# Area Contained in a Circle

This article presents multiple methods for finding the equation for the area contained within a circle. It is heavily inspired by integral calculus and is intended as a potential first encounter with the concept of a limit. It does not assume any previous knowledge of calculus though.

## Introduction

### Terminology

You will notice that I am taking great pains to say “area contained within a circle.” It would be simpler (and just as correct) to say “area of a disk.” The more common phrase “area of a circle” is incorrect. The circle refers only to the boundary, not the interior. When we say the area of a rectangle, that is fine because the word rectangle can refer to the whole shape.

Rectangles are defined as a special case of polygons, which have no agreed-upon definition. Three common definitions contradict each other: polygon refers only to the interior; polygon refers only to the boundary; polygon refers to both the interior and boundary. This is before dealing with the possibility of edges that cross and thus the possibility that only some interior parts are filled.

If we were to say “the area of the edges of a rectangle,” that would obviously be ridiculous. The edges of the rectangle are lines and thus infinitely thin. They have no area. A circle is the same. It is the edge surrounding a disk and is also made of an infinitely thin curve. It also cannot have an area.

I won’t be upset if you continue to use “area of a circle.” It is the most common way to refer to this concept. It is taught this way from as early as preschool (where rectangles, triangles, and circles are often treated as the same sort of thing even though they are often defined to be different creatures altogether). However, it is important that you know it is an incorrect shorthand. The distinction does make a difference in certain contexts, particularly if you want to distinguish between the interior (disk) and boundary (circle).

### Finding the Area Contained Within a Circle: Your First Encounter

To start, I’d like to invite you to try to come up with a solution on your own. Take a few minutes and think about how you have found equations for the area contained within other two-dimensional shapes (squares, rectangles, parallelograms, triangles, etc.). Can you think of any way to reuse or extend one of those techniques? Can you create your own method? Please don’t work too hard. If you find yourself getting frustrated, take a break or move on to the next section. There are some nice pointers to get you started. Please stop often and reassess if you have new ideas about how to solve the problem. The fewer hints you use, the stronger your creative and problem-solving skills will become. Remember that there are lots of ways to solve these problems. If you invent one for yourself that is not covered here, that is awesome!

## The Method of Boxes (Cartesian Integral)

### Setup

Have a look at this app. Leave the hint buttons alone for now. You can use the full-screen button if you like. Try setting the white and red number dials. What do you think they represent? How might this help you discover the area contained within the circle? Please be patient when you have both dials set to numbers above 50. The app may take a moment to process the data for you, especially on an older or handheld device. At the settings of 100 and 100, it is handling 400 million boxes for you!

## The Method of Segments (Polar Integral: Radius then Angle)

In the last section, you found a good guess for the equation that gives the area contained within a circle. In this section and the next one, you will confirm this guess using different approaches.

### Setup

Have a look at the app. Feel free to change the number of segments with the dial and use the full-screen button if you like. Please refrain from using the interact button or the hint button for now though. The app can struggle when showing more than 64 segments (especially on old or handheld devices). Please be patient.

Can you think of a way to find the equation for the area contained within a circle using this approach?

## The Method of Rings (Polar Integral: Angle then Radius)

Can you think of another way to find the area contained within a circle, perhaps one that has to do with rings?

### Setup

Take a look at the app. Feel free to change the number of rings with the dial and use the full-screen button if you like. However, please refrain from using the interact or hint buttons for now. For displays that are small or low resolution, strange patterns may occur if you choose to display many thin rings. If you start to see strange patterns (like stretched four-leaf clovers), please lower the number of rings to avoid confusion.

Can you think of a way to find the equation for the area contained within a circle using this approach?

#### NOTE

There is a small imperfection in the animation. Can you spot it? It is subtle and doesn’t affect the result, but it may bother your geometric intuition. If you do spot it, how might the animation be improved? We will discuss it in full at the end of this section.

#### NOTE

For those with exceptional geometric intuition, you may have noticed an important imperfection here. The start and end positions of the animation work out exactly right, but during the animation, there is some distortion. See if you can spot the problem and find a way to improve it. We will discuss it shortly.

### Discussion About the Imperfection

If you haven’t taken the time to try to spot the imperfection, please try now. If you run the animation, it is perfect at the beginning and end, but in between, there is an imperfection. The imperfection is at its worst at about halfway through the animation. Do you see it?

The transformation used does preserve area, but it doesn’t preserve lengths. While all the lengths are returned to their proper values at the end of the animation, in between, the length of the circumference shrinks by about a quarter and the widths of the ribbons (at their ends) widen to compensate. I could have fixed this, but I thought it was an interesting discussion to have and gives students a chance to really test their geometric intuition.

How do you think a perfect animation would have looked?

A perfect animation would have had the ribbons not change widths or lengths. This would have caused there to be gaps between the ribbons while they were unfurling.

For those who are interested, the transformation used was and also the horizontal direction was scaled up by . was the amount of the circle which was present (as a function of angle). It ranged from the full radians to only 1% that amount. was a scale factor that went between 1 and 100 in a nonlinear fashion.

## The Method of Inscribed Regular Polygons (Your Turn Again)

It’s your turn again. Have a look at this app. It inscribes regular polygons in a circle. Don’t use the hint buttons unless you really need to. If you do use the hint button, use the first (left) one and try to figure it out. If you still need help, try the second (right) hint button.

Can you find a way to confirm the equation for the area contained in a circle using this approach? Can you come up with more than one way? Would your approach change if you instead inscribed the circle in successive regular polygons? How would the diagram change in this case?

## For Teachers and Parents

### What is this Hint System

The hint system is designed to allow students to engage with the problems with little to no support (Discovery/Inquiry-Based Math). The hints gradually give more and more information until we end up with a full explanation (Traditional Lecture). Since Math is not a ‘spectator sport,’ it is hoped that repeated use of this hint system in multiple lessons will encourage students to use fewer and fewer hints and learn to solve problems creatively for themselves. This is, of course, only one possible arrangement of these hints. An instructor will be better able to adapt the presentation of the hints to the progress a specific student has made. This is beneficial to the student until they learn to use the system and self-regulate. If the student needs hints, they can choose to set aside their particular approach temporarily while filling in the gaps in the order the text presents them. Later they can analyze how the approach they took compares to that of the text.

### Isn't this Basically Integral Calculus?

Yes! I am using this lesson as a Trojan Horse. Exposing students to these ideas early so they have horizon knowledge about what is coming down the track later is important. It gives their subconscious extra years to digest the ideas before they become everyday companions.

### Why no Rigor?

Why no Rigor? Sorry, the treatment is not rigorous. To my Pure Math friends, sometimes rigor can get in the way of clarity when first encountering a concept. I promise a rigorous treatment of calculus, straight ‘outta’ Real Analysis, will come in future years.