In this explainer, we will learn how to use the Pythagorean theorem to find the length of the hypotenuse or a leg of a right triangle and its area. The Pythagorean theorem gives a relationship between the sides of any right triangle. Before we start, let’s remember what a right triangle is and how to recognize its hypotenuse. A right triangle is a triangle that has one right angle and always one longest side. This longest side is always the side that is opposite the right angle, while the other sides, called the legs, form the right angle. The longest side is called the hypotenuse. Let’s start by considering an isosceles right triangle, 𝑇, shown in the figure. Squares have been added to each side of 𝑇. As 𝑇 is isosceles, we see that the squares drawn at the legs are each made of two 𝑇s, and we also see that four 𝑇s fit in the bigger square. We deduce from this that area of the bigger square, 𝐴, is equal to the sum of the area of the two other squares, 𝐴 and 𝐴. We can write this as 𝐴+𝐴=𝐴. Let 𝑎 and 𝑏 be the lengths of the legs of the triangle (so, in this special case, 𝑎=𝑏) and 𝑐 be the length of the hypotenuse. Therefore, 𝐴=𝑎, 𝐴=𝑏, and 𝐴=𝑐, and by substituting these into the equation, we find that 𝑎+𝑏=𝑐. This result can be generalized to any right triangle, and this is the essence of the Pythagorean theorem. The Pythagorean theorem states that, in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. Writing 𝑐 for the length of the hypotenuse and 𝑎 and 𝑏 for the lengths of the legs, we can express the Pythagorean theorem algebraically as 𝑎+𝑏=𝑐. There are many proofs of the Pythagorean theorem. We are going to look at one of them. Let’s consider a square of length 𝑎 and another square of length 𝑏 that are placed in two opposite corners of a square of length 𝑎+𝑏 as shown in the diagram below. By expanding (𝑎+𝑏), we can find the area of the two little squares (shaded in blue and green) and of the yellow rectangles. We find (𝑎+𝑏)=𝑎+2𝑎𝑏+𝑏. This can be found as well by considering that the big square of length 𝑎+𝑏 is made of square of area 𝑎, another square of area 𝑏, and two rectangles of area 𝑎𝑏. Now, the blue square and the green square are removed from the big square, and the yellow rectangles are split along one of their diagnoals, creating four congruent right triangles. They are then placed in the corners of the big square, as shown in the figure. As the four yellow triangles are congruent, the four sides of the white shape at the center of the big square are of equal lengths. (They are the hypotenuses of the yellow right triangles.) Also, the angle of the white shape and the two non-right angles of the right triangle from a straight line. As the measure of the two non-right angles ofa right triangle add up to 90∘, the angle of the white shape is 90∘. Therefore, the white shape isa square. Let 𝑐 be the length of the white square’s side (and of the hypotenuses of the yellow triangles). Using the fact that the big square is made of the white square and the four yellow right triangles, we find triangles, we find that the area ofthe big square is 𝑐+4×𝑎𝑏2; that is, 𝑐+2𝑎𝑏. Since the big squares in both diagrams are congruent (with side 𝑎+𝑏), we find that 𝑎+2𝑎𝑏+𝑏=𝑐+2𝑎𝑏, and so 𝑎+𝑏=𝑐. This is ageometric proof of the Pythagorean theorem.
Now that we know the Pythagorean theorem, let’s look at an example.
Find 𝑥 in the right triangle shown.
Answer
We are given a right triangle and must start by identifying its hypotenuse and legs. The right angle is ∠𝐹, and the legs form the right angle, so they are the sides 𝐹𝐷 and 𝐹𝐸. The hypotenuse is the side opposite ∠𝐹, which is therefore 𝐷𝐸.
From the diagram, we have been given the lengths of the two legs and need to work out 𝑥, the length of the hypotenuse. Writing 𝑎 and 𝑏 for the lengths of the legs and 𝑐 for the length of the hypotenuse, we recall the Pythagorean theorem, which states that 𝑎+𝑏=𝑐.
Substituting for 𝑎, 𝑏, and 𝑐 with the actual values given in the diagram, we get 5+12=𝑥.
We must now solve this equation for 𝑥. Simplifying the left-hand side, we have 25+144=𝑥169=𝑥.
To find the value of 𝑥, we take the square roots of both sides, remembering that 𝑥 is positive because it is a length. Thus, 𝑥=√169=13.
In the first example, we were asked to find the length of the hypotenuse of a right triangle. Now, let’s see what to do when we are asked to find the length of one of the legs.
Find 𝑥 in the right triangle shown.
Answer
We are given a right triangle and must start by identifying its hypotenuse and legs. The right angle is ∠𝐾, and the legs form the right angle, so they are the sides 𝐾𝐼 and 𝐾𝐽. The hypotenuse is the side opposite ∠𝐾, which is therefore 𝐼𝐽.
From the diagram, we have been given the length of the hypotenuse and one leg, and we need to work out 𝑥, the length of the other leg, 𝐽𝐾. Writing 𝑎 and 𝑏 for the lengths of the legs and 𝑐 for the length of the hypotenuse, we recall the Pythagorean theorem, which states that 𝑎+𝑏=𝑐.
Substituting for 𝑎, 𝑏, and 𝑐 with the actual values given on the diagram, we get 15+𝑥=17.
To solve for 𝑥, we start by expanding the square numbers: 225+𝑥=289.
Then, we subtract 225 from both sides, which gives us 𝑥=289−225,𝑥=64.so
Finally, to find 𝑥, we take the square roots of both sides, remembering that 𝑥 is positive because it is a length. Thus, 𝑥=√64=8.
Let’s summarize how to use the Pythagorean theorem to find an unknown side of a right triangle.
- Identify the hypotenuse and the legs of the right triangle.
- With 𝑎 and 𝑏 as the legs of the right triangle and 𝑐 as the hypotenuse, write the Pythagorean theorem: 𝑎+𝑏=𝑐.
- Substitute 𝑎, 𝑏, and 𝑐 with their actual values, using 𝑥 for the unknown side, into the above equation.
- Solve for 𝑥.
Once we have learned how to find the length of the hypotenuse or a leg, we can also use the Pythagorean theorem to answer geometric questions expressed as word problems. Here is an example of this type.
Determine the diagonal length of the rectangle whose length is 48 cm and width is 20 cm.
Answer
Here, we are given the description of a rectangle and need to find its diagonal length. It helps to start by drawing a sketch of the situation.
The rectangle has length 48 cm and width 20 cm. Therefore, its diagonal length, which we have labeled as 𝑥 cm, will be the length of the hypotenuse of a right triangle with legs of length 48 cm and 20 cm.
Recall the Pythagorean theorem, which states that, in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides (the legs). Writing 𝑐 for the length of the hypotenuse and 𝑎 and 𝑏 for the lengths of the legs, this can be expressed algebraically as 𝑎+𝑏=𝑐.
Substituting for 𝑎, 𝑏, and 𝑐 with the values from the diagram, we have 48+20=𝑥.
To solve this equation for 𝑥, we start by writing 𝑥 on the left-hand side and simplifying the squares: 𝑥=2304+400=2704.
Then, we take the square roots of both sides, remembering that 𝑥 is positive because it is a length. Hence, we have 𝑥=√2704=52.
The dimensions of the rectangle are given in centimetres, so the diagonal length will also be in centimetres. We conclude that a rectangle of length 48 cm and width 20 cm has a diagonal length of 52 cm.
The following example is a slightly more complex question where we need to use the Pythagorean theorem.
Find the perimeter of 𝐴𝐵𝐶𝐷.
Answer
In this question, we need to find the perimeter of 𝐴𝐵𝐶𝐷, which is a quadrilateral made up of two right triangles, 𝐴𝐵𝐶 and 𝐴𝐶𝐷.
From the diagram, △𝐴𝐵𝐶 is a right triangle at 𝐵, and △𝐴𝐶𝐷 is a right triangle at 𝐶. We also know three of the four side lengths of the quadrilateral, namely 𝐴𝐵, 𝐵𝐶, and 𝐶𝐷. To calculate the perimeter of 𝐴𝐵𝐶𝐷, we need to find its missing side length, 𝐴𝐷.
Note that 𝐴𝐷 is the hypotenuse of △𝐴𝐶𝐷, but we do not know 𝐴𝐶. However, 𝐴𝐶 is the hypotenuse of △𝐴𝐵𝐶, where we know both 𝐴𝐵 and 𝐵𝐶.
Now, recall the Pythagorean theorem, which states that in a right triangle where 𝑎 and 𝑏 are the lengths of the legs and 𝑐 is the length of the hypotenuse, we have 𝑎+𝑏=𝑐.
Therefore, we will apply the Pythagorean theorem first in triangle 𝐴𝐵𝐶 to find 𝐴𝐶 and then in triangle 𝐴𝐶𝐷 to find 𝐴𝐷.
In triangle 𝐴𝐵𝐶, 𝐴𝐶 is the length of the hypotenuse, which we denote by 𝑥. Substituting for all three side lengths in the Pythagorean theorem and then simplifying, we get 20+48=𝑥400+2304=𝑥2704=𝑥.
As 𝑥 is a length, it is positive, so taking the square roots of both sides gives us 𝑥=√2704=52.cm
In triangle 𝐴𝐵𝐶, 𝐴𝐷 is the length of the hypotenuse, which we denote by 𝑦. Since we now know the lengths of both legs, we can substitute them into the Pythagorean theorem and then simplify to get 52+39=𝑦2304+1521=𝑦4225=𝑦.
As 𝑦 is a length, it is positive, so taking the square roots of both sides gives us 𝑦=√4225=65.cm
Finally, we can work out the perimeter of quadrilateral 𝐴𝐵𝐶𝐷 by summing its four side lengths: 𝐴𝐵+𝐵𝐶+𝐶𝐷+𝐴𝐷=20+48+39+65=172.
All lengths are given in centimetres, so the perimeter of 𝐴𝐵𝐶𝐷 is 172 cm.
The Pythagorean theorem can also be applied to help find the area of a right triangle as follows. When given the lengths of the hypotenuse and one leg, we can always use the Pythagorean theorem to work out the length of the other leg. Note that if the lengths of the legs are 𝑥 and 𝑦, then 𝑥𝑦 would represent the area of a rectangle with side lengths 𝑥 and 𝑦. Therefore, the quantity 𝑥𝑦2, which is half of this area, represents the area of the corresponding right triangle.
We will finish with an example that requires this step.
In the trapezoid 𝐴𝐵𝐶𝐷 below, 𝐴𝐷⫽𝐵𝐶 and 𝐴𝐸⟂𝐵𝐶. Find its area.
Answer
Here, we are given a trapezoid and must use information from the question to work out more details of its properties before finding its area.
The fact that 𝐴𝐸 is perpendicular to 𝐵𝐶 implies that 𝐴𝐵𝐸 is a right triangle with its right angle at 𝐸.
Similarly, since both 𝐴𝐸 and 𝐷𝐶 are perpendicular to 𝐵𝐶, then they must be parallel. When combined with the fact that 𝐴𝐷 is parallel to 𝐵𝐶 (and hence to 𝐸𝐶), this implies that 𝐴𝐸𝐶𝐷 is a rectangle.
Therefore, the area of the trapezoid will be the sum of the areas of right triangle 𝐴𝐵𝐸 and rectangle 𝐴𝐸𝐶𝐷.
First, consider 𝐴𝐵𝐸. We know that the hypotenuse 𝐴𝐵 has length 𝐴𝐵=15cm. In addition, we can work out the length of the leg 𝐵𝐸 because 𝐵𝐸=𝐵𝐶−𝐸𝐶=15−6=9cm.
Now, recall the Pythagorean theorem, which states that, in a right triangle where 𝑎 and 𝑏 are the lengths of the legs and 𝑐 is the length of the hypotenuse, we have 𝑎+𝑏=𝑐.
As we know two side lengths of the right triangle 𝐴𝐵𝐸, we can apply the Pythagorean theorem to find the missing length of leg 𝐴𝐸. Writing 𝑥 for this length and substituting for 𝑎, 𝑏, and 𝑐, we have 𝑥+9=15.
To solve for 𝑥, we start by expanding the square numbers: 𝑥+81=225.
Then, we subtract 81 from both sides, which gives us 𝑥=225−81,𝑥=144.so
To find 𝑥, we take the square roots of both sides, remembering that 𝑥 is positive because it is a length. Thus, 𝑥=√144=12.cm
Since we now know the lengths of the legs of right triangle 𝐴𝐵𝐸 are 9 cm and 12 cm, we can work out its area by multiplying these values and dividing by 2. Therefore, theareaof△𝐴𝐵𝐸=9×122=1082=54.
Secondly, consider rectangle 𝐴𝐸𝐶𝐷. Notice that its width is given by 𝐴𝐷=6cm. Moreover, we also know its height because it is the same as the missing length of leg of right triangle 𝐴𝐵𝐸 that we calculated above, which is 12 cm. Therefore, theareaofrectangle𝐴𝐸𝐶𝐷=6×12=72.
Finally, the area of the trapezoid is the sum of these two areas: 54+72=126. Since the lengths are given in centimetres then this area will be in square centimetres. The area of the trapezoid is 126 cm2.
Let’s finish by recapping some key concepts from this explainer.
- The Pythagorean theorem states that, in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides (called the legs).
- Writing 𝑐 for the length of the hypotenuse, and 𝑎 and 𝑏 for the lengths of the legs, we can express the Pythagorean theorem algebraically as 𝑎+𝑏=𝑐.
- We can use the Pythagorean theorem to find the length of the hypotenuse or a leg of a right triangle and to solve more complex geometric problems involving areas and perimeters of right triangles.