9.6 Solve Geometry Applications: Volume and Surface Area

Geometric Solids

Geometric solids are three-dimensional shapes with length, width, and height. Understanding how to calculate their volume (the space inside) and surface area (the total area covering the outside) lets you solve practical problems like finding how much water a tank holds or how much material you need to wrap a box.

Volume and Surface Area Formulas

Volume is always measured in cubic units (like cubic inches, $\text{in}^3$) because you're measuring 3D space. Surface area is measured in square units (like square feet, $\text{ft}^2$) because you're measuring flat area, just across all the faces.

Rectangular Prisms (think: boxes, rooms, bricks)

• Volume: $V = l \times w \times h$
• Surface area: $SA = 2(lw + lh + wh)$

The surface area formula works because a rectangular prism has three pairs of identical faces. You find the area of each pair ($lw$, $lh$, and $wh$) and multiply the total by 2.

Cylinders (think: soup cans, pipes, water tanks)

• Volume: $V = \pi r^2 h$
• Surface area: $SA = 2\pi r h + 2\pi r^2$

For volume, you're taking the area of the circular base ($\pi r^2$) and stacking it up through the height. For surface area, the $2\pi r h$ part is the curved side (imagine peeling the label off a can and flattening it into a rectangle), and $2\pi r^2$ covers the two circular ends.

Spheres (think: basketballs, globes)

• Volume: $V = \frac{4}{3}\pi r^3$
• Surface area: $SA = 4\pi r^2$

Both formulas depend only on the radius $r$, since a sphere is the same in every direction.

Cones (think: ice cream cones, traffic cones)

• Volume: $V = \frac{1}{3}\pi r^2 h$
• Surface area includes the circular base ($\pi r^2$) plus the lateral (sloped) surface, which uses the slant height

Cone vs. Cylinder Volume Comparison

A cone's volume is exactly one-third of a cylinder's volume when they share the same base radius and height. You can see this directly in the formulas:

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

This means it would take three cones of water to fill one cylinder of the same size. If you know a cylindrical cup holds 300 mL, a cone with the same radius and height holds 100 mL.

Key Vocabulary

• Polyhedron: A 3D solid with flat polygonal faces, straight edges, and vertices. Cubes, rectangular prisms, and pyramids are all polyhedra. Spheres, cylinders, and cones are not polyhedra because they have curved surfaces.
• Cross-section: The 2D shape you see when you slice through a solid with a flat cut. For example, slicing a cylinder horizontally gives you a circle; slicing it vertically gives you a rectangle.
• Pi ($\pi$): Approximately 3.14159. It shows up in every formula involving circles or curved surfaces.

Real-World Applications of Formulas

When you face a word problem involving volume or surface area, follow these steps:

1. Identify the shape. What geometric solid does the object resemble? A storage tank is usually a cylinder. A shipping box is a rectangular prism.
2. Pull out the measurements. Find the dimensions given in the problem (length, width, height, radius, diameter). If you're given a diameter, remember to divide by 2 to get the radius.
3. Pick the right formula. Choose volume if the problem asks about capacity, filling, or space inside. Choose surface area if it asks about covering, painting, or wrapping.
4. Plug in and calculate. Substitute your measurements into the formula and solve.
5. Check your units and interpret. Make sure your answer uses cubic units for volume or square units for surface area, and state what the number means in context.

Sketching 3D Objects from Measurements

Drawing a quick sketch helps you organize information and avoid mixing up dimensions. When you sketch:

• For rectangular prisms, draw the front face as a rectangle, then add depth lines to show the third dimension. Label length, width, and height.
• For cylinders and cones, draw the circular base (shown as an oval from an angled view) and the height. Label the radius and height.
• For spheres, draw a circle and mark the radius from the center to the edge.

Keep proportions roughly accurate. A cylinder described as 2 ft wide and 10 ft tall should look tall and narrow, not short and wide. Labeling every given measurement on your sketch makes it much easier to plug values into formulas without mistakes.

Problem-Solving Strategies

Applying Formulas Step by Step

1. Read the problem carefully. Decide whether it's asking for volume or surface area. Words like "fill," "hold," or "capacity" point to volume. Words like "cover," "wrap," or "paint" point to surface area.
2. Identify the solid. Match the real-world object to a geometric shape (a soccer ball is a sphere, a cereal box is a rectangular prism).
3. Sketch and label. Draw the shape and write in all given measurements. Convert diameter to radius if needed.
4. Write the formula before plugging anything in. This keeps your work organized.
5. Substitute and solve. Replace variables with your actual numbers and calculate step by step.
6. Check your units. Volume answers need cubic units; surface area answers need square units.
7. Answer the question. State your result in the context of the problem. "The tank holds approximately 226.2 cubic feet of water" is a complete answer; just writing "226.2" is not.