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?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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 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?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
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?
Can you find the area of a parallelogram defined by two vectors?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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 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.
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
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?
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.
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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. . . .
Can you work out the area of the inner square and give an
explanation of how you did it?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Explore one of these five pictures.
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?
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.
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
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
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
A follow-up activity to Tiles in the Garden.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
A task which depends on members of the group noticing the needs of
others and responding.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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!
Can you show that you can share a square pizza equally between two
people by cutting it four times using vertical, horizontal and
diagonal cuts through any point inside the square?
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.
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?
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
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
What is the same and what is different about these circle
questions? What connections can you make?
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 ?
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?
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. . . .
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).