Cross Sections and Solids of Revolution

Understanding how two-dimensional (2D) shapes and three-dimensional (3D) figures interact is a core part of geometry. We can explore this by slicing 3D objects to reveal 2D shapes, or by spinning 2D shapes to create 3D objects.

Cross Sections

A cross section is the 2D shape formed when a flat plane slices through a solid 3D figure. Think of it like cutting a piece of fruit and looking at the exposed face.

The shape of the cross section depends on the angle of the slice:

• Cone sliced parallel to the base: The cross section is a circle.
• Cone sliced perpendicular to the base (through the vertex): The cross section is a triangle.
• Cylinder sliced parallel to the base: The cross section is a circle.
• Cylinder sliced perpendicular to the base: The cross section is a rectangle.

Solids of Revolution

A solid of revolution is a 3D figure created by rotating a 2D shape around a straight line, called the axis of revolution.

Here are some common solids formed by revolution:

• Rotating a rectangle around one of its sides creates a cylinder.
• Rotating a right triangle around one of its legs creates a cone.
• Rotating a semicircle around its straight edge (diameter) creates a sphere.

Example: Finding the Volume of a Solid of Revolution

Problem: A right triangle with legs of length $3$ and $4$ is rotated about the longer leg. Describe the solid formed and find its volume.

Step 1: Describe the solid. When a right triangle is rotated around one of its legs, it forms a cone.

Step 2: Identify the dimensions. The triangle is rotated around the longer leg, which means the height of the cone is $h = 4$. The shorter leg sweeps out in a circle, becoming the radius of the cone's base, so $r = 3$.

Step 3: Calculate the volume. The formula for the volume of a cone is: $V = \frac{1}{3}\pi r^2 h$

Substitute the radius and height into the formula: $V = \frac{1}{3}\pi (3)^2 (4)$ $V = \frac{1}{3}\pi (9)(4)$ $V = 12\pi$

The solid is a cone with a volume of $12\pi$ cubic units.