Objective: Students will use their knowledge of area of two-dimensional surfaces to find the surface area of three-dimensional objects
CCSS.Math.Content.7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Note: Students should already be comfortable with finding area of two-dimensional shapes for this lesson.
Materials: boxes (cube, rectangular, or cylindrical for a challenge), scissors, tape, wrapping paper, rulers, work space for groups such as tables or desks, paper, pencils
Step One (Introduction):
Have a variety of cylindrical, triangular, rectangular or cube boxes on tables for students to examine, along with rolls of wrapping paper for display. Ask students to estimate how many square inches or feet they might need to wrap the boxes.
Here are some questions you can ask:
- What does the term “surface area" sound like it means?
- What are some situations where people might need to know the surface area of objects?
- How could you measure the surface area of an object, real life or theoretical?
Students should participate in the discussion and reach the conclusion that surface area means the area of the entire surface of the outside of an object. Students should discuss situations such as gift wrapping, sewing items in factories (such as tents), creating boxes and cans for products, upholstering furniture, buying paint for a house, or any other situations where it would be necessary for someone to know the area of the surface of an object. Students should also conclude that even if they do not have access to an object, it is possible to find its surface area given its dimensions.
Step Two: Investigation
Students will be working in groups. Ask students to take 1-3 boxes back to their desks or tables (depending on size of groups). Once they have their boxes and are in their groups, students will investigate each box and make notes on the following properties:
- Shape of box (Cube, Triangular, rectangular, cylinder for students who need a little challenge)
- Number of faces on each box
- Dimensions of each face (height and length, for rectangles and squares, base and height for triangles, diameter for circular faces and height for cylinders)
Circulate the room and assist students who need help identifying properties of their boxes.
After sufficient time for the investigation (about 20 minutes or so, depending on skill level of students), stop students for a quick discussion on what they found, and whether they have any ideas as to how to use the information they have to estimate how much wrapping paper they would need to wrap the boxes. Students should conclude that if they know the area of the faces of the box, they can add the area of each face to reach the total amount of units covered by the surface area of each box.
Step Three: Computation, Measurement, and Application
Briefly remind students of the formula for finding area of applicable two-dimensional shapes:
Squares and rectangles: A= l ∙ w
Triangles: A= ½ b ∙ h
Note: for cylinders, students need to find circumference of the face and height of the cylinder to find the area around the cylinder.
Students will continue to work in groups, focusing now on finding the area of each face of each object. They should use the formulas to compute the area of each face.
Students will then find the surface area of each object by adding up the area of each face. Briefly stop students to discuss the mathematical formula for surface area of objects (some students may have figured it out during the investigation; you may want to offer rewards to students who can cite the formula at this time).
- Surface area of rectangular prisms: A= 2(l∙w) + 2(w∙h) + 2(l∙h)
- Surface area of cubes: 6 (l ∙ w)
- Surface area of cylinders: A=2πr^2∙2πrh
After students have computed the surface area for their boxes (with teacher assistance as needed), they will begin estimating how many square inches or feet of wrapping paper they would need for each box (have them round up to nearest foot). They can then begin cutting out sections of wrapping paper to cover their boxes. They may cut and tape paper as needed to fit the boxes.
Step Four: Reflection and Assessment
After students have attempted to wrap the boxes, lead students in a brief discussion on the following points:
- What did they learn from the investigation and from wrapping the boxes?
- How did their estimates for the amount of wrapping paper needed compare to how much they used when wrapping the boxes? What might have accounted for some of the differences?
- How can students use the knowledge they gained today in real life situations?
- What can students do to help themselves remember the formulas needed for surface area?
If desired, give students a brief quiz on finding surface area of objects, and readdress the concept as needed.