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*.

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