Among all triangles, the 45-45-90 right triangle is the overachiever. Two equal legs, a hypotenuse locked to them by the square root of two, and angles so clean you can do most of the math in your head.

It shows up everywhere: the diagonal of a square, a folded paper corner, a roof pitch, a CNC toolpath. The 45 Degree Angle Calculator takes the one measurement you have — either a leg or the hypotenuse — and fills in everything else: both legs, the hypotenuse, the area, and the perimeter.

Two modes cover every case. If you know a leg, the calculator builds outward. If you know the hypotenuse, it works backward to the legs.

What Does the 45 Degree Angle Calculator Do?

This calculator solves a 45-45-90 right triangle completely from a single length. You tell it whether your number is a leg or the hypotenuse, and it returns all six facts: leg A, leg B, the hypotenuse, the area, the perimeter, and the three angles.

The angles never change — 45, 45, and 90 degrees — but the calculator lists them anyway so the result is a complete picture of the triangle.

All lengths are rounded to two decimals in the result cells. The angles are printed as 45° / 45° / 90°.

How to Use the 45 Degree Angle Calculator

Choose what you know from the dropdown: “One leg length” or “The hypotenuse.” This choice decides which formula the calculator applies.

Enter the length in the second field. It must be greater than zero, and it can carry decimals. The unit is whatever you are working in — inches, centimeters, feet — because the math is unit-free.

Press Calculate. Six result cells appear with every measurement rounded to two decimals.

Press Reset to reload the tool and clear the form before solving a different triangle.

Why the Two Legs Are Always Equal

In a 45-45-90 triangle, the two acute angles are both 45 degrees. Equal angles sit opposite equal sides, so the two legs must be identical. There is no 45-45-90 triangle with mismatched legs.

This is what makes the calculator’s job so simple. One leg determines the other leg instantly: it is the same number.

It also means the triangle is exactly half of a square cut along its diagonal. Picture the square and the formulas start to feel obvious.

From a Leg to the Hypotenuse

When you know a leg, the hypotenuse is the leg multiplied by the square root of two, about 1.41421. This comes straight from the Pythagorean theorem with both legs equal.

The formula is:

Hypotenuse = Leg × √2

A leg of 10 gives a hypotenuse of about 14.14. The hypotenuse is always the longest side, roughly 41 percent longer than either leg.

From the Hypotenuse to a Leg

When you know the hypotenuse, each leg is the hypotenuse multiplied by the square root of one half, about 0.70711. This is the same relationship run in reverse.

The formula is:

Leg = Hypotenuse × √(1/2)

A hypotenuse of 20 gives legs of about 14.14 each. Dividing by √2 would give the same answer; the calculator uses the equivalent multiply-by-√(1/2) form.

How the Area Is Found

The area of any right triangle is half the product of its legs. With both legs equal, that simplifies beautifully.

The formula is:

Area = Leg × Leg ÷ 2

Legs of 10 give an area of 50 square units. Because it is half a square, the area is exactly half the square of the leg.

How the Perimeter Is Found

The perimeter is the sum of all three sides: both legs plus the hypotenuse.

The formula is:

Perimeter = Leg + Leg + Hypotenuse

With a leg of 10 and a hypotenuse of 14.14, the perimeter is about 34.14 units. Every side counts once, including both identical legs.

The √2 Constant

The square root of two, about 1.41421, is the signature number of this triangle. It is the ratio of the hypotenuse to either leg, and it appears in every conversion the calculator performs.

It is an irrational number, so the calculator rounds results to two decimals. For most practical work — cutting wood, sizing a bracket — two decimals are plenty.

Draw a square and cut it along the diagonal. Each half is a 45-45-90 triangle, with the square’s sides as the legs and the diagonal as the hypotenuse.

This picture explains every formula at a glance. The area is half the square because the triangle is half the square. The hypotenuse is the diagonal because the hypotenuse is the diagonal.

It also explains why the legs must be equal. The square’s sides are equal, so the triangle’s legs are equal. There is no way to cut a square in half and get mismatched legs.

Whenever a formula feels abstract, return to the square. The geometry is doing the algebra for you.

Worked Example: Leg of 10 Units

First: choose “One leg length” and enter 10.

Leg B equals leg A, so both legs are 10.

Then: 10 × √2 ≈ 14.14.

The hypotenuse is about 14.14 units.

Then: 10 × 10 ÷ 2 = 50.

The area is 50 square units.

Then: 10 + 10 + 14.14 = 34.14.

The perimeter is 34.14 units, and the angles are 45° / 45° / 90°.

Worked Example: Hypotenuse of 20 Units

First: choose “The hypotenuse” and enter 20.

Then: 20 × √(1/2) ≈ 14.14.

Each leg is about 14.14 units.

Then: 14.14 × 14.14 ÷ 2 ≈ 100.

The area is 100 square units.

Then: 14.14 + 14.14 + 20 ≈ 48.28.

The perimeter is 48.28 units, and the angles are 45° / 45° / 90°.

Worked Example: Small Leg of 5 Units

First: choose “One leg length” and enter 5.

Both legs are 5.

Then: 5 × √2 ≈ 7.07.

The hypotenuse is about 7.07 units.

Then: 5 × 5 ÷ 2 = 12.5.

The area is 12.5 square units.

Then: 5 + 5 + 7.07 = 17.07.

The perimeter is 17.07 units.

Worked Example: Decimal Hypotenuse of 12.5 Units

First: choose “The hypotenuse” and enter 12.5.

Then: 12.5 × √(1/2) ≈ 8.84.

Each leg is about 8.84 units.

Then: 8.84 × 8.84 ÷ 2 ≈ 39.06.

The area is 39.06 square units.

Then: 8.84 + 8.84 + 12.5 ≈ 30.18.

The perimeter is 30.18 units.

The field accepts decimals, so real-world measurements like 12.5 work without rounding your input first.

The Wrong-Mode Trap

The dropdown is the most important input. Entering a hypotenuse while “One leg length” is selected treats your number as a leg, and every result comes out wrong.

The fix is simple: check the dropdown before pressing Calculate. If the hypotenuse comes out smaller than your input, you almost certainly chose the wrong mode.

A quick sanity check: the hypotenuse must always be the largest number in the result. If it is not, switch modes and recalculate.

Rounding and Precision

The calculator carries full precision internally and rounds only for display. That is why hand-checking with the rounded hypotenuse can give a slightly different perimeter.

For example, rounding 14.1421 to 14.14 and then adding by hand gives 34.14 either way here, but with other inputs the difference can reach a cent-sized rounding.

For shop work, two decimals are more than enough. For exactness, keep the √2 relationship symbolic until the final step.

Units: Inches In, Inches Out

The math is unit-free. Enter inches and every result is in inches; enter centimeters and every result is in centimeters.

The area comes out in square units of whatever you entered. A leg of 10 inches gives an area of 50 square inches.

The one rule is consistency: do not mix units. A leg in inches and a hypotenuse in feet would need conversion before entry.

The perimeter follows the input unit as well. There is no unit conversion anywhere in the tool, which is exactly why the rule matters.

Common 45-45-90 Mistakes

The most common mistake is mixing up the modes. Entering a hypotenuse while “One leg length” is selected treats your number as a leg, and every result comes out wrong. Check the dropdown first.

People also round too early. If you round the hypotenuse to 14.1 and then compute the perimeter by hand, you get a slightly different answer than the calculator’s 34.14. Let the tool carry full precision and round only at the end.

Another mistake is assuming the angles can vary. They cannot. Any triangle with angles other than 45, 45, and 90 is a different triangle with different formulas.

Finally, people enter zero or negative lengths. The calculator requires a length greater than 0, because nothing smaller can form a triangle.

Where These Calculations Are Useful

Carpenters meet this triangle in every square corner. The diagonal of a square tile, the brace across a square frame, and a 45-degree miter cut all follow the √2 ratio.

It appears in design and layout work too. A square rotated 45 degrees, a diamond pattern, or a diagonal banner all hide 45-45-90 triangles inside.

Students use it as the friendliest introduction to the Pythagorean theorem, because the equal legs keep the algebra clean and the answers checkable by hand.

It also shows up in electronics and machining. Diagonal measurements across square screens and square stock follow the same constant.

How to Interpret Your Result Correctly

The six cells are a complete description of one specific triangle. Leg A and leg B are always identical — if they ever differed, something went wrong.

The hypotenuse is always the largest number in the result. If your hypotenuse comes out smaller than a leg, you entered the value under the wrong mode.

All lengths share your input’s unit. Enter inches and every result is in inches; the area is in square inches.

The angles cell is a constant reminder. 45° / 45° / 90° never changes, which is what makes this triangle’s formulas so dependable.

Frequently Asked Questions

1. What is a 45-45-90 triangle?

A right triangle with two 45-degree acute angles. The equal angles force the two legs to be equal, and the hypotenuse is always the leg times √2.

2. What do I enter if I only know one side?

Enter it and select the matching mode: “One leg length” if it is a leg, “The hypotenuse” if it is the longest side. The calculator derives everything else.

3. Why are both legs always equal?

Equal angles sit opposite equal sides. Since both acute angles are 45 degrees, the sides opposite them — the two legs — must match.

4. What is the hypotenuse formula from a leg?

Hypotenuse = Leg × √2, where √2 ≈ 1.41421. A leg of 10 gives a hypotenuse of about 14.14.

5. What is the leg formula from a hypotenuse?

Leg = Hypotenuse × √(1/2), where √(1/2) ≈ 0.70711. A hypotenuse of 20 gives legs of about 14.14.

6. How is the area calculated?

Half the product of the legs: Area = Leg × Leg ÷ 2. It is exactly half the area of the square built on the leg.

7. How is the perimeter calculated?

By adding all three sides: Perimeter = Leg + Leg + Hypotenuse. Both equal legs count separately.

8. What units does the calculator use?

Whatever unit you enter. The math is unit-free, so inches in gives inches out, and the area comes out in square units of the same system.

9. Why are the results rounded to two decimals?

Because √2 is irrational and never terminates. Two decimals are precise enough for practical work while keeping the display clean.

10. Can the angles ever be different?

No. A 45-45-90 triangle always has angles of 45°, 45°, and 90°. Different angles mean a different triangle with different formulas.

11. What if I enter zero or a negative number?

The calculator asks for a length greater than 0. Zero and negative lengths are rejected because they cannot form a triangle.

12. Is the hypotenuse always the longest side?

Yes. At about 1.414 times either leg, the hypotenuse is always the largest of the three sides. Use that as a quick sanity check on your result.

13. How is this triangle related to a square?

It is exactly half a square cut along the diagonal. The legs are the square’s sides and the hypotenuse is its diagonal, which is why the √2 ratio appears.

14. Can I use decimals in the length field?

Yes. The field accepts any positive number, including decimals like 7.5, and the results keep two-decimal precision.

15. Does the calculator save my entries?

No. Your entries stay on the page only while it is open. Pressing Reset reloads the tool and clears the form.