In a regular quadrangular pyramid, the height is 12 cm and the apothem of the lateral face is 15 cm. Find the area of the lateral surface of the pyramid.
Consider a right-angled triangle KOM, in which the hypotenuse KM is equal to the apothem and is equal to 15 cm, and the leg KO is equal to the length of the height and is equal to 12 cm. By the Pythagorean theorem, we find the leg OM.
OM ^ 2 = KM ^ 2 – KO ^ 2 = 15 ^ 2 – 12 ^ 2 = 225 – 144 = 81. OM = 9 cm.
Then the side of the base of the pyramid is equal to AD = 2 * OM = 2 * 9 = 18 cm.
Let’s define the perimeter of the base. P = 4 * AD = 4 * 18 = 72 cm.
The area of the lateral surface of the pyramid is S side = 1/2 * P * KM = (72 * 15) / 2 = 540 cm2.
Answer: S side = 540 cm2.
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