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. . . .
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. . . .
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
What happens to the area and volume of 2D and 3D shapes when you
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. . . .
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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
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).
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.
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?
A task which depends on members of the group noticing the needs of
others and responding.
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
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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?
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Can you maximise the area available to a grazing goat?
Derive a formula for finding the area of any kite.
A follow-up activity to Tiles in the Garden.
Determine the total shaded area of the 'kissing triangles'.
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?
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.
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?
Explore one of these five pictures.
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. . . .
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?
How efficiently can you pack together disks?
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?
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?
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.
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
What is the same and what is different about these circle
questions? What connections can you make?
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
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.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
It is possible to dissect any square into smaller squares. What is
the minimum number of squares a 13 by 13 square can be dissected
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
What is the area of the quadrilateral APOQ? Working on the building
blocks will give you some insights that may help you to work it