# A square with a side of 2√2 is inscribed at the base of the cone. the height of the cone is 2√3

**A square with a side of 2√2 is inscribed at the base of the cone. the height of the cone is 2√3. Find the lateral surface area of the cone.**

Since a square is inscribed at the base of the cone, the diagonal of the square is equal to the diameter of the circle at the base of the cone, and then the radius of the circle is half the length of the diagonal of the square.

From the right-angled triangle ABC, we determine the length of the hypotenuse AB.

AB ^ 2 = AC ^ 2 + BC ^ 2 = 8 + 8 = 16.

AB = 4 cm.

Then R = ОА = ОВ = AB / 2 = 4/2 = 2 cm.

From the right-angled triangle OBD, we determine the length of the hypotenuse BD, which is the generatrix of the cone.

BD ^ 2 = OD ^ 2 + OB ^ 2 = 12 + 4 = 16.

ВD = L = 4 cm.

Let us determine the area of the lateral surface of the cone.

Side = n * R * L = n * 2 * 4 = 8 * n cm2.

Answer: The lateral surface area is 8 * n cm2.