Surface Area and Volume of Three-Dimensional Figures

Fundamental Concepts of Three-Dimensional Solids

• Definition and Classification of Polyhedrons:

  • Polyhedrons: These are three-dimensional solids with flat faces that are polygons. Examples include pyramids and prisms.
  • Non-Polyhedrons: These are three-dimensional solids that have curved surfaces. Examples include spheres, cylinders, and cones.

• Questions for Understanding Polyhedrons:

  • Naming Polyhedrons: Polyhedrons are typically named based on the shape of their bases (e.g., a triangular prism has triangular bases).
  • Identifying the Base: To determine which face of a polyhedron is a base, one must identify the parallel faces (in prisms) or the face opposite the vertex (in pyramids).
  • Labeling Units for Surface Area: Surface area is always measured and labeled in square units (e.g., $\text{in}^2$, $\text{cm}^2$, $\text{m}^2$).
  • Labeling Units for Volume: Volume is always measured and labeled in cubic units (e.g., $\text{in}^3$, $\text{cm}^3$, $\text{m}^3$).

• Identifying Solids (Check Your Understanding p. 79):

  • Example 1: A cylinder is classified as "Not a polyhedron."
  • Example 2: A pyramid is classified as a "Polyhedron." To identify a polyhedron fully, one must name its bases, faces, edges, and vertices (referenced via points $M$, $K$, $N$ in provided diagrams).

Geometric Formulas for Volume

• General Definitions:

  • Volume ($V$): This is a measure of the amount of space that a solid encloses. It is measured in cubic units.
  • $B$: Represents the area of the base of the solid.
  • $h$: Represents the height of the solid.
  • $r$: Represents the radius of the solid.

• Volume Formulas by Shape:

  • Prism: $V = B \times h$
  • Cylinder: $V = B \times h$
  • Pyramid: $V = \frac{1}{3} \times B \times h$
  • Cone: $V = \frac{1}{3} \times B \times h$
  • Sphere: V = \frac{4}{3} \times \text{\pi} \times r^3
  • Hemisphere: V = \frac{1}{2} \times (\frac{4}{3} \times \text{\pi} \times r^3) = \frac{2}{3} \times \text{\pi} \times r^3

Surface Area and Volume Calculations

• Check Your Understanding (Surface Area and Volume to the nearest tenth):

  • Example 3 (Triangular Prism): Base legs of $3.00\,\text{cm}$ and $4.00\,\text{cm}$ with a height of $3.00\,\text{cm}$.
  • Example 4 (Sphere/Hemisphere): A figure with a dimension of $6.00\,\text{in}$.
  • Example 5 (CUPCAKES Application): LaMea is icing cupcakes using a cone-shaped icing bag. The bag has the following dimensions:
    • Diameter: $3.5\,\text{in}$
    • Height: $5.0\,\text{in}$
    • Slant Height ($l$): $5.3\,\text{in}$
    • Note: The icing bag has no top (lateral area only).
    • Task A (Volume): Calculate the volume of icing that will fill the bag to the nearest tenth.
    • Task B (Surface Area): Calculate the area of plastic used to make the icing bag to the nearest tenth.

• Volume Practice Problems:

  • Cylinder 1: Radius = $6.00\,\text{in}$, Height = $8.00\,\text{in}$. Round results to the nearest hundredth.
  • Cylinder 2: Diameter = $4.00\,\text{cm}$, Height = $9.00\,\text{cm}$.
  • Prism 3: Base dimensions of $4.00\,\text{ft}$ and $2.00\,\text{ft}$ with a height of $12.00\,\text{in}$. (Note: Unit conversion between feet and inches may be required).
  • Solid 4: Dimension of $5.00\,\text{ft}$.
  • Prism 5 (Composite/Triangular): Dimensions include $9.00\,\text{m}$, $3.00\,\text{m}$, $4.00\,\text{m}$, $5.00\,\text{m}$, and a length/height of $5.00\,\text{m}$.
  • Rectangular Prism 6: Dimensions of $24.00\,\text{cm}$, $13.00\,\text{cm}$, and $5.00\,\text{cm}$.

Spheres and Hemispheres (Section 11-4)

• Formula Review (Spheres and Hemispheres):

  • Surface Area of a Sphere ($S$): S = 4 \times \text{\pi} \times r^2
  • Volume of a Sphere ($V$): V = \frac{4}{3} \times \text{\pi} \times r^3

• Surface Area Practice (Round to the nearest tenth):

  • Example 1 (Sphere): Dimension of $9.00\,\text{m}$.
  • Example 2 (Hemisphere): Diameter = $14.00\,\text{in}$.

• Volume Practice (Round to the nearest tenth):

  • Example 5 (Sphere): Radius = $10.00\,\text{ft}$.
  • Example 6 (Hemisphere): Diameter = $16.00\,\text{cm}$.

Real-World Applications of Volume

• Everyday Items and Units of Sale:
  • Many items are sold specifically by volume rather than mass or density because it is the space occupied that matters most to the consumer.
  • Gasoline: Sold by the gallon in the United States and by the liter in many other countries.
  • Grocery Items: Milk is frequently sold by the gallon; soda and other beverages are typically sold by the ounce or by the liter.
  • Gardening/Landscaping: Soil, specifically loam, is often sold by the cubic yard.
  • Core Principle: Volume focuses on the amount of space a given substance or solid encloses.