Lesson 9.2: Surface Area & Volume Applied Problems

Learning Objectives

• Solve surface area problems for prisms, cylinders, pyramids, cones, and spheres.
• Solve volume problems for various geometric solids.
• Apply formulas to real-world applied problems.

Topics Covered

• Surface area of prisms and cylinders
• Surface area of pyramids, cones, and spheres
• Volume of prisms and cylinders
• Volume of pyramids, cones, and spheres
• Applied problems in real-world contexts

Key Terms

• Prism: A solid with two parallel, congruent bases and rectangular sides.
• Cylinder: A solid with two parallel, congruent circular bases and a curved lateral surface.
• Pyramid: A solid with a polygonal base and triangular faces converging to a point.
• Cone: A solid with a circular base and a curved surface that converges to a point.
• Sphere: A perfectly round solid where every point on the surface is equidistant from the center.
• Surface Area: The total area of the exterior of a solid.
• Volume: The amount of space enclosed by a solid.

Lesson Examples

Example 1: Surface Area of a Cylinder

Find the surface area of a cylinder with radius 4 cm and height 10 cm.

1. Formula: \(SA = 2\pi r^2 + 2\pi r h\)
2. Compute: \(2\pi(4^2) + 2\pi(4)(10) = 32\pi + 80\pi = 112\pi\)
3. Approximate: \(SA \approx 351.86 \text{ cm}^2\)

Example 2: Volume of a Cone

Find the volume of a cone with radius 3 m and height 9 m.

1. Formula: \(V = \frac{1}{3}\pi r^2 h\)
2. Compute: \(\frac{1}{3}\pi (3^2)(9) = \frac{1}{3}\pi (81) = 27\pi\)
3. Approximate: \(V \approx 84.82 \text{ m}^3\)

Example 3: Applied Problem

A water tank is shaped like a rectangular prism with length 5 m, width 3 m, and height 2 m. Find its volume and surface area.

1. Volume: \(V = lwh = 5 \cdot 3 \cdot 2 = 30 \text{ m}^3\)
2. Surface Area: \(SA = 2(lw + lh + wh) = 2(15 + 10 + 6) = 2(31) = 62 \text{ m}^2\)

Practice Problems

1. Find the surface area of a sphere with radius 7 cm.
2. Find the volume of a cylinder with radius 5 m and height 12 m.
3. Find the surface area and volume of a pyramid with base length 6 m, width 4 m, and height 9 m.
4. A cone has a radius of 2 m and height of 5 m. Calculate its volume.
5. A rectangular prism has dimensions 4 cm × 5 cm × 6 cm. Find its surface area.

Tips & Common Mistakes

• Always identify the type of solid before applying a formula.
• Check units; convert if necessary to keep them consistent.
• For pyramids and cones, height is perpendicular to the base, not slant height (except for lateral area).
• Use \(Ï€ \approx 3.1416\) for numerical answers if needed.

Summary

Surface area and volume formulas allow us to solve applied problems involving geometric solids. Accurate identification of the solid and proper substitution of dimensions are key to solving real-world problems.

Challenge Problems

1. A sphere has a volume of 5000 cm³. Find its radius and surface area.
2. A cylindrical water tank has a diameter of 10 m and height 15 m. Calculate its surface area and volume.
3. Find the volume of a cone that has the same base radius and height as a cylinder of volume 300 m³.