Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
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?
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. . . .
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.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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. . . .
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
How efficiently can you pack together disks?
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).
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
If I print this page which shape will require the more yellow ink?
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?
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
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?
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.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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. . . .
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
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.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
What happens to the area and volume of 2D and 3D shapes when you
Analyse these beautiful biological images and attempt to rank them in size order.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
A task which depends on members of the group noticing the needs of
others and responding.
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Can you maximise the area available to a grazing goat?
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?
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 draw the height-time chart as this complicated vessel fills
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 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?
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.
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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. . . .
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
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. . . .
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
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
Derive a formula for finding the area of any kite.