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 the area of a parallelogram defined by two vectors?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
A follow-up activity to Tiles in the Garden.
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.
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?
Determine the total shaded area of the 'kissing triangles'.
What is the same and what is different about these circle
questions? What connections can you make?
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
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
Seven small rectangular pictures have one inch wide frames. The
frames are removed and the pictures are fitted together like a
jigsaw to make a rectangle of length 12 inches. Find the dimensions
of. . . .
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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.
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
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. . . .
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.
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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?
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.
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?
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?
Can you maximise the area available to a grazing goat?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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
How efficiently can you pack together disks?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
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!
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
A task which depends on members of the group noticing the needs of
others and responding.
What happens to the area and volume of 2D and 3D shapes when you
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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.
Analyse these beautiful biological images and attempt to rank them in size order.