• Student errors on the Surface Area and Volume worksheet will provide specific points to review.

[Note: It may be appropriate to review the difference between rational and irrational numbers either before or during the lesson. For helpful activities and resources, see unit M-8-5, “Rational and Irrational Numbers as Decimals and Fractions.”]

Use the following situation to introduce the concepts of surface area and volume. “Today we are party planners. We are going to plan for a celebration hosted in this classroom. To budget correctly, we need to find out how much it will cost to decorate the room by calculating measurements of our decorations. We will plan to paint the walls, paint piñatas to hang from the ceiling, and figure out how many balloons we will need to fill the room
2 feet deep.” Pass out the Party Planning Log (M-8-6-3_Party Planner.docx) to each student. Instruct students to use the handout to follow along with the lesson; they should complete the handout as necessary.

“Our first task as a group is to paint the walls. Paint is sold in one-gallon containers and we do not want to buy too much or too little. Each gallon of paint covers approximately
350 square feet of wall space. What do we need to know to determine the amount of paint that we need?” Students need to know the amount of wall space in the classroom to paint. “How do we figure out how much wall space there is?” The response should be area of the wall determined by height and width of each wall. This is the surface area of the inside of the classroom without the ceiling and floor.

Activity 1

Use the following activity to visualize what surface area of a solid means. “Find a partner and together complete the following activity. In front of you are six cardboard rectangles. Using the rulers in front of you, calculate the area of each rectangle. Once you have written the number on each cardboard piece, tape the sides together to make a rectangular prism. What is the surface area of each box?” Students will need to remember the meaning of rectangular prism and construct the right shape. Then, as pairs, they can add the areas of each side together to find the surface area of the box.

“Can we find an easier way to calculate the total area besides measuring each side and adding their areas together?”

Review the concept that Area = Length  Width. Then ask students to determine if any of the two sides are the same. Because each box is a rectangular prism, each side should be the same as its opposite. This means there are three pairs of two rectangles. Since two are the same, we can multiply their area by two in the final equation. Since there are three pairs, we must add three areas together, each multiplied by 2.

 = Area, where Length_a and Width_a are measurements of matching sides

“What if we were finding the surface area of a cube?

“A cube is equal on all sides, meaning that the length and width of each surface are the same. Plugging that into the equation above, 2 (Length₁ × Width₁) + 2 (Length₁ × Width₁) + 2 (Length₁ × Width₁) = 6 (Length₁ × Width₁).

“Now that we know how to calculate surface area, we can determine how much paint we need to decorate the classroom walls. The length of the room is 30 feet, the height of the room is 10 feet, and the width of the room is 25 feet. There are 72 square feet of windows and 78 square feet of doors and closets, neither of which we will paint. How many gallons of paint do we need to buy?”

Activity 2

The next activity will be to find the surface area of a sphere. “The next task for the party is to paint some spherical piñatas to use as decorations. Since the piñatas are smaller than the walls, we will buy paint in quarts. Each quart contains enough paint to cover 75 square feet of surface. How many piñatas with a circumference of 5 feet can we paint with one quart of paint?”

Review the measurements and equations for spheres. “Radius is the length from the center to any point on the surface of the sphere. Circumference is the length around any circle that passes through the center of the sphere and whose outer boundary matches that of the sphere.”

Draw a circle and label the radius, diameter (d = 2r), and circumference.

Write the following on the board:

Circumference = 2πr, where r is the radius of the circle

“The surface area of a sphere is the outside layer. On a globe, it is the layer painted with a map of the world. On a spherical light bulb, it is the area of the glass. On a basketball, it is the area of the leather material.” Show all of these objects to the class.

Write the equation for surface area of a sphere on the board for students to copy:

SA_sphere = 4πr², where r is the radius of the sphere

“What is the surface area of a basketball whose circumference is 30 inches? What do we have to determine first?” Students should respond that they need to determine the radius.

“Now that we have calculated the radius, how do we find the surface area?” Students should respond that they need to put the radius in the surface area equation.

SA ≈ 286.5 in.²

“To help visualize this measurement, think about one sheet of leather (material that covers the basketball) in the shape of a square. That square has a side length of about 17 inches (17  17 = 289).”

Draw a 17" by 17" square on the board to show the area in relation to the size of the basketball.

“Knowing what you now know, pair up and determine how many piñatas with a circumference of 5 feet we can decorate with 1 quart of paint that will cover 75 square feet.”

Activity 3

This activity focuses on filling a cylinder with spheres. “We have a bunch of vases in the shape of cylinders, and we want to buy round candies to fill them up. We will use the vases as centerpieces for the tables. What information are we going to need to know to determine how many candies we need to buy for each centerpiece?” Students should respond that they need to know the volume of each centerpiece (i.e., the radius and height) as well as the volume of each candy (i.e., the radius or diameter).

Tell students that the centerpieces are cylinders with a radius of 2 inches and a height of
10 inches. “How do we find the volume of a cylinder?” (multiply the area of the base (πr²) by the height (πr²h).) Write the formula to find the volume of a cylinder on the board:

Ask students to find the volume of each cylinder centerpiece; the approximate volume of each cylinder centerpiece is 125.66 inches³.

Ask students, “If I give you a spherical piece of candy, is it easy to measure the radius of the candy?” Remind students that the radius of a sphere is the distance from the exact center to the outside edge. Students should recognize that finding the exact center is difficult. “What measurement—radius, diameter, volume, circumference, etc.—is easy to measure for a sphere?” Students should recognize that circumference is the easiest. Tell students that the circumference of each candy is approximately 3.14 inches. “How can we use the circumference to determine the radius?” If necessary, remind students of the formula for circumference; students should determine that the radius is 0.5 inches.

“Now that we know the radius, how can we determine the volume?” Write the equation for volume of a sphere on the board:

Ask students to find the volume of a single piece of candy; the approximate volume of each candy is 0.523 in³.

“How many of these candies, each of which takes up 0.523 of a cubic inch, can we fit in a space that has a capacity of 125.66 cubic inches? How can we find out?” Students should recognize that they can divide 125.66 by 0.523 to arrive at an approximate answer of 240.3. “So do we really need 240 candies to fill up the centerpieces?” Ask students if they’ve ever looked at a stack of oranges in a supermarket or tennis balls stacked in a can. Are they stacked without any space in between? Point out to students that 240 candies would fill the prism completely,
i.e., with virtually no leftover space. “Since spheres don’t pack perfectly, we’ll need less than
240 pieces of candy for each centerpiece, but we can use 240 as a guide. The problem of determining exactly how many we need is much more difficult, so we’ll stick to an approximation for now.”

Activity 4

This part of the lesson will focus on volume of rectangles and spheres. “Our final decorating task is to buy enough balloons to fill the classroom to a depth of 2 feet. What information do we need to find this answer?” Students should respond that they need the volume of the balloons and the volume of the room at a height of two feet.

Do a short review of dimensions. “Notice that the exponent is the same as the exponent on the unit. Length, width, and height are one-dimensional; the exponent is 1. Since we do not write the one in exponential notation, it has no exponent. These measurements are reported in feet or inches or meters. Surface area takes two dimensions into account and is therefore represented in square feet or square meters. Frequently this is expressed as m². Volume has three dimensions; for a rectangular prism: length, width, and height. This measurement is then reported in cubic feet or cubic inches and is frequently expressed as ft³ or in³.”

• Circumference, 1-Dimensional (line), unit has a power of 1
• Surface Area, 2-Dimensional (paper, square, no depth), unit has a power of 2
• Volume, 3-Dimensional (solid), unit has a power of 3

After this review, have students calculate the volume of the boxes they built in Activity 1.

“What three measurements do we need to determine the total volume of this room?” Students should respond that we need height, length, and width. These measurements were given before to determine surface area. L = 30 ft., W = 25 ft., and H = 10 ft. “We multiply the length, width, and height to determine the volume. The length and width determine square footage. Multiplying this number by the height will give us the third dimension to create a solid. 30  25  10 = 7500 ft.³ If we only want to have the balloons two feet deep, what is the volume of the space they need to fill?” Have students find this number individually or in pairs.

“To determine the number of balloons we need to fill 1500 ft.³, we need to calculate the space each balloon will take up. In other words, we need to know each balloon’s volume. We are going to buy round balloons so we will use the formula for the volume of a sphere.” Write the equation for spherical volume on the board.

“When inflated, our balloons will have a circumference of 6 feet. Using the process that we followed earlier, pair up and solve the balloon problem.”

“When calculating the surface area and volume of a sphere, it is sometimes appropriate to leave the answer as a multiple of π. Because π is an irrational number, the answer will have to be rounded if it is written as a decimal. Leaving the solution as some number multiplied by π will give an exact answer to the problem.” Show the following example on the board, and then hand out the Surface Area and Volume worksheet (M-8-6-3_Surface Area and Volume and KEY.docx) for students to complete.

Example: Find the surface area and volume of a sphere that has a diameter of 6 inches.

“In this case the numbers are the same, but the unit is different because the surface area is two-dimensional and the volume is three-dimensional. Complete the surface area and volume worksheet. You may work in pairs or individually.”

Activity 5

“We will also create a large centerpiece for the main table. The centerpiece will be a large rectangular prism that is hollow inside. We’ll cover the outside of the centerpiece with colored paper, and fill the inside with water (so we can put flowers in it). Let’s start with covering the outside with colored paper. What do we need to know?” Students should respond that we need to know the dimensions of the prism. Tell them it is 4 inches wide,
6 inches long, and 14 inches tall. “Now, what measurement do we need to find if we want to know how much paper we need?” Students should respond that we need to know the surface area. Remind students that our centerpiece will not have a top on it (because we’re going to put flowers in it).

Ask students to work individually and determine the area of each of the five sides, and then find the surface area. (The surface area should be
(4 × 6) + (2 × 4 × 14) + (2 × 6 × 14) = 24 + 112 + 168 = 304 inches².)

“Now, what measurement do we need to find if we want to know how much water the centerpiece will hold?” Students should recognize that volume is needed. “Does it make a difference, for this calculation, that we are not putting a top on the centerpiece?” Students should recognize that this detail does not make a difference.

Ask students to work individually and determine the volume of the rectangular prism. (The volume should be 4 × 6 × 14 = 336 inches³.)

Extension:

Use the following strategies to tailor the lesson to meet the needs of your students throughout the year.

• The concepts covered in this lesson can be reinforced by having students start with the surface area and volume of spheres and calculate circumference, diameter, and radius. This will allow students to work more with the formulas but in reverse.
• Introduce the following two problems for students to work on in pairs.

1. What are the surface area, circumference, diameter, and radius of a sphere with a volume of 972π? (Encourage students to find the radius of the sphere because they can use that measurement to find all other measurements.)
   (r = 9, d = 18, C = 18π, SA = 324π)
2. Randy can fit 8 balls in a box with a capacity of 16 ft³. What is the diameter of each ball? (Have students start by determining the volume of each ball, and then calculate the radius to determine diameter.)