Area - circles, sectors and segments

  • Mean Geometrically
    problem
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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?
  • Quadarc
    problem
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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.
  • Approximating Pi
    problem
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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?
  • Spokes
    problem
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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.
  • Blue and White
    problem
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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?

  • An Unusual Shape
    problem
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    An Unusual Shape

    Age
    11 to 14
    Challenge level

    Can you maximise the area available to a grazing goat?

  • problem
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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?

  • Curvy areas
    problem
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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?

  • Gutter
    problem
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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?

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