The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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 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 fractions of the largest circle are the two shaded regions?
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.
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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
A follow-up activity to Tiles in the Garden.
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.
Explore one of these five pictures.
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?
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.
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?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
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 rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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
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?
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
What is the same and what is different about these circle
questions? What connections can you make?
A task which depends on members of the group noticing the needs of
others and responding.
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 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?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
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!
What happens to the area and volume of 2D and 3D shapes when you
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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
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
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?
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).
Can you work out the area of the inner square and give an
explanation of how you did it?
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. . . .
Determine the total shaded area of the 'kissing triangles'.
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.