Question

A=1/2aP a=apothem P=perimeter so A=? a square, side=24

Answer 1

Conditions

A=1/2aP a=apothem P=perimeter so A=? a square, side=24

Solution

If we have an apothem, then our object is a right square pyramid. All its faces are right triangles, its base is a square. So, P-perimeter is:

$$P = 4 \cdot 24 = 96$$

As an apothem is a height of face (drawn to the base side), we can find it by using the Pythagorean theorem:

$$\alpha^2 = 24^2 - 12^2 = 432$$ $$\alpha = \sqrt{432} = 12\sqrt{3}$$

So, A is:

$$A = \frac{1}{2} \cdot 12\sqrt{3} \cdot 96 = 576\sqrt{3}$$

Need to notice, that A is nothing else but the lateral surface area of our pyramid.

**Answer:** $576\sqrt{3}$