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A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

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Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

Six Discs

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?

This problem allows learners to see the value of estimating before making accurate calculations, and to see that sometimes an estimate is all that is needed. The problem also offers an opportunity to practise calculating areas and working out lengths accurately using trigonometry.

Possible approach

Show the image of the six boxes and explain that we're interested in comparing the areas of different boxes made to hold six circular discs. Ask learners to compare the areas of A and B, and allow everyone time to consider and then discuss in pairs which is bigger and why.

In discussing A and B, key ideas to consider include comparing the parts of the shapes which are the same, or comparing the size of the gaps between circles. Next, allow the class some time to discuss in pairs or small groups how to order all six shapes. Stress that the importance is not so much in the order the learners come up with but in their reasons for placing them in that order.

After allowing some time to work on estimating, suggest to the learners that some calculations may help them to put the shapes in order. Learners may decide they do not need to work out every area to be certain of the correct order - for example, if they are certain their estimate has identified the biggest or the smallest area they may choose not to calculate that one. They could then work in small groups to create a poster or presentation showing the correct order for the shapes and the justification and calculations they used to find it. The hint contains two diagrams which suggest an approach for working out the areas using trigonometry, which could be shared with the class if appropriate.

Key questions

Which shape do you think has the largest area and why?
What angles can we work out? What lengths do we know?
For which shapes do you think you need to work out the areas in order to be certain you had ordered the shapes correctly?

Possible extension

The problem Covering Cups provides another context for investigating packing circular shapes.
Learners could design their own shapes which contain six circles and challenge each other to estimate the areas.

Possible support

Some of the calculations for the exact areas require knowledge of trigonometry, but the problem could be tackled instead by constructing accurate diagrams of the boxes and measuring in order to calculate their areas.