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. . . .
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
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
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
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.
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. . . .
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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?
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. . . .
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 fractions of the largest circle are the two shaded regions?
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 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?
A follow-up activity to Tiles in the Garden.
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?
Explore one of these five pictures.
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?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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. . . .
If I print this page which shape will require the more yellow ink?
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?
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!
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?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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 white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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.
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
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.
What is the same and what is different about these circle
questions? What connections can you make?
Determine the total shaded area of the 'kissing triangles'.
Cut off three right angled isosceles triangles to produce a
pentagon. With two lines, cut the pentagon into three parts which
can be rearranged into another square.
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. . . .
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?
What happens to the area and volume of 2D and 3D shapes when you
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
Derive a formula for finding the area of any kite.
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
A task which depends on members of the group noticing the needs of
others and responding.
How efficiently can you pack together disks?
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.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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?