Surface Area of 3D Solids

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SWBAT find the surface area of polyhedra.

Big Idea

Students find the area of polyhedra faces and add.

Do Now

10 minutes

Students have worked on identifying polyhedrons in previous lessons.  The Do Now questions are an assessment of their knowledge.

Do Now

1.  Ashley has a polyhedron whose faces are all congruent, and it has 4 vertices. Which solid does Ashley have?

2.  Jon has 2 cubes. Henry has a square pyramid. How many faces do they have all together?

3.   How many edges does an octagonal pyramid have?

Students will have 5 minutes to answer the questions and then they will discuss their answers with their group.  If students are unsure about their answers, I will encourage them to use their notes to help them.  As a class we will review any unanswered questions.

Mini Lesson

15 minutes

For this lesson, students will be introduced to the concept of surface area and they will apply it to polyhedrons.

What is area?  Can we find the area of a polyhedron?

Students should recall that area is the amount of square units needed to cover a surface.  Some students may think we can find the area of a polyhedron, so I will pose the following questions.

When we had 2 dimensional figures, we were able to find the area.  What do you think it means to find the surface area of a polyhedron? 

Students will share their ideas. They may infer that since there are several faces to a polyhedron, we need to find the areas of those faces.  I will explain that surface area is the sum of the areas of all outside faces of a 3-D figure.  We will work through some examples together, so students have an understanding of how to organize their work. See Surface Area of Polyhedron Lesson for examples.

Example 1

What type of polyhedron is this?

Students should recognize this as a rectangular prism.

How many faces does it have?  What shape are the faces?  What does the net look like?

Students should identify that it has 6 rectangular faces.  We will draw the net.

Do all the faces have the same dimensions?  How can we organize our work so we can distinguish between the faces?

Students should notice that the surfaces have different dimensions and therefore we should label the faces.  We will label the faces: top, bottom, left, right, front, back. I will explain to students that they can also label the faces with numbers.

How can we find the area of each surface?  Do we have to find the area of both the front and back, left and right, or top and bottom faces?  

Students should recall the area formula for rectangles.   We will apply the formula to find the area of the faces.  Students should understand that we don't need to find the area of the congruent faces, therefore saving them time and work.  However, a common mistake is for students to forget to multiply the area of these congruent faces by two.

What is the next step to find the surface area of the polyhedron?

Students should realize that the final step is to add all of the areas together.

We will continue on to examples 2 and 3, with students leading the discussion of how to find the surface area.

Independent Practice

15 minutes

Students will be given a Surface Area of Polyhedrons Independent Practice Worksheet to complete.  Although it is independent work, I will encourage students to discuss their work with their group.  Students may have difficulty identifying the dimensions of the faces, especially for the triangular pyramids.  It is important that they remember that the base and height always form a right angle.

As students work, I will focus on the groups with lower level math students.  Since there are a lot of steps, it is important and helpful for them to organize their work.  I will also use polyhedron manipulatives to help them identify the faces.

After students have completed the worksheet, we will discuss their strategies and work.


Lesson Summary

5 minutes

To assess students' understanding of the concept of surface area, I will ask a series of questions.

When find the surface area of a polyhedron, will you ever be able to find the area of only one face?

Students should consider a cube, where all the faces have the same area.

How can you organize your work to show the steps of find surface area?

Students may suggest drawing a net, labeling the faces, labeling their work, ...

When you have congruent faces, what is a shortcut?

Students should suggest that you can multiply the area by the number of congruent faces?

When you find the area of all the faces, what should you do?

Student should remember to add the areas and label their answer square units.