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. . . .
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. . . .
What fractions of the largest circle are the two shaded regions?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Can you maximise the area available to a grazing goat?
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?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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.
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!
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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.
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
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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).
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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. . . .
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
A follow-up activity to Tiles in the Garden.
Derive a formula for finding the area of any kite.
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. . . .
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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?
Can you work out the area of the inner square and give an
explanation of how you did it?
A task which depends on members of the group noticing the needs of
others and responding.
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.
You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in
the middle. Cut the carpet into two pieces to make a 10 by 10 foot
Determine the total shaded area of the 'kissing triangles'.
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
Explore one of these five pictures.
A hallway floor is tiled and each tile is one foot square. Given
that the number of tiles around the perimeter is EXACTLY half the
total number of tiles, find the possible dimensions of the hallway.
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
How efficiently can you pack together disks?
What is the same and what is different about these circle
questions? What connections can you make?
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
Take any rectangle ABCD such that AB > BC. The point P is on AB
and Q is on CD. Show that there is exactly one position of P and Q
such that APCQ is a rhombus.
Can you find the area of a parallelogram defined by two vectors?
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?