45-45-90 Triangle Calculator

Side ratios

From the Pythagorean theorem, with both legs equal to L:

L^2 + L^2 = h^2 \quad\Rightarrow\quad h = L√(2)

Memory aid

Write down \textleg : \textleg : \texthyp = 1 : 1 : √(2). "x, x, x√2" is enough to solve any 45-45-90 problem — a square corner triangle made by halving a square along its diagonal.

Area and perimeter

With legs equal to L, the area is half the product of the legs:

A = \tfrac12 L^2, \qquad P = 2L + L√(2) = L\,(2 + √(2))

Connection to the square

A 45-45-90 triangle is half of a square cut along its diagonal. The hypotenuse is the diagonal; the legs are the sides. This geometric picture is the easiest way to derive the ratio without trigonometry.

Deriving the 1 : 1 : √2 Ratio

Start from the definition: two angles are 45°, so the triangle is isosceles and the two legs are equal. Call each leg a. Pythagoras gives h^2 = a^2 + a^2 = 2a^2, so h = a√(2).

That is the entire derivation, and it is worth committing to memory rather than the decimal. With √(2) ≈ 1.41421356, a leg of 5 gives a hypotenuse of about 7.071, and a hypotenuse of 10 gives legs of 10/√(2) = 5√(2) ≈ 7.071. Note the pleasing symmetry: dividing by √(2) and multiplying by √(2)/2 are the same operation, which is why the exact trigonometric values \sin 45° = \cos 45° = √(2)/2 appear in every table.

a : a : a√(2), \qquad \sin 45° = \cos 45° = (√(2))/(2), \quad \tan 45° = 1

Going Backwards from the Hypotenuse

Given the hypotenuse and needing the legs, divide by √(2) — or equivalently multiply by √(2)/2 ≈ 0.7071. Rationalising the denominator, h/√(2) = h√(2)/2, and the second form is preferred in written work because it avoids a surd in the denominator.

This is the direction that trips people up, because the instinct is to apply the same ×√(2) in both directions. A quick check settles it every time: the hypotenuse is the longest side of any right triangle, so going from hypotenuse to leg must make the number *smaller*. If your answer for a leg exceeds the hypotenuse, you have multiplied where you should have divided.

a = (h)/(√(2)) = (h√(2))/(2) ≈ 0.7071\,h

Area, Perimeter and Practical Appearances

With equal legs the area is simply A = a^2/2, and expressed through the hypotenuse it becomes A = h^2/4. The perimeter is a(2 + √(2)) ≈ 3.414a.

The shape is everywhere in practice. A square's diagonal — for checking that a frame, foundation or picture is truly square — is the hypotenuse of two of these triangles. The 45° mitre joint, where two pieces meet at a right angle, is cut along this triangle. Roof pitches described as "12 in 12" are 45°. In a square-section duct or a square baseplate, the bolt-circle diameter relates to the side by exactly this ratio. And in electronics, the 45° point on a phasor diagram is where resistance and reactance are equal, giving an impedance √(2) times either one — the half-power point that defines filter bandwidth.

A = (a^2)/(2) = (h^2)/(4), \qquad P = a(2 + √(2))

Worked Example

Inputs

• legLength: 1

Result: hypotenuse: 1.4142 area: 0.5 perimeter: 3.4142

Explanation

Legs = 1, hypotenuse = √2 ≈ 1.4142, area = ½ · 1² = 0.5, perimeter = 2 + √2 ≈ 3.4142.

Second Scenario

Inputs

• legLength: 2.25

Result: hypotenuse: 1.4142 area: 0.5 perimeter: 3.4142

Explanation

This scenario uses different inputs (legLength = 2.25) to show how changing one variable affects the 45-45-90 triangle result. Run the calculator above with these values to get the exact updated output with step-by-step work.