Prisms and Cylinders

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; are 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 inch². 

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 inch² 

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 = 2πr² + 2πrh (Formula for surface area of a cylinder) 

   = 2π(3²) + 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 = 2πr² + 2πrh (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 = 13πh (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.