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Compare Areas

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?

Take a Square

Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.


A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

Warmsnug Double Glazing

Age 14 to 16
Challenge Level


Why do this problem?

This problem is ideal for considering the variety of methods of generating and solving simultaneous equations within the context of calculating areas and perimeters of rectangles.

Mathematics lessons can sometimes feel neatly packaged, with information, techniques, patterns all readily accessible. But what do we do when the information arrives all jumbled up (with occasional errors) and we're asked to make sense of it? The problem offers students a chance to develop strategies for organising and understanding such situations. 


Possible approach

Project this image of the windows.
"Imagine you are the owner of a double glazing business. What variables do you think you would need to consider when deciding on the prices for your windows?"
Collect together ideas of the possible relevant variables.

"Warmsnug Double Glazing price their windows according to the area of glass used and the length of frame needed. Here is a worksheet showing the prices of different sized windows. Can you work out how Warmsnug arrived at the prices of these windows? Watch out - one of the windows has been priced incorrectly!"

Give students time to work together. While they are working, circulate and listen out for useful insights. If students are stuck, here are some helpful prompts:

  • Are there any windows that use the same amount of glass?
    How do their frame lengths differ?
  • Are there any windows that use the same amount of frame?
    How do their glass areas differ?

Once students think they have found a solution, challenge them to find a few different strategies for arriving at the pricing structure.

Give students time to work on this in pairs. While they are working, circulate and listen out for useful insights.

This worksheet has all the relevant data along with some efficient methods for finding the pricing structure. If your focus is on solving simultaneous equations give the class plenty of time to find as many different ways of finding the pricing structure as they can.

Bring the class together to share the new methods they have devised.

Next, hand out worksheet 1, worksheet 2, and worksheet 3, which become progressively more demanding. The problems on these sheets offer students the opportunity to apply and refine the methods they shared for the initial problem.
Alternatively, if a computer room is available students can work on this spreadsheet version.

Finally, use the spreadsheet to generate a Level 1, 2 or 3 problem and challenge the class to use an efficient method to work out the pricing structure and the incorrectly priced window.


Possible support

Along with the prompts above, suggest to students who are struggling with the large quantities of information that they initially ignore the windows with two panes.


Possible extension

For a follow-up problem on area and perimeter, see Changing Areas, Changing Perimeters.