3.1 Notes Key
3.1 ­ Nets to Surface Area
Obtain the three nets provided in class.
Carefully cut them out and try to make the three dimensional shape for each one.
Use very minimal tape so that we can take the 3D shapes back apart into 2D nets after.
What shape did you make from net...
#1) _______________________________
#2) ________________________________
#3) ________________________________
Obtain the three nets provided in class.
Carefully cut them out and try to make the three dimensional shape for each one.
Use very minimal tape so that we can take the 3D shapes back apart into 2D nets after.
Carefully undo the tape on your 3D objects to make 2D nets again. Try to keep the tape in working order so it can be used again.
What shape did you make from net...
#1) rectangular prism
#2) triangular prism
#3) cylinder
Activity for Net #1:
How many rectangles are in the net?
Take a ruler and measure the length and width of each rectangle. Then, find the area of each rectangle. Then, find the area of all rectangles together.
What do you notice about the areas of the rectangles?
How many different side lengths do you have?
Call one of your side lengths L, one of them W, and one of them H.
Construct your 2D net back into a 3D shape.
1
3.1 Notes Key
What do you notice about the areas of the rectangles?
there are 3 pairs of 2 identical rectangles
What would be the area of the front face of your rectangular prism? Use L, W, and/or H.
What other rectangle has this same area?
How many different side lengths do you have?
3 different side lengths in total
Call one of your side lengths L, one of them W, and one of them H.
What is the area of the side face?
What other rectangle has this same area?
What would be the area of the top face?
Construct your 2D net back into a 3D shape.
What other rectangle has this same area?
What would be the area of the front face of your rectangular prism? Use L, W, and/or H.
A = L x H
What other rectangle has this same area?
the back face
What is the area of the side face?
A = W x H
What other rectangle has this same area?
the other side
What would be the area of the top face?
A = L x W
What other rectangle has this same area?
the bottom
Combine the front face area and the back face area: (L x H) + (L x H) = 2(L x H) = 2LH
Combine the two side face areas:
(W x H) + (W x H) = 2(W x H) = 2WH
Combine the front face area and the back face area:
Combine the two side face areas:
Combine the top face area and the bottom face area:
Now, combine all three parts to make a formula for surface area of a rectangular prism.
Now, put your 2D net back into 3D
Using the rectangular prism formula we developed, find the Surface Area of the rectangular prism and see if it matches your calculations from earlier.
Combine the top face area and the bottom face area: (L x W) + (L x W) = 2(L x W) = 2LW
Now, combine all three parts to make a formula for surface area of a rectangular prism.
SA = 2LW + 2LH + 2WH
2
3.1 Notes Key
Now, put your 2D net back into 3D
Using the rectangular prism formula we developed, find the Surface Area of the rectangular prism and see if it matches your calculations from earlier.
h
l
h
l
Find the surface area if:
l = 4 cm
w = 3 cm
h = 7 cm
Find the surface area if:
l = 4 cm
w = 3 cm
h = 7 cm
w
Activity for Net #2:
Using your ruler, take the measurements that are necessary to find the area of each shape in the net, and then find the total area of the net.
w
SA = 2LW + 2LH + 2WH
= 2(4)(3) + 2(4)(7) + 2(3)(7)
= 24 + 56 + 42
= 122 cm2
How many total faces make up the net?
How many total faces make up the net?
5
What are the shapes of the faces?
What are the shapes of the faces?
2 triangles and 3 rectangles
Are there any faces that are identical?
Are there any faces that are identical?
the two triangles are always identical
Are the rectangles always identical? What does this depend on?
Are the rectangles always identical? What does this depend on? No. It depends on the side lengths of the triangle.
3
3.1 Notes Key
What is the formula for area of a triangle?
Since there are two identical triangles, what should you do?
What is the area formula for each rectangle?
What is the formula for area of a triangle?
b x h
2
Since there are two identical triangles, what should you do? multiply by 2, so 2(b x h)
2
What is the area formula for each rectangle?
L x W
What is the surface area formula for a triangular prism?
What is the surface area formula for a triangular prism?
SA = 2(b x h) + area of the 3 rectangles
2
Now, put your 2D net back into 3D and calculate the surface area of the triangular prism. See if it matches your calculation from earlier.
Now, put your 2D net back into 3D and calculate the surface area of the triangular prism. See if it matches your calculation from earlier.
Example: Calculate the surface area of the triangular prism.
Example: Calculate the surface area of the triangular prism.
SA = 2(b x h) + area of 2 three
rectangles
SA = 2(12 x 16) + (10)(20) + (12)(10) + (16)(10)
2
SA = 192 + 200 + 120 + 160 = 672 cm2
4
3.1 Notes Key
Activity for Net #3:
Using your ruler, take the measurements that are necessary to find the area of each shape in the net, and then find the total area of the net.
How many total faces make up the net?
What are the shapes of the different faces?
What is the formula for area of a circle?
Are the two circles identical? So what can you do with the formula?
How many total faces make up the net?
3
What are the shapes of the different faces?
2 circles and 1 rectangle
What is the formula for area of a circle?
A = πr2
Are the two circles identical? So what can you do with the formula?
Yes, so multiply formula by two
What is the formula for area of a rectangle?
Where does the width of the rectangle come from?
Where does the length of the rectangle come from? Put your net back into 3D briefly to help you answer this question.
What is the formula for this length?
two circles area = 2πr2
What is the formula for area of a rectangle?
A = L x W
Where does the width of the rectangle come from? it's the height of the cylinder, so W becomes H in this formula
Where does the length of the rectangle come from? Put your net back into 3D briefly to help you answer this question.
the length is the circumference of the circle
So, what is the formula for the area of the rectangle?
So, what is the complete formula for surface area of a cylinder?
What is the formula for this length?
C = 2πr so in this case, length = 2πr
5
3.1 Notes Key
So, what is the formula for the area of the rectangle? A = L x W = 2πr x h = 2πrh
Now, put your 2D net back into 3D and calculate the surface area of the triangular prism. See if it matches your calculation from earlier.
So, what is the complete formula for surface area of a cylinder?
SA = area of two circles + area of rectangle
SA = 2πr2 + 2πrh
Now, put your 2D net back into 3D and calculate the surface area of the triangular prism. See if it matches your calculation from earlier.
Example: Calculate the surface area of the cylinder to the nearest tenth.
8 m
1.8 m
Example: Calculate the surface area of the cylinder to the nearest tenth.
SA = 2πr2 + 2πrh
SA = 2π(1.8)2 + 2π(1.8)(8)
SA = 2π(3.24) + 2π(1.8)(8)
8 m
SA = 20.36 + 90.48
SA = 110.8 m2
Assignment:
­all of the 3.1 worksheet
1.8 m
6