Question

The square has a radius of 3 square root 2 what is the apothem?

Answer 1

The diagram of the square is shown in the first picture below 

We have the radius $3 \sqrt{2}$ and the triangle formed is an isosceles triangle 

We can start by finding out the length of one side of the square. A square has four sides of equal length and the length of the diagonal is twice the length of the radius. So, by using the Pythagoras theorem we have

$(2r)^{2} = a^{2}+ a^{2}$
$4 r^{2} =2 a^{2}$, substitute $r=3 \sqrt{2}$
$4 (3 \sqrt{2}) ^{2} = 2 a^{2}$
$4(18) = 2 a^{2}$
$72=2 a^{2}$
$a^{2}= \frac{72}{2}$
$a^{2}=36$
$a=6$

Then we'll use Pythagoras theorem again to find the length of the Apothem

We need to halve the length of one side of the square to obtain the other short side of the right angle triangle, so 6÷2 =3

apothem² = (3√2)² + 3²
apothem² = 18+9
apothem² = 27
apothem = √27 = 3√3