Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
Prove that the area of a quadrilateral is given by half the product of the lengths of the diagonals multiplied by the sine of the angle between the diagonals.
A farmer has a field which is the shape of a trapezium as
illustrated below. To increase his profits he wishes to grow two
different crops. To do this he would like to divide the field into
two. . . .
A trapezium is divided into four triangles by its diagonals.
Suppose the two triangles containing the parallel sides have areas
a and b, what is the area of the trapezium?
If I print this page which shape will require the more yellow ink?
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
Triangle ABC is right angled at A and semi circles are drawn on all three sides producing two 'crescents'. Show that the sum of the areas of the two crescents equals the area of triangle ABC.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
Draw two circles, each of radius 1 unit, so that each circle goes
through the centre of the other one. What is the area of the
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
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. . . .
What fractions of the largest circle are the two shaded regions?
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?
Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
Straight lines are drawn from each corner of a square to the mid
points of the opposite sides. Express the area of the octagon that
is formed at the centre as a fraction of the area of the square.
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
What happens to the area and volume of 2D and 3D shapes when you
Polygons drawn on square dotty paper have dots on their perimeter
(p) and often internal (i) ones as well. Find a relationship
between p, i and the area of the polygons.
Can you find the area of a parallelogram defined by two vectors?
A square of area 40 square cms is inscribed in a semicircle. Find
the area of the square that could be inscribed in a circle of the
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
How can you change the area of a shape but keep its perimeter the same? How can you change the perimeter but keep the area the same?
Explore one of these five pictures.
Can you draw the height-time chart as this complicated vessel fills
Bluey-green, white and transparent squares with a few odd bits of
shapes around the perimeter. But, how many squares are there of
each type in the complete circle? Study the picture and make. . . .
Analyse these beautiful biological images and attempt to rank them in size order.
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
How efficiently can you pack together disks?
A follow-up activity to Tiles in the Garden.
A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?
Three squares are drawn on the sides of a triangle ABC. Their areas
are respectively 18 000, 20 000 and 26 000 square centimetres. If
the outer vertices of the squares are joined, three more. . . .
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and. . . .
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you maximise the area available to a grazing goat?
What is the same and what is different about these circle
questions? What connections can you make?