Last updated: 2026-05-18

The 45-45-90 triangle

A right isosceles triangle has two equal legs, one right angle (90°), and two equal base angles (45°). It is the only triangle that is simultaneously isosceles and right-angled. Its side ratio is $1 : 1 : \sqrt{2}$, one of the two special-triangle ratios alongside $1 : \sqrt{3} : 2$ for the 30-60-90 triangle.

Relation to the square

Cutting a square diagonally produces two congruent 45-45-90 triangles. Each leg equals the square's side, and the hypotenuse equals the square's diagonal. This is the reason $\sqrt{2}$ appears so often in geometry: it is the diagonal-to-side ratio of a square.

Formulas

Let $l$ be the leg length.

Worked examples:

• $l = 5$: $c = 5\sqrt{2} \approx 7.071$, $S = 12.5$, $P = 5(2+\sqrt{2}) \approx 17.07$
• $c = 10$: $l = 10/\sqrt{2} \approx 7.071$, $S = 25$
• $S = 8$: $l = \sqrt{16} = 4$, $c = \sqrt{32} = 4\sqrt{2} \approx 5.657$

Derivation of the formulas

Hypotenuse: Apply the Pythagorean theorem to equal legs $l$: $c = \sqrt{l^2 + l^2} = l\sqrt{2}$

Area: Base $= l$, height $= l$ (both legs are perpendicular to each other): $S = \frac{1}{2} \times l \times l = \frac{l^2}{2}$

From hypotenuse: $l = c/\sqrt{2}$, so $S = l^2/2 = (c/\sqrt{2})^2/2 = c^2/4$.

From area: $S = l^2/2$ implies $l = \sqrt{2S}$, then $c = l\sqrt{2} = \sqrt{2S} \cdot \sqrt{2} = \sqrt{4S} = 2\sqrt{S}$.

Connections

• A 45-45-90 triangle is a special case of an isosceles triangle with apex angle 90°.
• The circumradius $R = c/2$ (half the hypotenuse, since the right angle subtends the diameter).
• The inradius $r = (l + l - c)/2 = l - c/2 = l(1 - \sqrt{2}/2) = l(2-\sqrt{2})/2$.

Applications

The 45-45-90 triangle appears throughout geometry and engineering. A square tile of side $l$ cut diagonally along its diagonal produces two such triangles, each with hypotenuse $l\sqrt{2}$. In drafting and carpentry, a 45° set square is a physical 45-45-90 triangle. In trigonometry, the exact values $\sin 45° = \cos 45° = \tfrac{\sqrt{2}}{2}$ and $\tan 45° = 1$ come directly from the $1:1:\sqrt{2}$ ratio. Use the Pythagorean Theorem Calculator to verify or extend the Pythagorean relationship, and the Triangle Area Calculator for the general area formula.