![]() You can also find this using our hypotenuse calculator. ![]() ![]() The hypotenuse c is equal to leg a times the square root of 2. However, since the ratio of the short legs to the hypotenuse in a 45 45 90 triangle is 1 : √2, the following is a simple formula to calculate the length of the hypotenuse: Note that in a 45 45 90 triangle legs a and b are the same length. The hypotenuse c is equal to the square root of leg a squared plus leg b squared. Thus, the formula to solve the hypotenuse is: However, the equation can be manipulated algebraically to isolate the hypotenuse, c. Where a and b are the two legs of the right triangle, and c is the hypotenuse. The Pythagorean Theorem is typically expressed as: In addition to the Pythagorean theorem, there are also a few simplified formulas that can be used on a 45 45 90 triangle as well, which allow you to solve for unknown side lengths given only one side. The tangent of the 45° angle is equal to its adjacent side divided by the opposite side, which is equal to 1.īecause a 45 45 90 triangle is a right triangle, you can use the Pythagorean theorem to solve for unknown side lengths. Since the ratios never change, regardless of the lengths of the three sides, the values of sine, cosine, and tangent evaluated at 45° will always be the same. The sine of an angle is equal to the side length opposite the angle divided by the hypotenuse, the cosine of an angle is equal to the side length adjacent to the angle divided by the hypotenuse, and the tangent of an angle is equal to the side length opposite the angle divided by the side length adjacent to the angle. Note, this ratio can help you evaluate the sine, cosine, and tangent of either of the 45° angles of a 45 45 90 triangle. The phrase “solving a triangle” typically refers to using given information to find unknown side lengths and angles. Using this ratio, you can solve for any side length if you know just one of the other side lengths. Keep in mind this ratio is structured as a : b : c, where a and b are the two shorter side lengths opposite the 45° angle (often called the legs), and c is the longest side length (called the hypotenuse). In a 45 45 90 triangle, the ratio of the side lengths is 1 : 1 : √2.
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