### Video Transcript

A square has an area of 349 square centimetres. Determine the length of its diagonal.

Let’s consider a square with length 𝑥 centimetres and area 349 square centimetres. The formula used to calculate the area of a square is 𝑥 squared. We multiply the length of the square by the width of the square.

When we draw on the diagonal of the square, we create two right angled triangles. To calculate the longest side or hypotenuse of a right angle triangle, we can use the Pythagorean theorem: 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑐 squared is the hypotenuse, in this case 𝑦, the longest side of the triangle.

Applying the Pythagorean theorem gives us 𝑥 squared plus 𝑥 squared equals 𝑦 squared. However, we already know that 𝑥 squared is equal to 349. Therefore, we have 349 plus another 349 equals 𝑦 squared. Adding 349 and 349 gives us 698. So 𝑦 squared equals 698.

Finally, if we square root both sides of our equation, we have 𝑦 equals the square root of 698. Therefore, the diagonal of a square with area 349 square centimetres is equal to the root of 698 centimetres.

You might notice here that the answer, the square root of 698, is double the area square rooted. As we see from the calculation, multiplying 349 by two gives us 698. And the diagonal of the square is the square root of 698.

In a similar way, if we had a different square with area nine square centimetres, the diagonal would be the square root of 18 centimetres as two multiplied by nine is 18. This method is a shortcut, which incorporates the Pythagorean theorem and the formula for the area of a square.