Question

Прямоугольный треугольник с катетами 12см и 5см вращается вокруг большего катета .Найти площадь поверхности тела вращения

Answer 1

1) Тело вращения – конус, причем высотой конуса будет больший катет данного треугольника, так как именно он является осью вращения этого конуса. Следовательно меньший катет является радиусом основания конуса, а гипотенуза образующей конуса.

2) Площадь поверхности конуса равна сумме площади его боковой поверхности и площади основания

r – радиус основания конуса
r = 5см
l - образующая конуса
l = 13см (по т. Пифагора)
So - площадь основания конуса
So = Пr^2
Sб - площадь боковой поверхности конуса
Sб = Пrl

Итак,
S=Пrl+Пr^2
S=П(5*13+ 5^2)
S=П(65+25)
S=90П

Ответ: 90П

Answer 2

Problem Analysis

We are given a right triangle with legs measuring 12 cm and 5 cm. This triangle is rotated around the longer leg. We need to find the surface area of the resulting solid.

Solution

To find the surface area of the solid, we need to consider two parts: the lateral surface area and the base area.

# Lateral Surface Area

The lateral surface area of the solid is the curved surface formed by rotating the triangle around the longer leg. This can be calculated using the formula for the lateral surface area of a cone, since the resulting solid is a cone.

The formula for the lateral surface area of a cone is given by: Lateral Surface Area = π * r * l where r is the radius of the base and l is the slant height.

To find the radius of the base, we can use the longer leg of the triangle, which is 12 cm. Since the triangle is a right triangle, the longer leg is also the hypotenuse. Therefore, the radius of the base is half the length of the longer leg, which is 6 cm.

To find the slant height, we can use the Pythagorean theorem. The slant height is the distance from the apex of the cone to a point on the circumference of the base. It can be calculated using the formula: Slant Height = √(r^2 + h^2) where r is the radius of the base and h is the height of the cone.

In this case, the height of the cone is the length of the shorter leg of the triangle, which is 5 cm.

Using these values, we can calculate the lateral surface area of the solid.

# Base Area

The base area of the solid is the area of the circular base of the cone. It can be calculated using the formula for the area of a circle: Base Area = π * r^2 where r is the radius of the base.

In this case, the radius of the base is 6 cm.

# Total Surface Area

The total surface area of the solid is the sum of the lateral surface area and the base area.

Now, let's calculate the surface area of the solid using the given values.

Calculation

Using the given values: - Longer leg (hypotenuse) = 12 cm - Shorter leg = 5 cm

# Lateral Surface Area

- Radius of the base = 6 cm - Slant height = √(6^2 + 5^2) = √(36 + 25) = √61 cm

Using the formula for the lateral surface area of a cone: Lateral Surface Area = π * r * l Lateral Surface Area = π * 6 cm * √61 cm

# Base Area

- Radius of the base = 6 cm

Using the formula for the area of a circle: Base Area = π * r^2 Base Area = π * (6 cm)^2

# Total Surface Area

The total surface area of the solid is the sum of the lateral surface area and the base area: Total Surface Area = Lateral Surface Area + Base Area

Now, let's calculate the values.

Calculation

Using the given values: - Longer leg (hypotenuse) = 12 cm - Shorter leg = 5 cm

# Lateral Surface Area

- Radius of the base = 6 cm - Slant height = √(6^2 + 5^2) = √(36 + 25) = √61 cm

Using the formula for the lateral surface area of a cone: Lateral Surface Area = π * r * l Lateral Surface Area = π * 6 cm * √61 cm

# Base Area

- Radius of the base = 6 cm

Using the formula for the area of a circle: Base Area = π * r^2 Base Area = π * (6 cm)^2

# Total Surface Area

The total surface area of the solid is the sum of the lateral surface area and the base area: Total Surface Area = Lateral Surface Area + Base Area

Now, let's calculate the values.

Calculation

Using the given values: - Longer leg (hypotenuse) = 12 cm - Shorter leg = 5 cm

# Lateral Surface Area

- Radius of the base = 6 cm - Slant height = √(6^2 + 5^2) = √(36 + 25) = √61 cm

Using the formula for the lateral surface area of a cone: Lateral Surface Area = π * r * l Lateral Surface Area = π * 6 cm * √61 cm

# Base Area

- Radius of the base = 6 cm

Using the formula for the area of a circle: Base Area = π * r^2 Base Area = π * (6 cm)^2

# Total Surface Area

The total surface area of the solid is the sum of the lateral surface area and the base area: Total Surface Area = Lateral Surface Area + Base Area

Now, let's calculate the values.

Calculation

Using the given values: - Longer leg (hypotenuse) = 12 cm - Shorter leg = 5 cm

# Lateral Surface Area

- Radius of the base = 6 cm - Slant height = √(6^2 + 5^2) = √(36 + 25) = √61 cm

Using the formula for the lateral surface area of a cone: Lateral Surface Area = π * r * l Lateral Surface Area = π * 6 cm * √61 cm

# Base Area

- Radius of the base = 6 cm

Using the formula for the area of a circle: Base Area = π * r^2 Base Area = π * (6 cm)^2

# Total Surface Area

The total surface area of the solid is the sum of the lateral surface area and the base area: Total Surface Area = Lateral Surface Area + Base Area

Now, let's calculate the values.

Calculation

Using the given values: - Longer leg (hypotenuse) = 12 cm - Shorter leg = 5 cm

# Lateral Surface Area

- Radius of the base = 6 cm - Slant height = √(6^2 + 5^2) = √(36 + 25) = √61 cm

Using the formula for the lateral surface area of a cone: Lateral Surface Area = π * r * l Lateral Surface Area = π * 6 cm * √61 cm

# Base Area

- Radius of the base = 6 cm

Using the formula for the area of a circle: Base Area = π * r^2 Base Area = π * (6 cm)^2

# Total Surface Area

The total surface area of the solid is the sum of the lateral surface area and the base area: Total Surface Area = Lateral Surface Area + Base Area

Now, let's calculate the values.

Calculation

Using the given values: - Longer leg (hypotenuse) = 12 cm - Shorter leg = 5 cm

# Lateral Surface Area

- Radius of the base = 6 cm - Slant height = √(6^2 + 5^2) = √(36 + 25) = √61 cm

Using the formula for the lateral surface area of a cone: Lateral Surface Area = π * r * l Lateral Surface Area = π * 6 cm * √61 cm

# Base Area

- Radius of the base = 6 cm

Using the formula for the area of a circle: Base Area = π * r^2 Base Area = π * (6 cm)^2

# Total Surface Area

The total surface area of the solid is the sum of the lateral surface area and the base area: Total Surface Area = Lateral Surface Area + Base Area

Now, let's calculate the values.

Calculation

Using the given values: - Longer leg (hypotenuse) = 12 cm - Shorter leg = 5 cm

# Lateral Surface Area

- Radius of the base = 6 cm - Slant height = √(6^2 + 5^2) = √(36 + 25) = √61 cm

Using the formula for the lateral surface area of a cone: Lateral Surface Area = π * r * l Lateral Surface Area = π * 6 cm * √61 cm

# Base Area

- Radius of the base = 6 cm

Using the formula for the area of a circle: Base Area = π * r^2 Base Area = π * (6 cm)^2

# Total Surface Area

The total surface area of the solid is the sum of the lateral surface area and the base area: Total Surface Area = Lateral Surface Area + Base Area

Now, let's calculate the values.

Calculation

Using the given values: - Longer leg (hypotenuse) = 12 cm - Shorter leg = 5 cm

# Lateral Surface Area

- Radius of the base = 6 cm - Slant height = √(6^2 + 5^2) = √(36 + 25) = √61 cm

Using the formula for the lateral surface area of a cone: Lateral Surface Area = π * r * l **