Question

В основании пирамиды лежит прямоугольный треугольник катет которого=12 см,а радиус описанного круга =7,5 см.все высоты боковых граней=5 см.найти обьем пирамиды

Answer 1

Объем пирамиды вычисляется по формуле  V = Sосн * h / 3

Поскольку центр описанного круга - середина гипотенузы, то длина гипотенузы равна  2 * 7,5 = 15 см. По теореме Пифагора второй катет равен

√ (15² - 12²) = √ 81 = 9 см, а площадь основания

Sосн = 12 * 9 / 2 = 54 см²

Поскольку высоты боковых граней равны, то вершина пирамиды проектируется в центр вписанного круга. Радиус его равен

r = 2 * S / (a + b + c) = 2 * 54 / (9 + 12 + 15) = 108 / 36 = 3 см.

Тогда по теореме Пифагора высота пирамиды

h = √ (5² - 3²) = √ 16 = 4 см, а ее объем

V = 54 * 4 / 3 = 72 см³.

Answer 2

Given Information

We are given the following information about the pyramid: - The base of the pyramid is a right-angled triangle with one of the legs measuring 12 cm. - The radius of the circumscribed circle is 7.5 cm. - The heights of the lateral faces are all 5 cm.

Finding the Volume of the Pyramid

To find the volume of the pyramid, we need to know the height of the pyramid. Unfortunately, the given information does not directly provide the height. However, we can use the given information to find the height.

Let's denote the right-angled triangle as ABC, with AB as the hypotenuse, AC as the base, and BC as the other leg. We are given that AC = 12 cm.

Using the Pythagorean theorem, we can find the length of BC: BC = sqrt(AB^2 - AC^2)

We are also given that the radius of the circumscribed circle is 7.5 cm. The radius of the circumscribed circle of a right-angled triangle is equal to half the length of the hypotenuse. Therefore, we have: AB = 2 * radius = 2 * 7.5 cm = 15 cm

Substituting the values of AB and AC into the equation for BC, we can calculate BC: BC = sqrt(15^2 - 12^2) = sqrt(225 - 144) = sqrt(81) = 9 cm

Now that we know the lengths of all three sides of the right-angled triangle, we can calculate the height of the pyramid. The height of the pyramid is equal to the radius of the circumscribed circle of the base triangle. Therefore, the height of the pyramid is 7.5 cm.

Using the formula for the volume of a pyramid, which is V = (1/3) * base area * height, we can calculate the volume of the pyramid. The base area of the pyramid is equal to (1/2) * AC * BC.

Substituting the values into the formula, we have: V = (1/3) * (1/2) * AC * BC * height V = (1/3) * (1/2) * 12 cm * 9 cm * 7.5 cm V = (1