missing side, or check if a triangle is right-angled
Pythagoras' theorem is one of the most fundamental concepts in geometry, stating that in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Written as a^2 + b^2 = c^2, this theorem has been used for thousands of years and remains a cornerstone of GCSE and A-level maths. Understanding Pythagoras' theorem is essential for solving problems involving distances, right-angled triangles, and coordinate geometry. It appears across multiple topics in the National Curriculum, from basic geometry through to 3D Pythagoras at Higher GCSE and A-level. Our calculator provides three modes: find the hypotenuse when you know both shorter sides, find a missing side when you know the hypotenuse and one side, and check whether three given side lengths form a right-angled triangle. Each calculation includes complete step-by-step working that mirrors the method expected in GCSE exam answers. Pythagorean triples are sets of whole numbers that satisfy the theorem. The most common are 3, 4, 5 and 5, 12, 13. Recognising these triples can save time in exams, as they appear frequently in GCSE and A-level questions. Any multiple of a Pythagorean triple also works; for example, 6, 8, 10 is simply double 3, 4, 5. In the UK maths curriculum, Pythagoras' theorem is first introduced at Key Stage 3 and is tested extensively at GCSE level (both Foundation and Higher tiers). At A-level, students apply the theorem in 3D space and combine it with trigonometry to solve more complex problems.
To use the Pythagoras calculator: 1. Select the calculation mode. Choose "Find hypotenuse" to calculate the longest side, "Find missing side" to calculate a shorter side, or "Check right angle" to verify if three sides form a right-angled triangle. 2. For "Find hypotenuse" mode, enter the two shorter sides (a and b). The calculator computes c = sqrt(a^2 + b^2). 3. For "Find missing side" mode, enter the hypotenuse (c) and one of the shorter sides (b). The calculator computes a = sqrt(c^2 - b^2). The hypotenuse must be larger than the known side. 4. For "Check right angle" mode, enter all three sides. The calculator checks whether a^2 + b^2 = c^2 (with floating-point tolerance) and tells you whether the triangle is right-angled. 5. Review the step-by-step working to see the substitution into the formula, the intermediate calculations, and the final answer rounded to 2 decimal places.
Pythagoras' theorem states: a^2 + b^2 = c^2 Where c is the hypotenuse (the side opposite the right angle, always the longest side), and a and b are the other two sides. To find the hypotenuse: c = sqrt(a^2 + b^2) Example: sides 3 and 4. c = sqrt(9 + 16) = sqrt(25) = 5. To find a missing side: a = sqrt(c^2 - b^2) Example: hypotenuse 13, side 5. a = sqrt(169 - 25) = sqrt(144) = 12. To check if a triangle is right-angled, verify that the squares of the two shorter sides sum to the square of the longest side. For example, 3, 4, 5: 9 + 16 = 25. Since 25 = 25, this is a right-angled triangle. Common Pythagorean triples to memorise for GCSE maths: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25. These appear regularly in exam questions set by AQA, Edexcel, and OCR boards. At A-level, Pythagoras extends to 3D space. To find the distance between two points in 3D, use d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2), which is an extension of the same principle applied across three perpendicular directions.