A regular quadrangular pyramid is inscribed in the cone. The height and radius of the cone

| April 25, 2021 | education

A regular quadrangular pyramid is inscribed in the cone. The height and radius of the cone are respectively equal to 15 cm and 8 cm. Find: 1) the side edge of the pyramid; 2) the side of the base of the pyramid; 3) the apothem of the pyramid; 4) the area of the base of the pyramid; 5) the area of the lateral surface of the pyramid.

The radius of the circle OA is half the diagonal of the square at the base of the pyramid.

In a right-angled triangle AOE, according to the Pythagorean theorem, we determine the length of the hypotenuse AE.

AE ^ 2 = OE ^ 2 + OA ^ 2 = 225 + 64 = 289.

AE = 17 cm.

The diagonals AC and BD of the square are equal and intersect at right angles, then triangle AOD is rectangular and isosceles.

AD ^ 2 = 2 * AO ^ 2 = 2 * 64.

AD = 8 * √2 cm.

Apothem EH is the height and median of an equilateral triangle AED, then AH = AD / 2 = 4 * √2 cm.

In a right-angled triangle AEН, according to the Pythagorean theorem, EH ^ 2 = AE ^ 2 – AH ^ 2 = 289 – 32 = 257.

EH = √257 cm.

Determine the area of ​​the base of the pyramid.

Sbn = АD ^ 2 = 128 cm2.

The lateral surface area is equal to the four areas of the triangle AED.

Side = 4 * Saed = AD * EH / 2 = 4 * 8 * √2 * √257 / 2 = 16 * √514 cm2.

Answer: 17 cm, 8 * √2 cm, √257 cm, 128 cm2, 16 * √514 cm2.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.