How to Find the Length of the Side of a Triangle If You Know the Other Two Sides

How to Find the Length of the Side of a Triangle If You Know the Other Two Sides
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Finding the measurement of the third side of a triangle when you know the measurement of the other two sides only works if you have a right triangle or the measurement of at least one other angle. Without this information you do not have enough data in order to find out the length of the third side. A right triangle has a built in third angle, as one of the angles has to be 90 degrees.

Right Triangle Using Pythagorean Theorem

    Draw the triangle on your paper labeling the two sides adjacent to the right angle, or legs, “​a​” and “​b​.” Label the hypotenuse, or third side “​c​.”

    Set up your equation so that

    a^2 + b^2 = c^2

    This is the Pythagorean Theorem used for solving for the unknown side.

    Fill in the lengths you know in the equation. The hypotenuse is always the longest side in a right triangle. This is a great way to check your work, because if either of the legs is longer than the hypotenuse, you know you have made an error.

    Solve for the unknown side. If you are solving for the hypotenuse, you fill in the “​a​” and “​b​,” square both numbers and then add the numbers together. Use your calculator to get the square root of the resulting sum to reach your answer. If you are solving for one of the legs of, you need to move the other leg to the same side as the “​c​” by subtracting. This leaves the remaining leg alone, allowing you to solve for it. This means you square the “​c​” number and the known leg. Subtract the squared leg value from the squared ​c​ value. Get the square root of the resulting number and you have your answer for the unknown leg.

Using the Law of Sines

    Set up the triangle so that the side opposite the angle is matched with the angle. Label the side opposite angle ​A​ as ​a​, the side across from angle ​B​ as ​b​ and the side opposite angle ​C​ as ​c​.

    Write the equation out to read

    \frac{a}{\sin A}= \frac{b}{\sin B} = \frac{c}{\sin C}

    This gives you the basics for solving for your unknown side.

    Take the angle you know and use the calculator to determine the sine of that angle. Most scientific calculators have you enter the angle number and then hit the button labeled “sin.” Write down the value.

    Divide the length of the side associated with the angle by the value of the sin of that angle. This gives you a number typically written as an approximation, as the decimal places go off indefinitely. Call this new number ​X​ for the purpose of this example.

    Take the value of the other known side and divide it by ​X​. This new number equals the sine of the new angle.

    Enter the number in the calculator and hit the “sin-1” to get the angle in degrees. You can now solve for the angle of the unknown side.

    Add the two known angles together and subtract the total from 180. All angles inside a triangle must add up to 180 degrees.

    Calculate the sine of the new angle by entering it in the calculator and hitting the “sin” button. Multiply the answer by X and this gives you the length of the unknown side.

    For an example using the Pythagorean Theorem as well as a new method, solving using the Law of Cosines, watch the video below:

    Tip:​ Law of Sines can be worked as stated or by inverting all of the information so that the sine of the angle is divided by the length of the side.

    Warning:​ Draw the problem to see what you are multiplying and dividing in order to ensure you understand how the problem is working. Remember, you must do the same thing to both sides of the equation in order to keep the sides equal.

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