45 45 90 Special Right Triangle Calculator
Enter any known value for a 45 45 90 triangle to calculate the edge lengths, altitude, area, perimeter, inradius, and circumradius.
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What is a 45 45 90 Triangle?
You can also think of a 45 45 90 triangle as half of a square divided from one corner to the opposite corner.
45 45 90 Triangle Formulas
Since a 45 45 90 triangle is a special right triangle, the formulas used to calculate parts of a right triangle can be used, substituting the angles measurements.
Solve the Hypotenuse
Given a known leg length, the hypotenuse can be solved using the Pythagorean theorem. Thus, the formula to solve the hypotenuse is:
c = a² + b²
The hypotenuse c is equal to the square root of leg a squared plus leg b squared. Note that in a 45 45 90 triangle legs a and b are the same length.
This formula can be simplified a bit to the following:
c = a√2
The hypotenuse c is equal to leg a times the square root of 2.
Solve the Leg
Given the hypotenuse, the legs can be solved using the Pythagorean theorem in a slightly different way.
a = b = c√2 / 2
The legs a and b are equal to the hypotenuse c times the square root of 2, divided by 2.
Alternatively, the legs can be solved using 45° angle and hypotenuse with the formula below.
a = b = c × sin(45°)
The legs a and b are equal to the hypotenuse c times the sin of the 45° angle.
Solve the Area
The area of a 45 45 90 triangle can be solved with the formula:
T = 2a / 2
Thus, the area T is equal to 2 times leg a, divided by 2
Solve the Perimeter
Use this formula to calculate the perimeter:
p = 2a + c
Thus, the perimeter p is equal to 2 times leg a plus hypotenuse c.
For more special right triangle formulas, try our 30 60 90 calculator.