3D Figures: Cross Sections, Rotations, Surface Area, Volume and Density

Cross Sections

• A cross-section is the intersection of a solid and a plane; it's a slice of a shape.
• 3-D shapes can have multiple cross-sections, including squares, rectangles, circles, ovals, and triangles.

Rotating 2-D Figures

• Rotating 2-D figures can create 3-D geometric solids.
• Analogy: A revolving door illustrates rotation around a central pole, creating a cylindrical path.

Solids of Rotation Examples

• Rotating a triangle creates a cone.
• Rotating a rectangle creates a cylinder.
• Rotating a trapezoid creates a frustum.
• Rotating a circle creates a sphere.

Surface Area

• Surface Area: The total area of the outside of a solid; measured in square units (e.g., ft²).

Surface Area Formulas

• Surface area of a Cube: $6s^2$ (where $s$ is the side length).
• Surface area of a Rectangular Prism: $2LW + 2WH + 2LH$ (where $L$ is length, $W$ is width, and $H$ is height).

Volume

• Volume: The amount of space an object occupies; measured in cubic units (e.g., in³).

Volume Formulas

• Volume of a Cube: $V = s^3$ (or $V = Bh$, where $B$ is the base area and $h$ is the height).
• Volume of a Rectangular Prism: $V = lwh$ (or $V = Bh$).
• Volume of a Triangular Prism: $V = Bh$ (area of base x height).

Volume of a Pyramid

• Volume of Pyramid: $V = \frac{1}{3}Bh$

Volume of a Cylinder

• Volume of Cylinder: $V = \pi r^2 h$

Volume of a Cone

• Volume of Cone: $V = \frac{1}{3} \pi r^2 h$

Volume of a Sphere

• Volume of Sphere: $V = \frac{4}{3} \pi r^3$

Density

• Density Formula: $Density = \frac{Mass}{Volume}$

Cavalieri's Principle

• Cavalieri's Principle: If 2 solids have the same base area and equal heights, the volumes are the same.
• Example 68: Two stacks of 23 quarters each are shown. One stack forms a cylinder, but the other stack does not form a cylinder. Use Cavalieri's principle to explain why the volumes of these two stacks of quarters are equal.