G7 Module 3 Lesson 21: Surface Area

G7

Module 3 Lesson 21: Surface Area

If you can find the area of basic shapes, you'll be fine.

Lesson 21 Goals

Students find the surface area of three-dimensional objects whose surface area is composed of triangles and quadrilaterals. 

They use polyhedron nets to understand that surface area is simply the sum of the area of the lateral faces and the area of the base(s).

Opening Exercise: 

Surface Area of a Right Rectangular Prism

On the provided grid, draw a net representing the surfaces of the right rectangular prism (assume each grid line represents 1 inch). Then, find the surface area of the prism by finding the area of the net.

 $A=bh$  

Find the area of all the shapes and add them together. Yes, it is that simple. 

Option 3

 $SA=2lw+2lh+2wh$  
 $SA=2\left(3\cdot4\right)+2\left(3\cdot6\right)+2\left(4\cdot6\right)$  
 $SA=24+36+48$  
 $SA=108\ in^2$  

Poll

 Marcus thinks that the surface area of the right triangular prism will be half that of the right rectangular prism and wants to use the modified formula$SA=\frac{1}{2}\left(2lw+2lh+2wh\right)$ Do you agree or disagree with Marcus?  

• Agree

• Disagree

Marcus is wrong....

The formula adds the areas of six rectangular faces. A right triangular prism only has three rectangular faces and also has two triangular faces (bases).

Do the math real quick to double check.

Poll

Would it be cool if we could make this even more simple?

• Yes, very much

• No, I like the hard way

• You lost me on the first slide, and I stopped trying.

Write an expression that represents the lateral area of the right triangular prism as the sum of the areas of its lateral faces.  

Write an expression that represents the lateral area of the right rectangular prism as the sum of the areas of its lateral faces.  

Write an expression that represents the lateral area of the right pentagonal prism as the sum of the areas of its lateral faces.  

What value appears often in each expression and why?

Right Triangular Prism Lateral Area:

 $a*h+b*h+c*h$  
Right Rectangular Prism Lateral Area:
 $a*h+b*h+a*h+b*h$  
Right Pentagonal Prism Lateral Area:

 $a*h+b*h+c*h+d*h+e*h$  

Rewrite each expression in factored form using the distributive property and the height of each lateral face.

Right Triangular Prism Lateral Area:

 $h\left(a+b+c\right)$  
Right Rectangular Prism Lateral Area:
 $h\left(a+b+a+b\right)$  
Right Pentagonal Prism Lateral Area:

 $h\left(a+b+c+d+e\right)$  

Poll

What do the parentheses in each case represent with respect to the right prisms?

• The perimeter of the base

• The area of the base

• The volume of the base

Lesson Summary

The surface area of a right prism can be obtained by adding the areas of the lateral faces to the area of the bases. The formula for the surface area of a right prism is  $SA=LA+2B$ , where  $SA$ represents the surface area of the prism,  $LA$  represents the area of the lateral faces, and  $B$  represents the area of one base. 

The lateral area  can be obtained by multiplying the perimeter of the base of the prism times the height of the prism.

Surface Area Formula

(for right prisms)

Surface are is the lateral area of the prism plus the area of both bases.

The lateral area of the prism is the perimeter of the base times the height.

 $SA=LA+2B$  
 $LA=P\cdot h$  

For each of the following nets, highlight the perimeter of the lateral area, draw the solid represented by the net, indicate the type of solid, and then find the solid’s surface area.

 $SA=LA+2B$  
 $LA=P\cdot h$  

P = ?

h = ?
LA = ?
B = ?
LA + 2B = ?

 $SA=LA+2B$  
 $LA=P\cdot h$  

For each of the following nets, highlight the perimeter of the lateral area, draw the solid represented by the net, indicate the type of solid, and then find the solid’s surface area.

 $SA=LA+2B$  
 $LA=P\cdot h$  

P = ?

h = ?
LA = ?
B = ?
LA + 2B = ?

 $SA=LA+2B$  
 $LA=P\cdot h$  

This is a cube with  $\frac{3}{4}$  inch sides

P = ?

h = ?
LA = ?
B = ?
LA + 2B = ?

What if the cube was 4 times larger?

 $SA=LA+2B$  
 $LA=P\cdot h$  

P = ?

h = ?
LA = ?
B = ?
LA + 2B = ?

 $SA=LA+2B$  
 $LA=P\cdot h$  

P = ?

h = ?
LA = ?
B = ?
LA + 2B = ?

 $SA=LA+2B$  
 $LA=P\cdot h$  

P = ?

h = ?
LA = ?
B = ?
LA + 2B = ?

 $SA=LA+2B$  
 $LA=P\cdot h$  

P = ?

h = ?
LA = ?
B = ?
LA + 2B = ?

A cube has a volume of  $64m^3$ . What is the cube’s surface area?

 $V=l\cdot w\cdot h$  
 $SA=LA+2B$  

Stept 1: draw a picture

The height of a right rectangular prism is  $SA=LA+2B$  ft. The length and width of the prism’s base are  $2$ ft  and  $1\ \frac{1}{2}$  ft. Use the formula  $SA=LA+2B$  to find the surface area of the right rectangular prism.

First write out the formula, then plug in what you know.

The surface area of a right rectangular prism is  $68\ \frac{2}{3}inches^2$ .  The dimensions of its base are $3\ inches$ and $7\ inches$ .

Use the formula $SA=LA+2B$ and  $LA=Ph$ to find the unknown height  of the prism.

That's all folks!

You don't have to do question 7, but if you do you'll get extra points.