Pythagoras
In a right triangle, the squares of the two short sides (the legs) add up to the square of the long side (the hypotenuse). If you know two of the sides, you can always find the third.
The explanation is shown. Quick: find a side, set out a right angle, check a room. Advanced: also the space diagonal — and angles, area and how sensitive the measurement is.
Your numbers
The 3-4-5 measure
Answer
Side c
5cm
How to work it out
- 1a² + b² = c²The two short sides (the legs) squared add up to the long side (the hypotenuse) squared.
- 23² + 4² = c²
- 39 + 16 = c²
- 4c² = 25
- 5c = √25 = 5 cmThe square root goes the other way: it finds the number that, multiplied by itself, gives c².
The formula
a² + b² = c²
Only applies in a right triangle. c is the hypotenuse — the long side opposite the right angle.
c = √(a² + b²)
When you know both legs.
a = √(c² − b²)
When you know the hypotenuse and one leg.
Tip from the workshop: the 3-4-5 rule. Measure 3 one way and 4 the other — if the diagonal is exactly 5, the corner is square.
In depth: more numbers and “what if”
More quantities
| Size | Value |
|---|---|
| Angle A (opposite a) | 36.87° |
| Angle B (opposite b) | 53.13° |
| Area (a · b / 2) | 6 cm² |
| Perimeter (a + b + c) | 12 cm |
Typical mistake: if you just add the sides, you get 7 cm. It is the sides SQUARED that are added.
More: Geometry
- Area and perimeterRectangle, triangle, trapezium, circle and composite shapes — area and perimeter.
- Volume and surface areaBox, cylinder, cone, sphere, pyramid, prism, pipe and frustum — also in litres.
- The circleRadius, circumference and area from one measurement — plus sector, annulus, chord and rise.
- Right triangleSine, cosine and tangent: all sides and angles from two known values.
- AnglesThe third angle in a triangle and the angle sum of a polygon.
- Points in the coordinate systemDistance, midpoint and the line between two points — drawn in.
- Similar figuresScale factor and missing sides — area is multiplied by k², volume by k³.
- Any triangleThe sine and cosine rules: find the rest from SSS, SAS, ASA or SSA.
- TransformationsReflect, rotate and translate a figure on a coordinate grid — before and after.