Calculate the legs, hypotenuse, area, and perimeter of a 45 45 90 triangle from any one known value.
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45 45 90 Triangle Formula
A 45 45 90 triangle is a right triangle with two 45 degree angles and one 90 degree angle. The two legs are equal, and the side ratio is always 1 : 1 : √2. Once you know any single measurement, every other property is fixed. The calculator uses the formulas below, where a is the length of each leg.
c = a * sqrt(2)
a = c / sqrt(2)
A = a^2 / 2
P = 2a + a * sqrt(2)
h = a / sqrt(2)
r = a * (2 - sqrt(2)) / 2
R = c / 2
- a = length of each leg (the two equal sides)
- c = length of the hypotenuse (the side opposite the right angle)
- A = area of the triangle
- P = perimeter of the triangle
- h = altitude drawn to the hypotenuse
- r = inradius (radius of the inscribed circle)
- R = circumradius (radius of the circumscribed circle)
The calculator lets you start from any one known value: a leg, the hypotenuse, the area, the perimeter, the altitude, the inradius, or the circumradius. It first works backward to the leg length a, then applies the formulas above to fill in every remaining property. The hypotenuse formula c = a√2 comes straight from the Pythagorean theorem applied to two equal legs. The altitude to the hypotenuse also equals exactly half the hypotenuse, since h = c / 2 in this triangle.
Side and Property Ratios for a 45 45 90 Triangle
Because the shape is fixed, every property is a constant multiple of the leg. The table below gives those multipliers so you can estimate values by hand or check the calculator output.
| Property | In terms of leg a | Decimal factor |
|---|---|---|
| Leg | a | 1.000 |
| Hypotenuse | a√2 | 1.414 |
| Area | a² / 2 | 0.500 × a² |
| Perimeter | a(2 + √2) | 3.414 |
| Altitude to hypotenuse | a / √2 | 0.707 |
| Inradius | a(2 – √2) / 2 | 0.293 |
| Circumradius | a√2 / 2 | 0.707 |
Common leg values and their matching hypotenuse are shown below for quick reference.
| Leg (a) | Hypotenuse (a√2) | Area |
|---|---|---|
| 1 | 1.414 | 0.500 |
| 5 | 7.071 | 12.500 |
| 10 | 14.142 | 50.000 |
| 12 | 16.971 | 72.000 |
Example Problems
Example 1: Known leg. A 45 45 90 triangle has legs of 6 cm. Find the hypotenuse, area, and perimeter.
- Hypotenuse: c = 6 × √2 = 8.485 cm
- Area: A = 6² / 2 = 18 cm²
- Perimeter: P = 2(6) + 8.485 = 20.485 cm
Example 2: Known hypotenuse. The hypotenuse of a 45 45 90 triangle is 10 in. Find each leg and the area.
- Leg: a = 10 / √2 = 7.071 in
- Area: A = 7.071² / 2 = 25 in²
Frequently Asked Questions
What is the rule for a 45 45 90 triangle?
The two legs are equal in length and the hypotenuse equals a leg multiplied by the square root of 2. The full side ratio is 1 : 1 : √2. This holds for every 45 45 90 triangle no matter how large or small it is.
How do you find the leg if you only know the hypotenuse?
Divide the hypotenuse by the square root of 2. For example, a hypotenuse of 14 gives a leg of 14 / √2 = 9.899. You can also multiply the hypotenuse by 0.707 to get the same result.
Is a 45 45 90 triangle always isosceles?
Yes. Because the two non right angles are both 45 degrees, the sides opposite them are equal, which makes the triangle isosceles. It is the only right triangle that is also isosceles.
