How to Use Pythagoras' Theorem to Find Side? [pptime time="1:00"](1:00) [/pptime]

You can use Pythagoras' Theorem to find the missing side in a right-angled triangle . For example, given a right-angled triangle with one side of 5 and hypotenuse of 13. Find the other side $x$. Step 1: Use Pythagoras' Theorem: $$ 5^2 + x^2 = 13^2 $$ Step 2: Simplifying the equations: $$ \begin{aligned} 25 + x^2 &= 169 \\ x^2 &= 144 \end{aligned} $$ Step 3: Take the positive square root (since hypotenuse $x$ cannot be negative): $$ x = \sqrt{144} = 12 $$ Thus, the other side is 12.

Applying Pythagoras' Theorem in 3D [pptime time="2:20"](2:20) [/pptime]

Pythagoras' Theorem can also be used in three dimensions to find the space diagonal of a cuboid—the longest diagonal. The trick is to apply Pythagoras' Theorem twice . For example, given a cuboid with side lengths 4, 2, and 3, find the space diagonal $x$: Step 1: First, find the diagonal of the base $d$ using Pythagoras' Theorem: \begin{aligned} d^2 &= 4^2 + 2^2 \\[0.5em] d^2 &= 16 + 4 \\[0.5em] d^2 &= 20 \end{aligned} Step 2: Now, use Pythagoras' Theorem again to find the full space diagonal $x$: $$ d^2 + 3^2 = x^2 $$ $$ 20 + 9 = x^2 $$ $$ x = \sqrt{29} $$ Thus, the space diagonal of the cuboid is $\sqrt{29}$.

Pythagoras Theorem

🎬 Video: Pythagoras Theorem Formula and Use

What is Pythagoras' Theorem? (0:01)

Pythagoras’ Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

The formula is:

$$ a^2 + b^2 = c^2 $$

where:

$a$ and $b$ are the two shorter sides (legs).
$c$ is the hypotenuse (the longest side opposite the right angle).

How to Use Pythagoras' Theorem to Find Side? (1:00)

You can use Pythagoras’ Theorem to find the missing side in a right-angled triangle.

For example, given a right-angled triangle with one side of 5 and hypotenuse of 13. Find the other side $x$.

Step 1: Use Pythagoras’ Theorem: $$ 5^2 + x^2 = 13^2 $$
Step 2: Simplifying the equations:

$$
\begin{aligned}
25 + x^2 &= 169 \\
x^2 &= 144
\end{aligned}
$$

Step 3: Take the positive square root (since hypotenuse $x$ cannot be negative):

$$ x = \sqrt{144} = 12 $$

Thus, the other side is 12.

Applying Pythagoras' Theorem in 3D (2:20)

Pythagoras’ Theorem can also be used in three dimensions to find the space diagonal of a cuboid—the longest diagonal. The trick is to apply Pythagoras’ Theorem twice.

For example, given a cuboid with side lengths 4, 2, and 3, find the space diagonal $x$: