45-45-90 Triangle Calculator

What a 45-45-90 triangle is

A 45-45-90 triangle is a right triangle whose two legs are the same length. Because the legs match, the two angles across from them match too, and since all three angles add to 180 degrees with the right angle already taking 90, each remaining angle comes out at 45 degrees. The easiest picture to keep in your head is a square cut corner to corner: every square splits into two of these triangles, which is why the shape shows up wherever squares and 45 degree lines do.

The matching legs force one fixed ratio on the sides. Call the leg x. The Pythagorean theorem says the hypotenuse squared is x² + x² = 2x², so the hypotenuse is x times √2, the square root of 2, about 1.4142. Every 45-45-90 triangle ever drawn has sides in the ratio x : x : x√2, no exceptions.

That single ratio is why the shape is classed as a special right triangle, alongside its cousin the 30-60-90 triangle. The University of Houston trigonometry notes state the same theorem, hypotenuse equals leg times √2, and the OpenStax Algebra and Trigonometry text builds it from right triangle trigonometry. Once the ratio is in your head, one side fixes all three sides, and a full triangle solver is overkill for this one shape.

The formulas, depending on which side you know

If you know a leg, multiply it by 1.4142 to get the hypotenuse. The area is leg × leg ÷ 2, because the two equal legs serve as base and height. The perimeter is leg + leg + hypotenuse, which works out to leg × 3.4142. The altitude dropped to the hypotenuse is always half the hypotenuse, so it equals leg × 0.7071.

If you know the hypotenuse instead, divide it by 1.4142, or multiply by 0.7071, to get each leg. Everything else then falls out of the leg: the area is leg² ÷ 2, which from the hypotenuse h is h² ÷ 4, and the altitude is h ÷ 2.

Worked out with whole numbers, for a leg of 7 inches: hypotenuse = 7 × 1.4142 = 9.8995 inches. Area = 7 × 7 ÷ 2 = 49 ÷ 2 = 24.5 square inches. Perimeter = 7 + 7 + 9.8995 = 23.8995 inches. Altitude = 9.8995 ÷ 2 = 4.9497 inches. One input, one multiplier, and the whole triangle is known.

Worked example: starting from the hypotenuse

The reverse direction matters just as often, because plans, screens and storage bins give you the diagonal and you need the sides. Suppose a square panel has a diagonal of 10 inches and you need the side length. The diagonal of a square is the hypotenuse of a 45-45-90 triangle, so the side is a leg.

Leg = 10 ÷ 1.4142 = 7.0711 inches. Check it with Pythagoras: 7.0711² = 50, and 50 + 50 = 100, which is 10², so the answer closes. Area = 7.0711 × 7.0711 ÷ 2 = 50 ÷ 2 = 25 square inches. Perimeter = 7.0711 + 7.0711 + 10 = 24.1421 inches. Altitude = 10 ÷ 2 = 5 inches, exactly half the diagonal.

Two moves in that list are worth rehearsing. Squaring the leg and doubling it must give back the squared hypotenuse, the fastest self-check in triangle math. And an altitude of exactly half the hypotenuse confirms the shape is a 45-45-90 and not some other right triangle.

The square diagonal shortcut

Since a square's diagonal splits it into two 45-45-90 triangles, the same math answers every square diagonal question: diagonal = side × 1.4142, and side = diagonal × 0.7071. It also gives a fast route to the area of a square measured corner to corner: area = diagonal² ÷ 2.

That comes up all over a house. A 36 inch square tabletop measures 36 × 1.4142 = 50.9117 inches corner to corner, so it will not pass flat through a 48 inch doorway. A 4 by 4 foot plywood sheet spans 48 × 1.4142 = 67.8823 inches on the diagonal, or 5.6569 feet, which tells you whether it fits in a truck bed laid flat.

Builders lean on the same relationship from the other side. In any rectangle the two diagonals are equal, so framing crews measure both diagonals of a door opening or a deck ledger; two matching numbers mean the corners are square, and a mismatch says which way it is racked. On a true square, reading the diagonal also hands you the side: 11.3137 inches across means 8 inch sides, because 11.3137 × 0.7071 = 8.

One family of diagonals does not belong in this table: televisions and monitors. A 27 inch screen is a 16:9 rectangle, about 23.5 inches wide and 13.2 inches tall, so no 45-45-90 math applies to it. The square diagonal rule is for square things only.

45 degree cuts, rafters and other shop uses

Walk into a shop and the ratio starts earning its keep. A 90 degree corner joint can be split into two 45 degree cuts so the seam runs down the middle of the trim, and the cut face across a board of width w is w × 1.4142, because the saw blade travels the hypotenuse of a 45-45-90 triangle. A 1x6 board, 5.5 inches wide, needs a cut face of 5.5 × 1.4142 = 7.7782 inches, which is why miter cuts eat more length off a board than people expect.

Roof framing is the same triangle standing on end. A 12/12 pitch means 12 inches of rise for every 12 inches of run, which is exactly 45 degrees, so each foot of run costs √2 feet of rafter. A run of 12 feet needs a rafter of 12 × 1.4142 = 16.9706 feet before any overhang, and carpenters remember that as 17 inches of rafter per foot of run.

Any half-right-angle layout problem is this triangle in disguise: a corner brace reaching 24 inches along each wall is 24 × 1.4142 = 33.9411 inches long end to end, a square sign cover 22 inches on a side spans 22 × 1.4142 = 31.1127 inches across, and picture frames, decks and layout jigs all reuse the one multiplier.

Rounding the square root of 2

√2 = 1.41421356..., a decimal that never ends, so every hand calculation picks how many digits to keep. The cost of rounding depends on the size of the number, and the table below prices it out for a leg of 7 and a hypotenuse of 10.

For pencil-and-paper work, 1.414 loses at most about 0.0015 on room sized numbers, far under the 0.0625 inch of a 1/16 inch mark, which is about the best a saw cut can hold anyway. Keeping two digits, 1.41, drifts by about 0.02 of an inch on the divide side, more than ten times the error of 1.414. For anything going to a saw, 1.414 is enough; for drawings and machining, keep 1.4142 and let the calculator carry the rest.

Note that division is slightly less forgiving than multiplication, because dividing by a too-small multiplier lands the error on the same side twice. When a plan says 10 ÷ √2, writing 7.07 keeps you inside a hundredth of an inch even before you round to tape measure marks.

Mistakes people make with this triangle

• Grabbing the 30-60-90 ratios. The other special right triangle has sides x, x√3 (about 1.7321) and 2x. If a problem says the hypotenuse is exactly twice a leg, that is a 30-60-90 triangle, never a 45-45-90.
• Multiplying when you should divide. Going from leg to hypotenuse multiplies by 1.4142; going from hypotenuse to leg multiplies by 0.7071. A quick sanity check: the hypotenuse must come out longer than either leg.
• Using two different legs. If the legs differ, the shape is a plain right triangle, not a 45-45-90, and the ratio does not hold. Use the Pythagorean theorem directly, or the Pythagorean theorem calculator.
• Squaring the hypotenuse for the area. Area from the hypotenuse h is h² ÷ 4, not h² ÷ 2. For h = 10 that is 25, not 50.

The calculator and the tools next door

The calculator above runs the whole ratio in one step: type a leg or a hypotenuse, and it returns both legs, the hypotenuse, the area, the perimeter and the altitude, all straight from x : x : x√2. It never needs the angle, because in this triangle the angles are already known.

For shapes outside the special-ratio club, the right triangle calculator solves from any two sides, the Pythagorean theorem calculator covers the leg-leg-hypotenuse relationship on its own, and the triangle calculator takes three sides of any triangle at all. For pitched roofs specifically, the roof pitch calculator converts rise and run into angle and rafter length, and the equilateral triangle calculator handles the other classic special triangle.