Audited 29 Jul 2026·Last updated 31 Jul 2026·4 citations·Tier 2·0 uses

45-45-90 Triangle Calculator

Enter one measurement of a 45-45-90 triangle — a leg, the hypotenuse, the area or the perimeter — and get every side plus area, perimeter and inradius.

45-45-90 Triangle Calculator

Which measurement do you already know?
Any unit you like — millimetres, inches, feet, metres. Lengths come back in the same unit and the area comes back in that unit squared. Must be greater than zero.
Hypotenuse
14.1421
The longest side, equal to the leg × √2 ≈ 1.4142135624. It is also the diagonal of the square this triangle is half of. Because √2 is irrational, this figure is an approximation unless it is the number you typed — exact to the 10 decimal places shown and no further.
Leg (both are equal)
10
Area
50sq units
Perimeter
34.1421
Altitude to the hypotenuse
7.0711
Inradius
2.9289

Background.

The Quanta 45-45-90 triangle calculator takes any single measurement of an isosceles right triangle and returns everything else: the leg, the hypotenuse, the area, the perimeter, the altitude from the right angle onto the hypotenuse, and the inradius. Enter a leg, the hypotenuse, the area or the perimeter — four modes, one number each. With the default leg of 10, the hypotenuse comes back as 14.1421356237, the area as exactly 50, the perimeter as 34.1421356237, the altitude as 7.0710678119 and the inradius as 2.9289321881.

A 45-45-90 triangle has exactly one degree of freedom. Its angles are fixed, so its side ratios are fixed too — 1 : 1 : √2 — and a single measurement determines the whole figure. That is the difference between this page and a general Pythagorean solver, which needs two sides before it can produce a third. It can handle one direction of this problem if you already know both legs are equal and type the same number twice; it cannot go the other way at all, because working back from the hypotenuse to a leg requires knowing a leg, which is exactly what you were asking for.

The ratio comes from cutting a square in half. Euclid's Elements Book I, Proposition 46 shows how to describe a square on a given line; draw either diagonal and the square falls into two identical right triangles. Each has two sides equal to the square's side, so its two non-right angles are equal, and since they must sum to 90° each one is 45°. The hypotenuse is the square's diagonal, which Euclid Book I, Proposition 47 — the Pythagorean theorem — fixes at √(L² + L²) = √(2L²) = L√2. So leg, leg and hypotenuse stand in the ratio 1 : 1 : √2 at every scale, and the calculator's hypotenuse output is also, exactly, the answer to 'what is the diagonal of a square of side L?'

Two facts are worth memorising. To go from a leg to the hypotenuse, multiply by about 1.4142. To go back, divide by the same number — many textbooks write that as multiplying by √2 ÷ 2 ≈ 0.7071, which is identical. Everything else on this page follows: the area is leg² ÷ 2, the perimeter is leg × (2 + √2), the altitude onto the hypotenuse is exactly half the hypotenuse, and the inradius is leg × (2 − √2) ÷ 2.

One caveat belongs beside the answer, not three screens below it. √2 is irrational, so a 45-45-90 triangle can never have all three sides as whole numbers: fix the leg at 10 and the hypotenuse runs 14.142135623730950488… forever, and fix the hypotenuse at a round number and the legs do the same. Every length here except the one you typed is exact only to the ten decimal places shown. The area is the happy exception — leg² ÷ 2 is rational whenever the leg is, which is why 10 gives exactly 50 and not an approximation. That irrationality also means a 45-45-90 triangle can never be a Pythagorean triple, the exact complement of the triples page on this site.

And a scope limit: this page assumes your triangle really is 45-45-90. If the angles differ even slightly the answers will be close but wrong, and the triangle solver is the right tool for the measurements you actually have.

What is 45-45-90 triangle calculator?

A 45-45-90 triangle — also called an isosceles right triangle — is a right triangle whose other two angles are both 45°. Wolfram MathWorld's entry describes it as having 'angles of 45°, 45°, and 90°' and states that for legs of length a 'the hypotenuse has length √2 a', with area ½a². Because the two non-right angles are equal, the two sides opposite them are equal as well, which is what makes the triangle isosceles and what reduces it to a single degree of freedom. The classical derivation needs no trigonometry. Take a square, which Euclid constructs in Elements Book I, Proposition 46, and draw one of its diagonals. The square splits into two congruent triangles, each with two sides equal to the square's side L and one right angle between them; the remaining two angles are equal and sum to 90°, so each is 45°. The hypotenuse of each triangle is the square's diagonal, and the Pythagorean theorem (Elements Book I, Proposition 47) gives it as √(L² + L²) = L√2. Hence the ratio 1 : 1 : √2, where √2 = 1.4142135623… — a number the Pythagoreans famously proved irrational, and which MathWorld's Pythagoras's Constant entry defines as one that 'cannot be expressed as a ratio p/q of integers p and q'. Together with the 30-60-90 triangle, the 45-45-90 is one of the two 'special right triangles' every geometry course covers, and it is the source of the exact values sin 45° = cos 45° = √2 ÷ 2 and tan 45° = 1. Practically, it turns up wherever a square does: the diagonal of a square tile, a floor, a screen, a picture frame or a piece of sheet material is the hypotenuse of a 45-45-90 triangle, and mitre joints for a square corner are cut at 45°.

How to use this calculator.

  1. Choose which measurement you already have from the dropdown: a leg, the hypotenuse, the area or the perimeter. Any one fixes the whole triangle.
  2. Note that there is a single leg option, not two. Both legs of a 45-45-90 triangle are the same length, so measuring one measures both.
  3. Enter the value in any unit. Lengths come back in the same unit; the area comes back in that unit squared.
  4. Read the hypotenuse at the top. In the default leg mode it is the number most people came for, and it is also the diagonal of a square whose side is that leg.
  5. Check the area if you want a quick sanity test: it should be exactly leg² ÷ 2, with no decimals at all when the leg is a whole number. If your area looks like a long decimal in leg mode, you probably typed the hypotenuse instead.
  6. Use the altitude to the hypotenuse when you need the height measured from the long edge rather than from a leg — for a mitred corner brace, a gusset, or a cut list. It is exactly half the hypotenuse.
  7. Use the inradius for inscribed-circle problems: the largest circle that fits inside the triangle has that radius.
  8. If your triangle is not exactly 45-45-90, do not use this page. Enter the measurements you actually have into the triangle solver instead.

The formula.

leg : leg : hyp = 1 : 1 : √2 , A = L² ⁄ 2 , P = L(2 + √2)
L · leg — both legs are equalL√2 · hypotenuse, also the diagonal of a square of side LA · area, P · perimeteraltitude to the hypotenuse = L ⁄ √2 = hypotenuse ⁄ 2inradius = L(2 − √2) ⁄ 2 ≈ 0.2928932188 L

Every mode on this page solves for the leg L first, then rebuilds the rest of the triangle from it.

WHERE THE RATIO COMES FROM. Describe a square on a given straight line of length L (Euclid, Elements Book I, Proposition 46) and draw a diagonal. The square splits into two congruent right triangles, each with two sides of length L meeting at a right angle. The other two angles are equal — the triangle is isosceles — and they must sum to 90°, so each is 45°. The hypotenuse is the diagonal, and the Pythagorean theorem (Elements Book I, Proposition 47) gives hypotenuse² = L² + L² = 2L², so hypotenuse = L√2. Hence 1 : 1 : √2.

THE FOUR INVERSIONS. Leg mode is trivial: L is what you typed. Hypotenuse mode divides by √2, since hyp = L√2; equivalently, multiply by √2 ÷ 2 ≈ 0.7071067812, which is the same operation rationalised. Area mode uses A = L² ÷ 2, so L = √(2A). Perimeter mode uses P = 2L + L√2 = L(2 + √2), so L = P ÷ (2 + √2), where 2 + √2 ≈ 3.4142135624.

WORKING THE DEFAULT ALL THE WAY THROUGH. With L = 10: the hypotenuse is 10 × 1.414213562373095049 = 14.142135623730950488, displayed as 14.1421356237. The area is 10² ÷ 2 = 50 exactly. The perimeter is 10 + 10 + 14.142135623730950488 = 34.142135623730950488, displayed as 34.1421356237. The altitude to the hypotenuse is (10 × 10) ÷ 14.142135623730950488 = 7.071067811865475244, displayed as 7.0710678119 — exactly half the hypotenuse, as it must be. The inradius is (10 + 10 − 14.142135623730950488) ÷ 2 = 5.857864376269049512 ÷ 2 = 2.928932188134524756, displayed as 2.9289321881.

TWO CROSS-CHECKS THAT USE DIFFERENT FORMULAS. The parent square has side 10 and area 100, and this triangle is exactly half of it — 50, matching the area output. And the diagonal of a square of side 10 is 14.142135623730950488, matching the hypotenuse. Neither route uses the ½ × base × height or the L√2 formula the module runs internally, and both checks are in the test suite rather than only in this paragraph.

WHY THE ALTITUDE AND THE CIRCUMRADIUS ARE THE SAME NUMBER. For any right triangle the circumcentre sits at the midpoint of the hypotenuse, so the circumradius is half the hypotenuse. For any right triangle the altitude onto the hypotenuse is (leg₁ × leg₂) ÷ hypotenuse; here that is L² ÷ (L√2) = L ÷ √2, which is also half of L√2. The two coincide only because the legs are equal. Rather than ship the same figure under two labels, this page reports the altitude and explains the coincidence.

ROUNDING STAGE. All arithmetic runs at 40 significant digits in Decimal, and √2 is computed from scratch rather than entered as the literal 1.4142135624 — a literal would inject error into the ninth decimal place of the perimeter and the inradius. The leg is never rounded before the other five outputs are derived from it. Rounding happens once, at the return boundary, to 10 decimal places. Nothing on this page is sorted into bands or graded, so no threshold depends on display rounding.

A worked example.

Example

A tiler needs the diagonal of a 10-centimetre square tile so they can dry-lay a diamond pattern and check the setting-out. Leaving the mode on leg and entering 10 returns a hypotenuse of 14.1421356237 centimetres, an area of exactly 50 square centimetres, a perimeter of 34.1421356237 centimetres, an altitude to the hypotenuse of 7.0710678119 centimetres and an inradius of 2.9289321881 centimetres. In practical terms the diagonal is 141.42 millimetres, which on a tape reading to the millimetre is 141 mm and a bit — and no unit you pick will ever make it come out round, because √2 is irrational. The area of exactly 50 is worth pausing on: the tile itself is 100 square centimetres, this triangle is half of it, and that halving is exact with no decimals at all. That is the one quantity in a 45-45-90 triangle that stays rational, and it is a useful sanity check — if the area comes back as a long decimal while you are in leg mode, you almost certainly typed the hypotenuse by mistake. The altitude to the hypotenuse, 7.0710678119 centimetres, is the perpendicular distance from the square corner to the middle of the diagonal, and it is exactly half of 14.1421356237, which is a consequence of the two legs being equal rather than a coincidence. The inradius of 2.9289321881 centimetres is the radius of the biggest circle that will sit inside the half-tile without crossing an edge. Switching to hypotenuse mode and entering 14.1421356237 gives the leg back as 10, which is the calculation to run when you are working from a measured diagonal — a mitred frame, a screen size, or a square opening whose sides you cannot reach. Cross-checking a different way: the parent square of side 10 has a diagonal of √(10² + 10²) = √200 = 14.142135623730950488, which is the same number the calculator returned, arrived at without using the 1 : 1 : √2 ratio at all.

value10
known Quantityleg

Frequently asked questions.

What are the rules for a 45-45-90 triangle?
Two rules, and they are all you need. Both legs are equal. And the hypotenuse is a leg times √2, about 1.4142135624. Written as a ratio, leg : leg : hypotenuse = 1 : 1 : √2. Everything else follows: the area is leg² ÷ 2, the perimeter is leg × (2 + √2), the altitude from the right angle onto the hypotenuse is exactly half the hypotenuse, and the inradius is leg × (2 − √2) ÷ 2. Because the two legs are equal, this is also called an isosceles right triangle, and it is exactly half of a square cut along its diagonal.
How do I find the leg when I know the hypotenuse?
Divide the hypotenuse by √2. If the hypotenuse is 14.1421356237, the leg is 14.1421356237 ÷ 1.4142135624 = 10. Most textbooks write the same operation as multiplying by √2 ÷ 2 ≈ 0.7071067812, which is the rationalised form and gives an identical answer. This is the direction a general Pythagorean-theorem calculator cannot help with: to find a missing leg it needs the hypotenuse and the other leg, and if you already had a leg you would not be asking. Select 'Hypotenuse' from the dropdown and this page does it in one step, along with the area, perimeter, altitude and inradius.
Is the hypotenuse of a 45-45-90 triangle the same as the diagonal of a square?
Yes, exactly. Cut a square of side L along a diagonal and each half is a 45-45-90 triangle whose legs are the square's sides and whose hypotenuse is the diagonal. So the diagonal of a square of side L is L√2, and this calculator answers the square-diagonal question as a side effect: enter the square's side as the leg and read the hypotenuse. Going the other way works too — enter a measured diagonal as the hypotenuse and the leg output is the square's side, which is how you size a square opening you can only measure corner to corner.
What is the area of a 45-45-90 triangle?
Half the square of a leg: A = leg² ÷ 2. For a leg of 10 that is exactly 50. The formula is just ½ × base × height with the two legs playing both roles, since they meet at the right angle. If you only know the hypotenuse h, substitute leg = h ÷ √2 to get A = h² ÷ 4 — the area is a quarter of the square of the hypotenuse. The area is the only quantity in this triangle that stays rational: a whole-number leg always gives an exact area, even though the hypotenuse never is.
Can a 45-45-90 triangle have whole-number sides?
No. If the legs are a whole number L, the hypotenuse is L√2 and √2 is irrational — MathWorld's Pythagoras's Constant entry puts it as a number that 'cannot be expressed as a ratio p/q of integers p and q', a fact the Pythagoreans proved and, according to the story, were rather upset about. Turn it round and the same problem appears: a whole-number hypotenuse gives legs of h ÷ √2, also irrational. So a 45-45-90 triangle can never be a Pythagorean triple. Triples such as 3-4-5 and 5-12-13 have whole-number sides and untidy angles; this triangle has tidy angles and irrational sides. You cannot have both, and any 45-45-90 you actually cut is an approximation.
How is this different from the 30-60-90 triangle calculator?
Different shape, different constant, different question. A 30-60-90 triangle has three sides all of different lengths, in the ratio 1 : √3 : 2, and its hypotenuse is exactly twice its short leg. A 45-45-90 has two equal legs and a hypotenuse of leg × √2. The two are the only 'special right triangles' taught at school, and between them they supply the exact trigonometric values at 30°, 45° and 60°, but neither is a special case of the other. This page asks which of four measurements you have; the 30-60-90 page asks which of five, because it has to know which of its two unequal legs you are holding.
Why are the altitude to the hypotenuse and the circumradius the same number here?
Two separate facts happen to coincide for this triangle. First, the circumcentre of any right triangle sits at the midpoint of the hypotenuse, so the circumradius is always half the hypotenuse. Second, the altitude from the right angle onto the hypotenuse is (leg₁ × leg₂) ÷ hypotenuse for any right triangle, and when the two legs are equal that becomes L² ÷ (L√2) = L ÷ √2, which is also half of L√2. In a 30-60-90 triangle the two are different — the altitude is (s × s√3) ÷ 2s = s√3 ÷ 2 while the circumradius is s. This page reports the altitude and explains the coincidence rather than shipping the same figure twice under two labels.

References& sources.

  1. [1]Wolfram Research, MathWorld — 'Isosceles Right Triangle' (live revision retrieved 2026-07-29). States the angles of 45°, 45° and 90°, that for legs of length a 'the hypotenuse has length √2 a', and the area ½a². Open access.
  2. [2]Euclid, Elements, Book I, Proposition 46 — 'To describe a square on a given straight line.' The parent figure of the diagonal-bisection derivation. David E. Joyce web edition of the Heath translation, Clark University; statement confirmed verbatim on retrieval 2026-07-29. Open access.
  3. [3]Euclid, Elements, Book I, Proposition 47 — 'In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.' Applied to the halved square it gives the hypotenuse as √(2L²) = L√2, an entirely geometric derivation that uses no trigonometry and serves as the independent check on this page's constant. Joyce web edition; confirmed verbatim on retrieval 2026-07-29. Open access.
  4. [4]Wolfram Research, MathWorld — 'Pythagoras's Constant' (live revision retrieved 2026-07-29). Gives the expansion √2 = 1.4142135623… and states that the irrationality of √2 'means that it cannot be expressed as a ratio p/q of integers p and q' — the basis for this page's scope limit that no 45-45-90 triangle has all-rational sides. Open access.
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