Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
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?
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. . . .
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?
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).
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?
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.
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
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?
How efficiently can you pack together disks?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
If I print this page which shape will require the more yellow ink?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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. . . .
Derive a formula for finding the area of any kite.
Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!
The area of a square inscribed in a circle with a unit radius is,
satisfyingly, 2. What is the area of a regular hexagon inscribed in
a circle with a unit radius?
Can you maximise the area available to a grazing goat?
Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?
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
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
A tower of squares is built inside a right angled isosceles
triangle. The largest square stands on the hypotenuse. What
fraction of the area of the triangle is covered by the series of
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
Determine the total shaded area of the 'kissing triangles'.
Analyse these beautiful biological images and attempt to rank them in size order.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
What happens to the area and volume of 2D and 3D shapes when you
A task which depends on members of the group noticing the needs of
others and responding.
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?
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
Can you draw the height-time chart as this complicated vessel fills
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
You can move the 4 pieces of the jigsaw and fit them into both
outlines. Explain what has happened to the missing one unit of
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .
Can you find rectangles where the value of the area is the same as the value of the perimeter?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?