Pythagorean Theorem
Pythagoras' theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, expressed as a² + b² = c². This 2500-year-old mathematical relationship appears in Year 8 of the UK National Curriculum and forms the foundation for countless calculations in geometry, physics, and engineering. The theorem applies exclusively to right triangles, where one angle measures exactly 90 degrees.
Why it matters
Pythagoras' theorem underpins navigation systems, construction projects, and screen technology. GPS devices calculate distances using triangulation based on this principle, whilst builders use the 3-4-5 triangle to ensure walls meet at perfect right angles. Television manufacturers rely on the theorem to determine screen sizes—a 42-inch TV means the diagonal measures 42 inches, calculated from the width and height using c² = w² + h². Engineers designing bridges calculate structural loads using Pythagorean relationships, and astronomers measure distances between celestial objects. The theorem also appears in GCSE coursework for calculating distances on coordinate grids, finding the magnitude of vectors, and solving trigonometry problems. Students encounter it again in A-level mathematics when working with 3D coordinates and complex numbers.
How to solve pythagorean theorem
Pythagorean Theorem
- In a right triangle: a² + b² = c² (c = hypotenuse).
- To find hypotenuse: c = √(a² + b²).
- To find a leg: a = √(c² − b²).
Example: Legs 3, 4: c = √(9+16) = √25 = 5.
Worked examples
Two corners of a park are 3 m apart east-west and 4 m apart north-south. What is the straight-line distance between them?
Answer: 5
- Identify the right triangle → legs = 3, 4; hypotenuse = ? — A right triangle has one 90-degree corner, like the corner of a book. The two shorter sides next to that corner are the 'legs', and the long side across from it is the 'hypotenuse'.
- Write the Pythagorean theorem: a² + b² = c² → 3² + 4² = c² — This famous formula says: if you draw a square on each side of a right triangle, the two smaller squares together have the same area as the big square. Think of it like two small pizza boxes fitting perfectly into one large one.
- Plug in the known values and calculate the squares → 3² + 4² = 9 + 16 = 25 — Squaring means multiplying a number by itself: 3 x 3 = 9 and 4 x 4 = 16. Then add them: 9 + 16 = 25.
- Take the square root to find c → c = sqrt(25) = 5 — The square root 'undoes' the squaring. We need the number that, multiplied by itself, gives 25. That number is 5. It's like asking: 'what size square has an area of 25?' Answer: 5 x 5.
- Verify: does a² + b² = c²? → 3² + 4² = 9 + 16 = 25 = 5² ✓ — Always check your work! Plug the answer back in to make sure both sides are equal. This is like double-checking your change at the store.
A right triangle has legs 7 and 24. Find the hypotenuse.
Answer: 25
- Identify the right triangle → legs = 7, 24; hypotenuse = ? — A right triangle has one 90-degree corner, like the corner of a book. The two shorter sides next to that corner are the 'legs', and the long side across from it is the 'hypotenuse'.
- Write the Pythagorean theorem: a² + b² = c² → 7² + 24² = c² — This famous formula says: if you draw a square on each side of a right triangle, the two smaller squares together have the same area as the big square. Think of it like two small pizza boxes fitting perfectly into one large one.
- Plug in the known values and calculate the squares → 7² + 24² = 49 + 576 = 625 — Squaring means multiplying a number by itself: 7 x 7 = 49 and 24 x 24 = 576. Then add them: 49 + 576 = 625.
- Take the square root to find c → c = sqrt(625) = 25 — The square root 'undoes' the squaring. We need the number that, multiplied by itself, gives 625. That number is 25. It's like asking: 'what size square has an area of 625?' Answer: 25 x 25.
- Verify: does a² + b² = c²? → 7² + 24² = 49 + 576 = 625 = 25² ✓ — Always check your work! Plug the answer back in to make sure both sides are equal. This is like double-checking your change at the store.
A right triangle has hypotenuse 122 and one leg 22. Find the other leg.
Answer: 120
- Identify the right triangle and label the sides → known leg = 22, hypotenuse = 122, missing leg = ? — The hypotenuse is always the longest side (across from the right angle). We know one leg and the hypotenuse, and we need to find the other leg.
- Write the Pythagorean theorem and rearrange for the missing leg → a² + b² = c² => x² = c² - known² — Since a² + b² = c², we can move the known leg to the other side by subtracting. It's like a balance scale: if you take something off one side, you must take the same off the other.
- Plug in the known values → x² = 122² - 22² = 14884 - 484 = 14400 — Square the hypotenuse: 122 × 122 = 14884. Square the known leg: 22 × 22 = 484. Subtract: 14884 - 484 = 14400.
- Take the square root → x = √14400 = 120 — The square root of 14400 is 120 because 120 × 120 = 14400. The missing leg is 120.
- Verify: does a² + b² = c²? → 120² + 22² = 14400 + 484 = 14884 = 122² ✓ — Check by squaring all sides and confirming the equation balances. Good habit!
Common mistakes
- Confusing which side is the hypotenuse leads to errors like calculating 5² + 13² = c² when finding the leg of a 5-12-13 triangle, giving c = √194 ≈ 13.9 instead of the correct answer 12.
- Forgetting to take the square root after adding the squares produces incorrect results, such as writing 3² + 4² = 25 as the final answer instead of c = √25 = 5.
- Applying the theorem to non-right triangles gives wrong measurements—using a² + b² = c² on an equilateral triangle with sides of 6 would incorrectly suggest the third side is √72 ≈ 8.5 rather than 6.