🎯 Learning Goal: Find the total area of ALL the faces of a rectangular prism
Remember Jerry's baseball boxes? Today we'll figure out which box design uses the least cardboard!
Area = the amount of space inside a flat shape
We measure it in square units (square inches, square feet, square centimeters, etc.)
Example: A rectangle that is 5 cm by 3 cm
Area = 5 × 3 = 15 square cm
This flat rectangle could cover 15 square centimeters of your desk
Volume = the amount of space inside a 3D shape
We measure it in cubic units (cubic inches, cubic feet, cubic centimeters, etc.)
Jerry's Baseball Boxes 🎾
Remember: All of Jerry's boxes hold 24 baseballs
That means they all have a volume of 24 cubic units
Examples: 4×3×2, 6×2×2, 12×2×1, 24×1×1
Jerry needs to order cardboard to make his boxes.
All the boxes hold 24 baseballs (same volume)...
But do they all need the SAME AMOUNT of cardboard? 🤔
Let's think about it...
A box that is 12×2×1 is long and flat
A box that is 3×4×2 is more like a cube
Do they use the same amount of cardboard? 🤷
Today we'll find out! 🎯
Surface Area = the TOTAL area of ALL the faces (sides) of a 3D shape
It tells us how much material we need to cover the OUTSIDE of the shape
🌎 Real World: Surface area tells us how much cardboard, wrapping paper, paint, or material we need to cover something!
Think about wrapping a present 🎁
You need to cover the top, bottom, front, back, left side, and right side
Surface area = the total amount of wrapping paper you need!
Every rectangular prism (box) has 6 faces
These 6 faces come in 3 pairs of matching rectangles:
TOP
length × width
BOTTOM
length × width
FRONT
length × height
BACK
length × height
LEFT
width × height
RIGHT
width × height
Steps to Find Surface Area:
1 Find the area of the TOP face (length × width)
Multiply by 2 because the BOTTOM is the same!
2 Find the area of the FRONT face (length × height)
Multiply by 2 because the BACK is the same!
3 Find the area of the LEFT face (width × height)
Multiply by 2 because the RIGHT is the same!
4 ADD all those areas together = SURFACE AREA!
Where:
💡 Remember: We multiply by 2 for each pair because opposite faces are identical!
One of Jerry's box designs:
Question: How much cardboard does Jerry need to make this box?
(In other words: What is the surface area?)
First, let's check: Does this box hold 24 baseballs?
Volume = 4 × 3 × 2 = 24 cubic units ✓
Box: 4 units × 3 units × 2 units
1 Find the area of TOP and BOTTOM
Top face = length × width
Top face = 4 × 3 = 12 square units
Both top and bottom = 2 × 12 = 24 square units
🌎 This means we need 12 square units of cardboard for the top, and 12 more for the bottom = 24 total so far
Box: 4 units × 3 units × 2 units
Running total so far: 24 square units
2 Find the area of FRONT and BACK
Front face = length × height
Front face = 4 × 2 = 8 square units
Both front and back = 2 × 8 = 16 square units
🌎 We need 8 square units for the front, and 8 more for the back = 16 total
Box: 4 units × 3 units × 2 units
Running total so far: 24 + 16 = 40 square units
3 Find the area of LEFT and RIGHT sides
Left side = width × height
Left side = 3 × 2 = 6 square units
Both left and right = 2 × 6 = 12 square units
🌎 We need 6 square units for the left side, and 6 more for the right = 12 total
Box: 4 units × 3 units × 2 units
4 Add all the areas together
Top & Bottom: 24 square units
Front & Back: 16 square units
Left & Right: 12 square units
TOTAL = 24 + 16 + 12 = 52 square units
Surface Area = 52 square units
🌎 Jerry needs 52 square units of cardboard to make this box!
Another box Jerry could use:
Find the surface area of this box!
Work through the 4 steps with your partner, then we'll check together
Box: 6 units × 2 units × 2 units
Step 1: Top & Bottom = 2(6 × 2) = 2(12) = 24 sq units
Step 2: Front & Back = 2(6 × 2) = 2(12) = 24 sq units
Step 3: Left & Right = 2(2 × 2) = 2(4) = 8 sq units
Step 4: Total = 24 + 24 + 8 = 56 sq units
Surface Area = 56 square units
💡 Interesting! This box holds the same 24 baseballs but needs MORE cardboard (56) than the first box (52)!
A more cube-shaped box:
Find the surface area of this box!
Use the 4 steps we learned
Box: 3 units × 2 units × 4 units
Step 1: Top & Bottom = 2(3 × 2) = 2(6) = 12 sq units
Step 2: Front & Back = 2(3 × 4) = 2(12) = 24 sq units
Step 3: Left & Right = 2(2 × 4) = 2(8) = 16 sq units
Step 4: Total = 12 + 24 + 16 = 52 sq units
Surface Area = 52 square units
🤯 WOW! This box uses the same amount of cardboard (52) as our very first example, even though it's a different shape!
Remember: All these boxes hold 24 baseballs (volume = 24 cubic units)
Your Task:
Box A
12 × 2 × 1
(long and very flat)
Box B
8 × 3 × 1
(wide and flat)
Box C
6 × 4 × 1
(flatter box)
Box D
24 × 1 × 1
(VERY long and flat!)
✏️ Materials needed: Paper for calculations, pencil, and your math journal for notes