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At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

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Six Discs

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

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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.


Stage: 4 Challenge Level: Challenge Level:1

Why do this problem?

The solution uses the formula for the area of a circle and gives practice in geometrical reasoning and manipulating algebraic expressions.

Possible approach

This printable worksheet may be useful: Salinon.

Start with a class discussion about the radii of the 4 semi-circles and the circle on AB as diameter. When the students have understood that there are only two unknowns involved, and the other radii can all be expressed in terms of the two unknowns, then they are ready to write down the areas and answer the question which they could do individually.

Dynamic geometry software could be a useful tool to aid investigation but this is not essential.

Key questions

If the blue semicircles have radii a and b what can you say about the other radii?

Possible support

Try Round and Round

Possible extension

Partly Circles is a similar problem.

Learners could make up their own problems of this type.