Volume of a Cylinder and Cone Composite Solid (Interactive 3D)

Grades 8-9

Question: Create an interactive 3D graph. Draw a solid made of a cylinder and a cone. The cylinder has radius 3 cm and height 2 cm. The cone is on top of the cylinder. The cone has radius 3 cm and height 5 cm. Show the rotation axis on the left side. Label the radius as 3 cm, the cylinder height as 2 cm, and the cone height as 5 cm. Find the total volume step by step.

To find the total volume of a composite solid made of a cylinder and a cone, we calculate the volume of each part separately and then add them together.

1. Volume of the Cylinder

The cylinder at the base has a radius

r = 3

cm and a height

h_1 = 2

cm. The formula for the volume of a cylinder is:

V_{cylinder} = \pi \cdot r^2 \cdot h_1

V_{cylinder} = \pi \cdot 3^2 \cdot 2

V_{cylinder} = \pi \cdot 9 \cdot 2

V_{cylinder} = 18\pi \text{ cm}^3 \approx 56.55 \text{ cm}^3

2. Volume of the Cone

The cone on top shares the same radius

r = 3

cm and has a height

h_2 = 5

cm. The formula for the volume of a cone is:

V_{cone} = \frac{1}{3} \cdot \pi \cdot r^2 \cdot h_2

V_{cone} = \frac{1}{3} \cdot \pi \cdot 3^2 \cdot 5

V_{cone} = \frac{1}{3} \cdot \pi \cdot 9 \cdot 5

V_{cone} = 3 \cdot \pi \cdot 5

V_{cone} = 15\pi \text{ cm}^3 \approx 47.12 \text{ cm}^3

3. Total Volume

To find the total volume, we sum the two individual volumes:

V_{total} = V_{cylinder} + V_{cone}

V_{total} = 18\pi + 15\pi

V_{total} = 33\pi \text{ cm}^3

Using

\pi \approx 3.14159

V_{total} \approx 33 \cdot 3.14159

V_{total} \approx 103.67 \text{ cm}^3

Visual Description

In the 3D model below:

The blue solid represents the cylinder with a height of 2 cm.
The orange solid represents the cone sitting directly on top with a height of 5 cm.
The red dashed line passing through the center is the axis of rotation.
The labeled segments indicate the dimensions: the radius ( $r = 3$ ), the base height ( $h = 2$ ), and the top height ( $h = 5$ ).