Surface Area
Learning Objectives
- Understand nets.
- Use nets to represent prisms.
- Find the surface area of a prism.
- Find the surface area of cylinders.
- Find the surface area of a sphere.
Nets
One final way to represent a solid is to use a net. If you cut out a net you can fold it into a model of a figure. Nets can also be used to analyze a single solid. Here is an example of a net for a cube.
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Example 1
What kind of figure does the net create? Draw the figure.
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Example 2
What kind of net can you draw to represent the figure shown? Draw the net.
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Review Exercises
Draw a net for each of the following:Answers
Prisms
A prism is a three-dimensional figure with a pair of parallel and congruent ends, or bases. The sides of a prism are parallelograms. Prisms are identified by their bases.
Surface Area of a Prism Using Nets
The prisms above are right prisms. In a right prism, the lateral sides are perpendicular to the bases of prism. Compare a right prism to an oblique prism, in which sides and bases are not perpendicular.
Area Congruence Postulate: If two polygons (or plane figures) are congruent, then their areas are congruent.
Area Addition Postulate: The surface area of a three-dimensional figure is the sum of the areas of all of its non-overlapping parts.
You can use a net and the Area Addition Postulate to find the surface area of a right prism.
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A = l · w A = 10 · 5 = 50 square units
Similarly, the areas of the other rectangles are inserted back into the equation above.Total surface area = area A + area B + area C + area D + area E + area F Total surface area = (10 · 5) + (10 · 3) + (10 · 5) + (10 · 3) + (5 · 3) + (5 · 3) Total surface area = 50 + 30 + 50 + 30 + 15 + 15 Total surface area = 190 square units
Example 3
Use a net to find the surface area of the prism.
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where h is the height of the triangle and b is its base.
Note that triangles A and E are congruent so we can multiply the area of triangle A by 2.Thus, the surface area is 324 square units.
Review Exercises
For each of the following find the surface are a using the method of nets and the perimeter- The base of a prism is a right triangle whose legs are 3 and 4 and show height is 20. What is the total area of the prism?
- A right hexagonal prism is 24 inches tall and has bases that are regular hexagons measuring 8 inches on a side. What is the total surface area?
- What is the volume of the prism in problem #4?
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- How many square feet (excluding the roof) are there on the surface of the barn to be painted?
- If a gallon of paint covers 250 square feet, how many gallons of paint are needed to paint the barn?
- A cardboard box is a perfect cube with an edge measuring 17 inches. How many cubic feet can it hold?
- A swimming pool is 16 feet wide, 32 feet long and is uniformly 4 feet deep. How many cubic feet of water can it hold?
- A cereal box has length 25 cm, width 9 cm and height 30 cm. How much cereal can it hold?
Answers
Nets and perimeter:- 40.5 in2
- 838 cm2
- 252 square units
- 1484.6 square units
- 3990.7 cubic inches
- 2450 square feet
- 10 gallons of paint
- 2.85 cubic feet (be careful here. The units in the problem are given in inches but the question asks for feet.)
- 2048 cubic feet
- 6750 cm3
Cylinders
A cylinder is a three-dimensional figure with a pair of parallel and congruent circular ends, or bases. A cylinder has a single curved side that forms a rectangle when laid out flat. As with prisms, cylinders can be right or oblique. The side of a right cylinder is perpendicular to its circular bases. The side of an oblique cylinder is not perpendicular to its bases.
Surface Area of a Cylinder Using Nets
You can deconstruct a cylinder into a net.
The area of the rectangular lateral area L is given by the product of a width and height. The height is given as 24. You can see that the width of the area is equal to the circumference of the circular base.
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Now we can find the area of the entire cylinder using A = (area of two bases) + (area of lateral side).
You can see that the formula we used to find the total surface area can be used for any right cylinder.
Area of a Right Cylinder: The surface area of a right cylinder, with radius and height h is given by A = 2B + L, where B is the area of each base of the cylinder and L is the lateral area of the cylinder.
Example 4
Use a net to find the surface area of the cylinder.
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Calculate L.
Find the area of the entire cylinder.
Thus, the total surface area is approximately 854.08 square units
Surface Area of a Cylinder Using a Formula
You have seen how to use nets to find the total surface area of a cylinder. The postulate can be broken down to create a general formula for all right cylinders. A = 2B + L
Putting the two equations together we get: Factoring out a from the equation gives:
The Surface Area of a Right Cylinder: A right cylinder with radius r and height h can be expressed as:
A = 2πr2 + 2πrh
or:
A = 2πr(r + h)
You can use the formulas to find the area of any right cylinder.
Example 5
Use the formula to find the surface area of the cylinder.
Example 6
Find the surface area of the cylinder.
Example 7
Find the height of a cylinder that has radius 4cm and surface area of 226.08 sq cm. Write the formula with the given information and solve for h.
Spheres
A sphere is a three-dimensional figure that has the shape of a ball.
- A sphere is the set of all points that lie a fixed distance r from a single center point O.
- A sphere is the surface that results when a circle is rotated about any of its diameters.
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Surface Area of a Sphere
You can infer the formula for the surface area of a sphere by taking measurements of spheres and cylinders. Here we show a sphere with a radius of 3 and a right cylinder with both a radius and a height of 3 and express the area in terms of π.
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The Surface Area of a Sphere is given by: A = 4πr2
Example 8
Find the surface area of a sphere with a radius of 14 feet. Use the formula.
Example 9
Find the surface area of the following figure in terms of π.
Now find the area of the cylinder without its top.
Thus, the total surface area is
Review Exercises
- Find the radius of the sphere that has a volume of 335 cm3.
- Determine the surface area and volume of this shape:
- The radius of a sphere is 4. Find its volume and total surface area.
- A sphere has a radius of 5. A right cylinder, having the same radius has the same volume. Find the height and total surface area of the cylinder.
- Sphere: volume = 296 cm3. Find the Diameter.
- Sphere: surface area is 179 in2. Find the Radius.
- Tennis balls with a diameter of 3.5 inches are sold in cans of three. The can is a cylinder. Assume the balls touch the can on the sides, top and bottom. What is the volume of the space not occupied by the tennis balls?
- A sphere has surface area of 36π in2. Find its volume.
- A giant scoop, operated by a crane, is in the shape of a hemisphere of radius = 21 inches. The scoop is filled with melted hot steel. When the steel is poured into a cylindrical storage tank that has a radius of 28 inches, the melted steel will rise to a height of how many inches?
Answers
Note that these problems use π not 3.14.- 1. Radius = 4.31 cm
- Surface area = 706.86 cm2 Volume = 1767.15 cm3
- Volume = 268.08 units3 Surface area = 201.06 units2
- Height = 20/3 units total surface area = 366.52 units2
- Diameter = 8.27cm
- Radius = 3.77 inches
- Volume of cylinder = 32.16π in3 volume of tennis balls = 21.44π in3 Volume of space not occupied by tennis balls = 33.68 in3
- Volume = 113.10 in3
- Height of molten steel in cylinder will be 7.88 inches