How to Use the Pythagorean Theorem
Ensure that your triangle is a right triangle., Assign the variables a, b, and c to the sides of your triangle., Determine which side(s) of the triangle you are solving for., Plug your two known values into the equation., Calculate the squares...
Step-by-Step Guide
-
Step 1: Ensure that your triangle is a right triangle.
The Pythagorean Theorem is applicable only to right triangles, so, before proceeding, it's important to make sure your triangle fits the definition of a right triangle.
Luckily, there is only one qualifying factor
- to be a right triangle, your triangle must contain one angle of exactly 90 degrees.
As a form of visual shorthand, right angles are often marked with a small square, rather than a rounded "curve"
to identify them as such.
Look for this special mark in the corner of your triangle. -
Step 2: Assign the variables a
In the Pythagorean Theorem, the variables a and b refer to the sides that meet in a right angle, while the variable c refers to the hypotenuse
- the longest side which is always opposite the right angle.
So, to begin, assign the shorter sides of your triangle the variables a and b (it doesn’t matter which side is labeled 'a' or 'b'), and assign the hypotenuse the variable c. , The Pythagorean Theorem allows mathematicians to find the length of any one of a right triangle's sides as long as they know the lengths of the other two sides.
Determine which of your sides has an unknown length
- a, b, and/or c.
If the length of only one of your sides is unknown, you're ready to proceed.
Let's say, for example, that we know that our hypotenuse has a length of 5 and one of the other sides has a length of 3, but we're not sure what the length of the third side is.
In this case, we know we're solving for the length of the third side, and, because we know the lengths of the other two, we're ready to go! We'll return to this example problem in the following steps.
If the lengths of two of your sides are unknown, you'll need to determine the length of one more side to use the Pythagorean Theorem.
Basic trigonometry functions can help you here if you know one of the non-right angles in the triangle. , Insert your values for the lengths of the sides of your triangle into the equation a2 + b2 = c2.
Remember that a and b are the non hypotenuse sides, while c is the hypotenuse.
In our example, we know the length of one side and the hypotenuse (3 & 5), so we would write our equation as 3² + b² = 5² , To solve your equation, begin by taking the square of each of your known sides.
Alternatively, if you find it easier, you may leave your side lengths in the exponent form, then square them later.
In our example, we would square 3 and 5 to get 9 and 25, respectively.
We can rewrite our equation as 9 + b² =
25. , If necessary, use basic algebra operations to get your unknown variable on one side of the equals sign and your two squares on the other side of the equals sign.
If you're solving for the hypotenuse, c will already be isolated, so you won't need to do anything to isolate it.
In our example, our current equation is 9 + b² =
25.
To isolate b², let's subtract 9 from both sides of the equation.
This leaves us with b² =
16. , You should now be left with one variable squared on one side of the equation and a number on the other side.
Simply take the square root of both sides to find the length of your unknown side.
In our example, b² = 16, taking the square root of both sides gives us b =
4.
Thus, we can say that the length of the unknown side of our triangle is
4. , The reason that the Pythagorean Theorem is so widely used today is that it's applicable in countless practical situations.
Learn to recognize right triangles in real life
- in any situation where two straight objects or lines meet at a right angle and a third line or object stretches diagonally across from the right angle, you can use the Pythagorean Theorem to find the length of one of the sides, given the length of the other two.
Let's try real-world example that's a little more difficult.
A ladder is leaning against a building.
The base of the ladder is 5 meters (16.4 ft) from the bottom of the wall.
The ladder reaches 20 meters (65.6 ft) up the wall of the building.
How long is the ladder? "5 meters (16.4 ft) from the bottom of wall” and "20 meters (65.6 ft) up the wall” clue us into the lengths of the sides of our triangle.
Since the wall and the ground (presumably) meet at a right angle and the ladder leans diagonally against the wall, we can think of this arrangement as a right triangle with sides of length a = 5 and b =
20.
The length of the ladder is the hypotenuse, so c is our unknown.
Let's use the Pythagorean Theorem: a² + b² = c² (5)² + (20)² = c² 25 + 400 = c² 425 = c² sqrt(425) = c c =
20.6 .
The approximate length of the ladder is
20.6 meters (67.6 ft). -
Step 3: and c to the sides of your triangle.
-
Step 4: Determine which side(s) of the triangle you are solving for.
-
Step 5: Plug your two known values into the equation.
-
Step 6: Calculate the squares.
-
Step 7: Isolate your unknown variable on one side of the equals sign.
-
Step 8: Take the square root of both sides of the equation.
-
Step 9: Use the Pythagorean Theorem to find the sides of real-world right triangles.
Detailed Guide
The Pythagorean Theorem is applicable only to right triangles, so, before proceeding, it's important to make sure your triangle fits the definition of a right triangle.
Luckily, there is only one qualifying factor
- to be a right triangle, your triangle must contain one angle of exactly 90 degrees.
As a form of visual shorthand, right angles are often marked with a small square, rather than a rounded "curve"
to identify them as such.
Look for this special mark in the corner of your triangle.
In the Pythagorean Theorem, the variables a and b refer to the sides that meet in a right angle, while the variable c refers to the hypotenuse
- the longest side which is always opposite the right angle.
So, to begin, assign the shorter sides of your triangle the variables a and b (it doesn’t matter which side is labeled 'a' or 'b'), and assign the hypotenuse the variable c. , The Pythagorean Theorem allows mathematicians to find the length of any one of a right triangle's sides as long as they know the lengths of the other two sides.
Determine which of your sides has an unknown length
- a, b, and/or c.
If the length of only one of your sides is unknown, you're ready to proceed.
Let's say, for example, that we know that our hypotenuse has a length of 5 and one of the other sides has a length of 3, but we're not sure what the length of the third side is.
In this case, we know we're solving for the length of the third side, and, because we know the lengths of the other two, we're ready to go! We'll return to this example problem in the following steps.
If the lengths of two of your sides are unknown, you'll need to determine the length of one more side to use the Pythagorean Theorem.
Basic trigonometry functions can help you here if you know one of the non-right angles in the triangle. , Insert your values for the lengths of the sides of your triangle into the equation a2 + b2 = c2.
Remember that a and b are the non hypotenuse sides, while c is the hypotenuse.
In our example, we know the length of one side and the hypotenuse (3 & 5), so we would write our equation as 3² + b² = 5² , To solve your equation, begin by taking the square of each of your known sides.
Alternatively, if you find it easier, you may leave your side lengths in the exponent form, then square them later.
In our example, we would square 3 and 5 to get 9 and 25, respectively.
We can rewrite our equation as 9 + b² =
25. , If necessary, use basic algebra operations to get your unknown variable on one side of the equals sign and your two squares on the other side of the equals sign.
If you're solving for the hypotenuse, c will already be isolated, so you won't need to do anything to isolate it.
In our example, our current equation is 9 + b² =
25.
To isolate b², let's subtract 9 from both sides of the equation.
This leaves us with b² =
16. , You should now be left with one variable squared on one side of the equation and a number on the other side.
Simply take the square root of both sides to find the length of your unknown side.
In our example, b² = 16, taking the square root of both sides gives us b =
4.
Thus, we can say that the length of the unknown side of our triangle is
4. , The reason that the Pythagorean Theorem is so widely used today is that it's applicable in countless practical situations.
Learn to recognize right triangles in real life
- in any situation where two straight objects or lines meet at a right angle and a third line or object stretches diagonally across from the right angle, you can use the Pythagorean Theorem to find the length of one of the sides, given the length of the other two.
Let's try real-world example that's a little more difficult.
A ladder is leaning against a building.
The base of the ladder is 5 meters (16.4 ft) from the bottom of the wall.
The ladder reaches 20 meters (65.6 ft) up the wall of the building.
How long is the ladder? "5 meters (16.4 ft) from the bottom of wall” and "20 meters (65.6 ft) up the wall” clue us into the lengths of the sides of our triangle.
Since the wall and the ground (presumably) meet at a right angle and the ladder leans diagonally against the wall, we can think of this arrangement as a right triangle with sides of length a = 5 and b =
20.
The length of the ladder is the hypotenuse, so c is our unknown.
Let's use the Pythagorean Theorem: a² + b² = c² (5)² + (20)² = c² 25 + 400 = c² 425 = c² sqrt(425) = c c =
20.6 .
The approximate length of the ladder is
20.6 meters (67.6 ft).
About the Author
Teresa Hart
Writer and educator with a focus on practical hobbies knowledge.
Rate This Guide
How helpful was this guide? Click to rate: