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.
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