45 45 90 Triangle Calculator 

A 45-45-90 triangle is the isosceles right triangle: two equal legs, a right angle, and a hypotenuse that is always the leg times √2. Because the 1: 1: √2 ratio never changes, one known side fixes everything else. Enter a leg or the hypotenuse below and the calculator returns both legs, the hypotenuse, the area, the perimeter, the altitude to the hypotenuse, the inradius and the circumradius, with the exact radical form (like 10√2) shown next to the rounded decimal so a homework answer never gets lost to rounding.

Quick Answer

How to use the 45-45-90 triangle calculator

Pick which side you know, type its length, choose its unit, and click Calculate. The results stay hidden until you calculate, and all nine outputs appear together.

1. Say which side you know: one of the two equal legs, or the hypotenuse. The legs meet at the right angle; the hypotenuse sits opposite it and is always the longest side.
2. Enter the length: any positive number, decimals included. The page is pre-filled with the worked example, a leg of 10 cm.
3. Pick the unit you measured in: mm, cm, m, km, in, ft, yd or mi, or leave it unitless for pure homework numbers. Lengths come back in that unit and the area in that unit squared.
4. Read the Exact form line as well as the decimals: it writes the unknown side as a multiple of √2 (for a leg of 10, hypotenuse = 10.0000√2). The reading box under the results then checks the answer against Pythagoras with your own numbers, so you can see the solution verify itself.

Two honest limits. The tool starts from a side only: if your problem gives the area or the perimeter instead, the section below shows the one-step algebra that turns either into a leg you can enter. And every decimal is rounded to 4 places, which is why the exact form matters when the next step of a problem feeds on this one.

The 1: 1: √2 ratio and every formula the tool uses

Every output is a one-line consequence of the ratio 1: 1: √2, which itself comes straight from the Pythagorean theorem: with two equal legs L, the hypotenuse squared is L² + L² = 2L², so the hypotenuse is L√2. Nothing in the results box is a black box.

Perimeter is the row students most often rush: it is all three sides added, 2L + L√2, not just the two legs. For perimeter questions on other shapes, the perimeter formula calculator writes out the formula for each shape with your numbers substituted. The inradius and circumradius rows are the ones general triangle tools tend to skip, and they are covered in their own section below.

Worked example: a leg of 10 cm, then a hypotenuse of 14 cm

Start from a leg and the whole triangle falls out in four steps. Starting from the hypotenuse is the same chain run backwards, and both directions are checked in the table under the steps.

Now the reverse direction the tool also handles. Given a hypotenuse of 14 cm, each leg is 14 ÷ √2 = 9.8995 cm, exactly 7√2 cm, the area is 49.0000 cm² and the perimeter is 33.7990 cm. That 9.8995 is worth a second look: a widely copied figure for this classic example says 9.8997, which is a rounding slip. 7√2 = 9.8994949…, so 9.8995 is the correct 4-decimal value, and it is what this calculator returns.

Notice the last row: a hypotenuse of 20 gives legs of 14.1421, which is exactly the first row scaled by √2. Similar triangles keep the same ratios at every size, so any row of this table can be checked against any other by scaling.

Starting from the area or perimeter instead of a side

The tool accepts a leg or the hypotenuse only, but an area or perimeter converts to a leg in one step, and then the tool takes over. From area A, the leg is √(2A). From perimeter P, the leg is P ÷ (2 + √2).

Both inversions are exact, not approximations. Area 50 cm² gives leg = √100 = 10 cm, which is the worked example above. Perimeter 34.1421 cm gives leg = 34.1421 ÷ 3.4142 = 10 cm. Problems phrased around a square are the same trick in disguise: the diagonal of a square is a 45-45-90 hypotenuse, so a square with side 10 has diagonal 10√2 = 14.1421, and a square whose diagonal is 14 has side 7√2 = 9.8995. If a worksheet gives you the diagonal, enter it as the hypotenuse and read the side off as the leg.

Altitude, inradius and circumradius: why two of them always match

The altitude to the hypotenuse and the circumradius are the same number in every 45-45-90 triangle, both equal to half the hypotenuse. The altitude from the right-angle corner splits the hypotenuse into two equal parts, and the circumcenter of a right triangle sits at the midpoint of its hypotenuse, so both lengths are H/2 = L/√2.

The inradius is the odd one out: r = L(2 − √2)/2, about 0.2929 × L, the radius of the largest circle that fits inside the triangle and touches all three sides. For the circle geometry behind inscribed and circumscribed circles in general, the circle formula calculator covers radius, diameter, circumference and area from any one of them. Use the matching altitude and circumradius as a free error check: the calculator prints both, with the formula it used beside each, so a mistyped input shows up as two numbers that should agree but do not.

Where 45-45-90 triangles show up: square diagonals and 45° cuts

Cut a square along its diagonal and you get two 45-45-90 triangles, which is why this shape runs tiling, framing and trim work. Any 45° miter cut, any square tile cut corner to corner, and any brace set at 45° is this triangle with a tape-measure number attached.

The fixed angles also fix the trigonometry at 45°: sin 45° = cos 45° = √2/2 ≈ 0.7071, and tan 45° = 1 exactly. Those are the values behind the altitude result: the altitude is the leg times sin 45°, which is L × √2/2 = L/√2. If your problem starts from an angle and a side in a general right triangle rather than this special one, the cosine calculator works the ratio directly. And the sibling worth knowing is the 30-60-90 calculator: the other special right triangle, with ratio 1: √3: 2, which covers the 30° and 60° cases this page deliberately does not.

What competing calculators miss

General right-triangle solvers handle this shape, but they make you work like it is any other triangle. The gaps that matter for this specific search:

• Omni Calculator's 45-45-90 tool is the strongest rival and does accept area, perimeter and altitude as starting inputs. Its results, though, lead with decimals: the exact surd form a teacher wants (7√2, not 9.8995) is not the headline, and on the worked example here the exact form is printed as its own result line with the formula beside it.
• Single-purpose tools such as StatGarden's isosceles right triangle page stop at sides, area and perimeter. No altitude, no inradius, no circumradius, so the three outputs that separate a full solve from a partial one are missing entirely.
• Akyno and ToolDone-style pages list separate boxes for leg A, leg B and the hypotenuse, inviting you to fill in values you do not have, when the whole point of the 1: 1: √2 ratio is that one side is enough. This tool asks for exactly one length and one decision: leg or hypotenuse.
• The gap this page fills: one input, nine outputs, decimals and exact radical forms side by side, the radii included, and a built-in Pythagoras check written in your own numbers under every result, so the answer arrives already verified.

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