The height of a regular quadrangular pyramid is 5 cm and makes an angle of 60 degrees with the side face.

The height of a regular quadrangular pyramid is 5 cm and makes an angle of 60 degrees with the side face. Find the total surface area of the pyramid.

The height of the pyramid forms a right-angled triangle with an apothem and a segment at the base of the pyramid, which is half the length of the base.

Let’s find the length of the apothem.

L = 5 cm ÷ cos60 ° = 10 cm.

Let’s find the length of the base.

a = 2 * L * sin60 ° = 2 * 10 cm * sin60 ° = 17 cm.

Let’s find the area of the base.

Sо = a² = (17 cm) ² = 289 cm².

Let’s find the area of the lateral surface.

St = 4 * L * a ÷ 2 = 2 * L * a.

St = 2 * 10 cm * 17 cm = 340 cm.

Let’s find the total area.

S = 289 cm² + 340 cm² = 629 cm².

Answer: the area of the base of the pyramid is 629 cm².