Pythagorean theorem
Hypotenuse or missing leg of a right triangle, or the right-angle test on three lengths (converse).
Result
5
- Calculation
- c = √(3² + 4²) = 5
- Area
- 6
- Perimeter
- 12
Note
The hypotenuse measures 5.
Calculation, assumptions and sources
OverviewPythagoras’ theorem in its three uses: the hypotenuse from the two legs, a leg from the other leg and the hypotenuse, and the converse, which tells whether three lengths form a right, acute or obtuse triangle.
Pythagoras’ theorem in its three uses: the hypotenuse from the two legs, a leg from the other leg and the hypotenuse, and the converse, which tells whether three lengths form a right, acute or obtuse triangle. The triangle is drawn to scale.
Method & formulasc = √(a² + b²)
In a right triangle, the square of the hypotenuse is the sum of the squares of the other two sides.
- Hypotenuse
- the side opposite the right angle, the longest of the three. The other two are the legs (catheti).
- Converse
- if a² + b² = c², the angle opposite c is right. It checks a right angle from three measured lengths.
- Pythagorean triple
- three integers satisfying a² + b² = c²: 3-4-5 and its multiples, 5-12-13, 8-15-17, 7-24-25…
- Acute, obtuse
- if the sum of the squares of the two shorter sides exceeds the square of the longest, all angles are acute; if it is smaller, one angle is obtuse.
Validity domainThe theorem holds in the Euclidean plane.
The theorem holds in the Euclidean plane. Applied to measured lengths, it inherits their uncertainties, and the converse compares with a relative tolerance. On a sphere (large terrestrial distances), the angle sum exceeds 180° and the theorem no longer applies as is.
The thirteen-knot ropeTo check a right angle on site, mark 60 cm on one wall, 80 cm on the other, and measure the diagonal: at 100 cm, the angle is right; above, it is obtuse; below, acute.
To check a right angle on site, mark 60 cm on one wall, 80 cm on the other, and measure the diagonal: at 100 cm, the angle is right; above, it is obtuse; below, acute. The closed rope with thirteen evenly spaced knots, stretched into a 3-4-5 triangle, served the same purpose without a square.
Reading the resultThe area of a right triangle is half the product of the legs.
The area of a right triangle is half the product of the legs. In converse mode, the relative gap between a² + b² and c² measures the departure from a right triangle; on lengths measured to the millimetre, a gap under one percent comes from the measurement.