A right-angled triangle (or right triangle) is a triangle in which one of the angles is $90^\circ$90° - that is, a right angle. In diagrams, the $90^\circ$90° angle is indicated by a small square. The longest side of a right-angled triangle is always the one which is opposite the right angle (as this is always the largest internal angle), and it is called the hypotenuse.
Pythagoras' theorem
Pythagoras was an ancient Greek philosopher who lived about $2500$2500 years ago. It was thought that either he or one of his followers proved a theorem relating the three sides of a right-angled triangle. Hence the theorem is named after Pythagoras. However, the same concept was actually known and used by ancient Babylonians, Hindus and Chinese centuries before his time.
Pythagoras' theorem tells us that the three sides of any right-angled triangle are related in the following way:
For a right-angled triangle with sides measuring $a$a, $b$b and $c$c:
$a^2+b^2=c^2$a2+b2=c2
where $c$c represents the length of the triangle's hypotenuse (the longest side) and $a$a and $b$b are the lengths of the two short sides.
This theorem tells us that the sum of the squares of the short sides $\left(a^2+b^2\right)$(a2+b2) is equal to the square of the hypotenuse $\left(c^2\right)$(c2).
We can also think of this in terms of area. The following applet shows that the area of a square with side length equal to the hypotenuse is equal to the sum of areas of the squares with side lengths equal to the two shorter sides.
Pythagoras' theorem can be used to find the length of the sides of a right-angled triangle if two of the sides are known.
Finding the length of the hypotenuse
When we are asked to find the hypotenuse, or longest side, of a right-angled triangle we can use Pythagoras' theorem directly if we know the length of the two short sides.
Worked examples
example 1
Find the length of the hypotenuse of a right-angled triangle whose two other sides measure $10$10 cm and $12$12 cm.
Think: Here we want to find the length of the hypotenuse (the longest side), and are given the lengths of the two shorter sides $10$10 and $12$12. Using Pythagoras' theorem, we call the hypotenuse $c$c and the two shorter sides $a$a and $b$b.
Do:
| $c^2$c2 | $=$= | $a^2+b^2$a2+b2 |
start with the formula |
| $c^2$c2 | $=$= | $10^2+12^2$102+122 |
fill in the values for $a$a and $b$b |
| $c^2$c2 | $=$= | $100+144$100+144 |
evaluate the squares |
| $c^2$c2 | $=$= | $244$244 |
add the $100$100 and $144$144 together |
| $c$c | $=$= | $\sqrt{244}$√244 |
take the square root of both sides |
| $c$c | $=$= | $15.62$15.62 cm |
rounding to $2$2 decimal places |
Therefore, the length of the hypotenuse is $15.62$15.62 cm (to two decimal places). It is worth checking our working by ensuring that the length of the hypotenuse we have calculated is longer than the two shorter sides.
example 2
Find the length of the hypotenuse of a right-angled triangle whose two other sides measure $3$3 cm and $4$4 cm.
Think: Here we want to find $c$c the hypotenuse, and are given the lengths of the two shorter sides $a$a and $b$b.
Do: We will substitute the values we know in the formula and then solve to find $c$c.
| $c^2$c2 | $=$= | $3^2+4^2$32+42 |
fill in the values for $a$a and $b$b |
| $c^2$c2 | $=$= | $9+16$9+16 |
evaluate the squares |
| $c^2$c2 | $=$= | $25$25 |
add the numbers together |
| $c$c | $=$= | $\sqrt{25}$√25 |
take the square root of both sides |
| $c$c | $=$= | $5$5 cm |
|
Reflect: The numbers $3$3, $4$4, and $5$5 form a Pythagorean Triad, sometimes called a Pythagorean Triple.
Practice questions
Question 1
Which side of the triangle in the diagram is the hypotenuse?
$AB$AB $CA$CA $BC$BC
Question 2
Find the length of the hypotenuse, $c$c in this triangle.
Finding the length of a short side
When we are asked to find the length of one of the short sides of a right-angled triangle we can use Pythagoras' theorem if we know the length of the hypotenuse and the other short side. We can manipulate the formula to enable us to find the length of either short side as follows:
$b^2=c^2-a^2$b2=c2−a2
$a^2=c^2-b^2$a2=c2−b2
Notice that, when determining a short side, the formula always involves subtracting the square of the other short side from the square of the hypotenuse and taking the square root of the result.
Worked example
Example 3
Find the length of unknown side $b$b of a right-angled triangle whose hypotenuse is $6$6 mm and one other side is $4$4 mm.
Think: Here we want to find $b$b, the length of a shorter side.
Do:
| $b^2$b2 | $=$= | $c^2+a^2$c2+a2 |
| $b^2$b2 | $=$= | $6^2-4^2$62−42 |
| $b$b | $=$= | $\sqrt{6^2-4^2}$√62−42 |
| $b$b | $=$= | $4.47$4.47 mm ($2$2 d.p.) |
Practice questions
Question 3
Calculate the value of $b$b in the triangle below. 
Question 4
Find $b$b, where $b$b cm is the length of one side of a right-angled triangle whose hypotenuse is $3$3 cm in length and whose other side is $2$2 cm in length. Round your answer to two decimal places.
Is the triangle right-angled?
The theorem can also be used to determine whether a triangle is right-angled.
For example, if a triangle has side lengths measuring $6$6 cm, $8$8 cm and $10$10 cm, we can check if it satisfies Pythagoras' theorem:
$c^2=a^2+b^2$c2=a2+b2
$RHS$RHS$=$=$10^2=100$102=100 and $LHS$LHS$=$=$6^2+8^2=100$62+82=100
So $10^2=6^2+8^2$102=62+82, and a triangle with sides $6$6, $8$8 and $10$10 units is a right-angled triangle.
Practice question
Question 5
Use Pythagoras' theorem to determine whether this is a right-angled triangle. Let $a$a and $b$b represent the two shorter side lengths. First find the value of $a^2+b^2$a2+b2. Let $c$c represent the length of the longest side. Find the value of $c^2$c2. Is the triangle a right-angled triangle? Yes No
Problem solving with Pythagoras' theorem
Pythagoras' theorem can be used in many practical applications to either determine unknown lengths of right-angled triangles (as long as we know two out of three side lengths) or check if a triangle is right-angled. For example, to check that a rectangular building frame is square (that is, has right-angled corners), a builder could measure two adjacent side lengths and a diagonal that form a triangle and test if they satisfy Pythagoras' theorem. If they don't then the frame is not square.
A useful strategy to solve practical problems involving Pythagoras' theorem is to:
- Draw a diagram of the situation
- Identify any right-angled triangles
- Identify known sides of the right-angled triangles and always label the hypotenuse as side $c$c
- Use Pythagoras' theorem to work out the unknown side/s
Practice questions
Question 6
The screen on a handheld device has dimensions $9$9 cm by $6$6 cm, and a diagonal of length $x$x cm. What is the value of $x$x? Round your answer to two decimal places.
Question 7
The top of a flag pole is $4$4 metres above the ground and the shadow cast by the flag pole is $9$9 metres long. The distance from the top of the flag pole to the end of its shadow is $d$d m. Find $d$d, rounded to two decimal places.
Question 8
A farmer wants to build a fence around the entire perimeter of his land, as shown in the diagram. The fencing costs $\$37$$37 per metre. What is the value of $x$x? Round your answer correct to two decimal places. What is the value of $y$y? Round your answer correct to two decimal places. How many whole metres of fencing does the farmer need to purchase, if fencing is sold by the metre? At $\$37$$37 per metre of fencing, how much will it cost him to build the fence along the entire perimeter of the land?