## Key Concepts

- Use the net of a prism.
- Find the surface area of a right prism.
- Find the surface area of a cylinder.
- Find the height of a cylinder.

## Introduction

### Prism

Prism is a polyhedron with two congruent faces, called *bases, *that lie in parallel planes.

#### Lateral faces

All faces other than the bases; parallelograms formed by connecting the corresponding vertices of the bases.

#### Lateral edges

Segments connecting the vertices of the bases and lateral faces.

**Surface area: (of a polyhedron) **

Sum of the areas of its faces.

**Lateral area: (of a polyhedron) **

Sum of the area of its lateral faces.

**Net:**** **Two-dimensional representation of the faces.

**Right prism: **Each lateral edge is perpendicular to both bases.

**Oblique prism:**** **Lateral edges are not perpendicular to the bases.

### Use the net of a prism

**Example 1:**

Find the surface area of a rectangular prism with a height of 3 inches, length 7 inches, and width 4 inches using a net.

**Solution:**

**Step 1:** Sketch the prism. Imagine unfolding it to make a net.

**STEP 2:** Find the areas of the rectangles that form the faces of the prism.

**STEP 3:** Add the areas of all the faces to find the surface area.

The surface area of the prism is S = 2(21) + 2(12) + 2(28) = 122 inch^{2}.

### Find the surface area of a right prism

The surface area of a prism is

- S = Ph + 2B, where
- P = perimeter of the base
- h = height
- B = area of the base

**Example 2:**

Find the surface area of a right rectangular prism with a height of 3 inches, length 7 inches, and width 4 inches using the formula for the surface area of a right prism.

**Solution:**

Surface area = 2B + Ph

B = 7 × 4 = 28

P = 2L + 2W

= 2(7) + 2(4)

= 22

h = 3

S.A = 2(28) + 22(3)

= 122 inch^{2}

### Find the surface area of a cylinder

**Cylinder: **Solid with congruent circular bases that lie in parallel planes. The height is the perpendicular distance between its bases. The radius of a base is the radius of the cylinder.

**Right cylinder**: The segment joining the centers of the bases is perpendicular to the bases.

**Lateral Surface Area**: The lateral area of a cylinder is the area of its curved surface. It is equal to the product of the circumference and the height, or 2πrh.

**Surface Area**: The surface area of a cylinder is equal to the sum of the lateral area and the areas of the two bases.

**Example 3:**

Find the surface area of the right cylinder.

**Solution:**

*S* = 2π*r*^{2} + 2π*rh* (Formula for surface area of a cylinder)

= 2π(3^{2}) + 2π(3)(4) (Substitute known values)

= 18π + 24π (Simplify)

= 42π

= 131.88 (Use a calculator)

**The surface area is about 131.88 square feet. **

### Find the height of a cylinder

**Example 4:**

Find the height of a cylinder that has a radius of 6.5 centimeters and a surface area of 592.19 square centimeters.

**Solution: **

Substitute known values in the formula for the surface area of a right cylinder and solve for the height *h*.

S = 2πr^{2} + 2π*rh* (Surface area of a cylinder)

592.19 = 2π(6.5)^{2} + 2π(6.5)h (Substitute known values)

592.19 = 265.33+ 13πh (Simplify)

592.19 – 265.33 = 13π*h* (Subtract 265.33 from each side)

326.86 = 40.82h (Simplify. Use a calculator)

8 ≈ *h* (Divide each side by 40.82)

**The height of the cylinder is about 8 centimeters. **

## Exercise

- The surface area of a polyhedron is the sum of the areas of its _____________.
- The lateral area of a polyhedron is the sum of the areas of its ______________________.
- Find the height of the right cylinder shown, which has a surface area of 157.08 square meters.

- Find the surface area of a box of cereal with a height of 15 inches, a length of 8 inches, and a width of 4 inches.
- Find the surface area of a right rectangular prism with a height of 8 inches, a length of 3 inches, and a width of 5 inches.
- Find the lateral area and surface area of the cylinder below.

- Find the surface area of the figure below.

- Find the lateral and surface area of the figure below.

- Draw a net of a triangular prism.
- You are wrapping a stack of 20 compact discs using a shrink wrap. Each disc is cylindrical with a height of 1.2 millimeters and a radius of 60 millimeters. What is the minimum amount of shrink wrap needed to cover the stack of 20 discs?

### Concept Map

### What have we learned

- Use the net of a prism to find the surface area.
- Find the surface area of a right prism by using the formula.
- Find the surface area of a cylinder by using the formula.
- Find the height of a cylinder by using the formula.

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