Area - triangles, quadrilaterals, compound shapes

problem
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Inscribed in a circle

Age

14 to 16

Challenge level

The area of a square inscribed in a circle with a unit radius is 2. What is the area of these other regular polygons inscribed in a circle with a unit radius?
problem
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Areas of parallelograms

Age

14 to 16

Challenge level

Can you find the area of a parallelogram defined by two vectors?
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Trapezium four

Age

14 to 16

Challenge level

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
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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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Towering trapeziums

Age

14 to 16

Challenge level

Can you find the areas of the trapezia in this sequence?
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Six discs

Age

14 to 16

Challenge level

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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Bicentric quadrilaterals

Age

14 to 16

Challenge level

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.
problem
Favourite

Quad in quad

Age

14 to 18

Challenge level

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
problem
Favourite

Kite in a square

Age

14 to 18

Challenge level

Can you make sense of the three methods to work out what fraction of the total area is shaded?
problem

Halving the triangle

Age

16 to 18

Challenge level

Draw any triangle PQR. Find points A, B and C, one on each side of the triangle, such that the area of triangle ABC is a given fraction of the area of triangle PQR.