Regular polygons and circles

There are 118 NRICH Mathematical resources connected to Regular polygons and circles
Sweets in a box
problem
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Sweets in a box

Age
7 to 11
Challenge level
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How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

Quadarc
problem
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Quadarc

Age
14 to 16
Challenge level
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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.
Shapes on the Playground
problem
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Shapes on the playground

Age
7 to 11
Challenge level
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Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?
Rolling Around
problem
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Rolling around

Age
11 to 14
Challenge level
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A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
Round and round the circle
problem
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Round and round the circle

Age
7 to 11
Challenge level
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What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Where are they?
problem
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Where are they?

Age
7 to 11
Challenge level
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Use the isometric grid paper to find the different polygons.

What Shape and Colour?
problem
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What shape and colour?

Age
5 to 7
Challenge level
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Can you fill in the empty boxes in the grid with the right shape and colour?

Salinon
problem
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Salinon

Age
14 to 16
Challenge level
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This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
2 Rings
problem
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2 rings

Age
5 to 7
Challenge level
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The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?
Baby Circle
problem
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Baby circle

Age
16 to 18
Challenge level
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A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?