The 45-45-90 Triangle

A 45-45-90 triangle is an isosceles right triangle: its two legs are equal and its two acute angles are both 45°. The sides are always in the ratio 1 : 1 : √2. If each leg is s, the hypotenuse is s√2; if you know the hypotenuse, divide it by √2 to get each leg.

That single rule solves every 45-45-90 problem, and it comes straight from the Pythagorean theorem. This page shows where the ratio comes from, the rule for each side you might start with, how to tidy answers such as 10 over √2, and where the triangle turns up outside the classroom.

What makes a triangle 45-45-90

The name lists the three angles. The angles of any triangle add up to 180°, so once one angle is 90°, the other two share the remaining 90°. In a 45-45-90 triangle they share it equally.

Equal angles always sit opposite equal sides, so the two legs (the sides that form the right angle) have the same length. The reverse is true as well: a right triangle with equal legs must have two 45° angles. "45-45-90 triangle" and "isosceles right triangle" are two names for one shape. If you need a refresher on legs, the hypotenuse and the right-angle mark, see parts of a right triangle.

The quickest way to make one is to cut a square along a diagonal. The cut splits two of the square's 90° corners into 45° angles, and the sides of the square become the legs.

This means every fact about 45-45-90 triangles is also a fact about squares. The hypotenuse of the triangle is the diagonal of the square, and the triangle's area is half the square's area, s² over 2.

Where the 1 : 1 : √2 ratio comes from

Call each leg s and the hypotenuse c. The Pythagorean theorem says the squares of the two legs add up to the square of the hypotenuse, so

The hypotenuse is always √2 ≈ 1.414 times as long as a leg, whatever the size of the triangle. The sides are therefore in the ratio s : s : s√2, which simplifies to 1 : 1 : √2 when you divide by s.

The picture proves the same thing with no algebra at all. The square on each leg splits into two copies of the triangle, while the square on the hypotenuse holds four. Two plus two is four, so the hypotenuse square has exactly twice the area of a leg square. The same idea appears in Plato's dialogue Meno, where Socrates shows that the square built on the diagonal of a square has double the original area.

Because √2 is irrational, a 45-45-90 triangle can never have three whole-number sides: whole-number legs give an irrational hypotenuse, and a whole-number hypotenuse gives irrational legs. That is why no Pythagorean triple has two equal legs. The proof that √2 cannot be written as a fraction is on why √2 is irrational.

45-45-90 triangle rules

There are only two directions to travel: from a leg to the hypotenuse, or from the hypotenuse back to a leg.

Decimals in the table are rounded to the nearest hundredth.

From a leg to the hypotenuse

Multiply the leg by √2. A leg of 7 gives a hypotenuse of 7√2. Leave the answer in that exact form unless the question asks for a decimal; then 7√2 ≈ 9.90.

From the hypotenuse to a leg: divide and rationalize

Go the other way by dividing by √2. A hypotenuse of 10 gives legs of 10 over √2. That is correct, but most teachers want no square root in the denominator, so you rationalize: multiply the top and bottom by √2, which is the same as multiplying by 1.

The result is worth memorizing as a shortcut: a leg is half the hypotenuse, times √2. For a hypotenuse of 10 that is 5√2 in one step. When the hypotenuse already contains √2, the division is even easier: a hypotenuse of 8√2 gives legs of exactly 8. For more practice with radicals, see square roots and simplifying radicals.

Diagonal of a square: d = s√2

A diagonal cuts a square into two 45-45-90 triangles, with the square's sides as legs and the diagonal as the hypotenuse. So the diagonal of a square with side s is

Turn the formula around to get the side from the diagonal, s = d over √2, and square that to get the area: A = s² = d² over 2. Half the diagonal squared is handy when the diagonal is the only thing you can measure. For a rectangle, whose diagonal is √l² + w², use the rectangle diagonal calculator.

Worked examples

To check your own numbers, the special right triangle calculator gives exact and decimal answers for any side you enter.

Where 45-45-90 triangles show up

• Anything square. The diagonal of a tile, a chessboard or a square table is the side times √2. A diagonal brace across a square frame, such as a gate or a shelf, is cut to that length. Builders use the same idea to check that corners are square; see the theorem in construction.
• Grids and games. On a square grid a diagonal step covers √2 ≈ 1.414 times the distance of a straight step, so many grid-based games and path-finding programs count a diagonal move as about 1.4 straight moves. More on this in the theorem in computer science.
• Drawing tools. The two classic set squares used in technical drawing are a 45-45-90 triangle and a 30-60-90 triangle.
• Paper sizes. ISO paper such as A4 (210 mm × 297 mm) has sides in the ratio 1 : √2, rounded to the nearest millimeter. The long side is almost exactly the diagonal of a square built on the short side, because 210√2 ≈ 296.98.
• Trigonometry. With legs 1 and hypotenuse √2, you can read off sin45° = cos45° = 1 over √2 = √2 over 2 ≈ 0.7071 and tan45° = 1. These values fit the Pythagorean identity: 1 over 2 + 1 over 2 = 1.
• Constructing √2. A 45-45-90 triangle with legs 1 is the first step of the spiral of Theodorus, which builds √2, √3, √4, … one triangle at a time.

Special right triangles are also a favorite on standardized tests. The SAT reference sheet shows the 45-45-90 and 30-60-90 side ratios, while the ACT gives no formulas, so it pays to know them by heart; see the theorem on the SAT and ACT.

Common mistakes

• Multiplying when you should divide. To get a leg from the hypotenuse you divide by √2. Multiplying gives a "leg" longer than the hypotenuse, which is impossible.
• Using the rule on the wrong triangle. The 1 : 1 : √2 ratio only works when the triangle has a right angle and two equal legs (or, equivalently, a 45° angle). For any other right triangle, use a² + b² = c² directly.
• Writing √2s instead of s√2. They are different numbers: for s = 8, 8√2 ≈ 11.31 but √16 = 4.
• Rounding √2 too early. Using 1.4 for a 100-foot square gives a diagonal of 140 feet instead of about 141.42 feet. Keep √2 exact until the last step.
• Mixing up the two special triangles. The 45-45-90 triangle uses √2; the 30-60-90 triangle uses √3 and 2. More traps are listed on the common mistakes page.