Pythagoras’ theorem

Hypotenuse of a right-angled triangle from its two shorter sides.

Ref: FMA-12
Rev: v1
Field: Foundations: Maths
Result: c [m]

a^{2} + b^{2} = c^{2}

c is the hypotenuse, the side opposite the right angle. a and b are the other two sides.

In a right-angled triangle, the square on the longest side equals the sum of the squares on the other two.

Calculator

Solve for

cHypotenuseComputed

aSide a

0.5100

bSide b

0.5100

Angle opposite a36.87 °
Area of the triangle6 m²

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Hypotenuse (m) current value 5 m

bSide bKey driver

3.04100

aSide a

4.03100

If one side is much shorter, the hypotenuse is almost equal to the longer side.
It gives straight-line distance from grid offsets.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (m)	Local step	Change in result
b Side b	0.5 to 100 m	3.041	100	97	+10%	+6.51 %
a Side a	0.5 to 100 m	4.031	100.1	96.05	+10%	+3.71 %

Sweep of Side a

Side a (m)	Hypotenuse (m)
0.5	4.031
0.9696	4.116
1.88	4.42
3.646	5.413
7.071	8.124
13.71	14.28
26.59	26.89
51.57	51.72
100	100.1

Sweep one variable

Vary

Hypotenuse (m)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

Hypotenuse

Result · m

What the equation gives you. Everything else feeds it.

Side a

m · typical 0.5 to 100

One of the two shorter sides. Raises c, with most effect when a is the longer of the two.

Side b

m · typical 0.5 to 100

The other shorter side. Raises c, with most effect when b is the longer of the two.

Worked example with the values above

Squares: 3² + 4² = 9 + 16 = 25.
Take the square root.
c = 5 m.

Assumptions and limits

The triangle has a right angle.
Flat (plane) geometry.

Common mistakes

Using it for non-right triangles.
Adding the sides instead of the squares.

Statistics and uncertainty

For two independent position errors the distance error is about the larger error scaled by the direction: δc ≈ √[(aδa)² + (bδb)²]/c.

Where it is used

Straight-line distance between map points.
Slant range from altitude and ground range.

Maths you need

Squares and roots: x² and √x
Right angles: perpendicular sides

Related formulas

Practise this formula