Area, perimeter, and circumference cause more confusion among students than necessary.
The problem is that we dive in with formulae before students have their bearings. Let your students get their hands dirty with geometry. They’ve got to play with the shapes and see why these magical formulae work.
Beginning adders and subtractors work with manipulatives before they delve into abstract arithmetic. Older students are still novices at geometry. Give them a chance to touch the math and have some fun.
Area’s Relationship to Perimeter
Let’s set up this lesson by creating some conflict around a common misconception.
Students, my square has a perimeter of four meters. I’m going to double it. Will my area double also?
No! Doubling perimeter does not double area. But I bet your students think it does.
Introduce this as an inductive exploration. Set up a pattern by starting with a perimeter of four square feet.
| Side Length | Perimeter | Area |
|---|---|---|
| 1 m | 4 m | 1 sq m |
| 2 m | 8 m | 4 sq m |
| 3 m | 12 m | 9 sq m |
| 4 m | 16 m | 16 sq m |
| 5 m | 20 m | 25 sq m |
Stop here, because several interesting patterns have arisen. Ask for observations. Here are a couple:
- At 4 m per side, perimeter and area intersect.
- The area increases much faster than the length of the perimeter (you could graph this and discuss linear versus exponential growth).
- Sides increase by one, perimeters increase by four, and area increases by three, five, seven, then nine! I wonder if it continues…
Show how dramatic the increase is with larger numbers:
| Side Length | Perimeter | Area |
|---|---|---|
| 10 m | 40 m | 100 sq m |
| 50 m | 200 m | 2500 sq m |
| 250 m | 1000 m | 62500 sq m |
Squares Versus Rectangles
Continue to expand their perceptions of the relationship between area and perimeter. Ask, “Will a perimeter of 20 m always lead to an area of 25 sq m?”
Uncertain nods
Let’s build some rectangles instead of squares and surround them with 20 m of fence. Ask for all the combinations that give 20 m:
|Length | Width | Perimeter | Area
|——–|–|–|
| 1 m | 9 m | 20 m | 9 sq m |
| 2 m | 8 m | 20 m | 16 sq m |
| 3 m | 7 m | 20 m | 21 sq m |
| 4 m | 6 m | 20 m | 24 sq m |
| 5 m | 5 m | 20 m | 25 sq m |
Look at the remarkable difference in areas! A square is nearly three times more efficient than a skinny rectangle.
What if you included decimals? How low could the area go? A 0.1 m by 9.9 m rectangle has a shockingly small area.
But is this always true?
Try it with some other perimeters. Lead students to the generalization: a square uses perimeter more efficiently than any other rectangle.
And The Circle Rolls In
So, the square may be the most efficient rectangle, but is it the most efficient of all shapes?
Roll a circle in and have a competition.
Will a circle with circumference 20 m beat the square’s area of 25 sq m?
If your students have gotten their feet wet in algebra, they can find the radius of a circle by starting with the circumference.
Find the radius of the circle:
- c = πd (formula of circumference of a circle)
- 20 = πd (circumference given)
- 20 / π = d (divide both sides by π
- ~ 6.37 = d (approximate radius)
Halve that diameter to get a radius of ~ 3.185 m.
Using the radius, find the area of the circle:
- A = πr² (formula of area of a circle)
- A = π * 10.144 (square the radius we found)
- A = ~ 31.85 (approximate area of the circle)
So the square was king of the quadrilaterals with 25 sq m, however the circle handily beats that with an area of nearly 32 sq m.
But… Why?
Now, if you’ve got a bunch of gifted minds in your classroom, someone is going to ask the age-old question: why?
Why is the circle so efficient?
The short answer: given the same perimeter, a regular figure with more sides will cover more area.
But don’t just tell them! Show it using this calculator. Slowly (and dramatically!) build a table like so:
| # Sides | Side Length | Area |
|---|---|---|
| 4 | 5 m | 25 sq m |
| 5 | 4 m | ~ 27.52 sq m |
| 6 | 3.33 m | ~ 28.86 sq m |
| 8 | 2.5 m | ~ 30.177 sq m |
| 10 | 2 m | ~ 30.77 sq m |
I bet they’ll be disappointed when you stop at the decagon. Feel free to go further, or give it as some optional homework.
So a hexagon will beat a square, an octagon will beat a hexagon, and so on.
Guess which shape has the most sides? That’s right, the circle with infinity sides will always win.
Finally
While you’re discussing polygons, please play this video for your students: Nonagon.