Find the area of a circle......without using pi

I haven't had much time to write on the blog lately as we are busy assessing and writing reports. Don't you love end-of-year?

As part of the assessment process, we used a pencil and paper test to confirm what we were thinking about Maths skills and concepts.

We had done a bit of inquiry into find the area of 2D shapes with straight sides. We hadn't looked at circles.

So I thought I'd throw in a question about finding the area of a circle. I knew that some of the kids knew the formula so I made the rule that they were not allowed to use pi in any of their calculations.

Just wanted to see what they would do.

So here is the question:

You are given a circle with a diameter of 10cm. Find the area of the circle by dividing it up into smaller shapes. (Do not use pi in any of your calculations.)

(I cheated - I used pi and got an area of about 78.54cm2..  I wanted to have some "real" value to assess the accuracy of the students and their methods against.)


And here a some of the solutions that the kids came up with...

Method 1 - Break it up

Nice idea. Using regular shapes such as squares and rectangles, we can get an approximation of what the area of the circle might be. It is a bit complicated in that it involves lots of different calculations, but the solution presented here was pretty close to our target.

Same strategy but choosing different shapes. This attempt is a bit less detailed so it loses a bit of accuracy but the end solution is also pretty close to out target. The students quickly saw that since the shape is symmetrical, some of the calculations only need to be done once then multiplied by the number of times that shape appears in the diagram.

Method 2 - Subtract the corners from a bigger square

Seeing that the circle has a diameter of 10cm, it would fit into a square with sides 10cm x 10cm. We could then subtract the "corner" triangles to get an approximation. I'm not sure that the student measures and calculated the triangle accurately as their solution was a bit out but the strategy and process was excellent.

Same general idea but this time using rectangles in the corners instead of triangles. 

Method 3 - Using a grid

This student drew up a 1cm grid and used this to count the number of squares. They worked out the number of complete squares and then added half the number of incomplete squares to get a pretty accurate result.

Using a grid for a quarter of the circle means you only have to count a quarter of the number of squares to calculate an approximate area that is very close to our target.

And finally, this student chose to use a 4cm grid to get their solution which was slightly higher than the others but definitely demonstrated a good understanding of the problem and what to do to solve it.



Not everyone got a solution to this question. Several were not sure what to do or failed to use their knowledge to calculate accurately and as a result their solution were way out. 

But all of this provided some useful information - either confirming what I already suspected about each student or giving me an insight into the way they thought.

So....back to the report writing!