Area - circles, sectors and segments

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Mean Geometrically

Age

16 to 18

Challenge level

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?
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Quadarc

Age

14 to 16

Challenge level

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.
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Approximating Pi

Age

14 to 18

Challenge level

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?
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Spokes

Age

16 to 18

Challenge level

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.
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Blue and White

Age

11 to 14

Challenge level

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
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An Unusual Shape

Age

11 to 14

Challenge level

Can you maximise the area available to a grazing goat?
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Salinon

Age

14 to 16

Challenge level

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?
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Curvy Areas

Age

14 to 16

Challenge level

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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Triangles and Petals

Age

14 to 16

Challenge level

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?
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Gutter

Age

14 to 16

Challenge level

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?