An equilateral triangle has an apothem of 5 cm. find the perimeter of the triangle to the nearest centimeter.

The apothem of the triangle is equal to the height.  By the Pythagorean Theorem:

s^2=h^2+(b/2)^2

s^2=h^2+b^2/4

s^2=(4h^2+b^2)/4

4s^2=4h^2+b^2,  since this is an equilateral triangle s=b so

4s^2=4h^2+s^2

3s^2=4h^2

s^2=4h^2/3

s=√(4h^2/3), and the perimeter of an equilateral triangle is just 3s so

p=3√(4h^2/3)

p=√(12h^2), we were told that h=5cm so

p=√(12*25)

p=√300 cm

p=10√3 cm

p≈17.32 cm (to nearest hundredth of a centimeter)