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 prove this formula for finding the area of a quadrilateral from its diagonals?
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. . . .
Triangle ABC is right angled at A and semi circles are drawn on all three sides producing two 'crescents'. Show that the sum of the areas of the two crescents equals the area of triangle ABC.
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. . . .
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?
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
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?
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?
If I print this page which shape will require the more yellow ink?
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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
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?
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 have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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 follow-up activity to Tiles in the Garden.
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
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Make an eight by eight square, the layout is the same as a
chessboard. You can print out and use the square below. What is the
area of the square? Divide the square in the way shown by the red
dashed. . . .
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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. . . .
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.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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.
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!
Explore one of these five pictures.
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?
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.
Determine the total shaded area of the 'kissing triangles'.
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).
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
Derive a formula for finding the area of any kite.
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
Three squares are drawn on the sides of a triangle ABC. Their areas
are respectively 18 000, 20 000 and 26 000 square centimetres. If
the outer vertices of the squares are joined, three more. . . .
Can you work out the area of the inner square and give an
explanation of how you did it?
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. . . .
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. . . .
This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.
What is the same and what is different about these circle
questions? What connections can you make?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?