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?
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
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. . . .
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
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.
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 the area of a parallelogram defined by two vectors?
Can you maximise the area available to a grazing goat?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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 same and what is different about these circle
questions? What connections can you make?
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 are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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.
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?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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.
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
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.
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
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
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
Determine the total shaded area of the 'kissing triangles'.
Can you work out the area of the inner square and give an
explanation of how you did it?
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?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Can you draw the height-time chart as this complicated vessel fills
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Derive a formula for finding the area of any kite.
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. . . .
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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?
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!
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.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Do you know how to find the area of a triangle? You can count the
squares. What happens if we turn the triangle on end? Press the
button and see. Try counting the number of units in the triangle
now. . . .
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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).
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?
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
Explore one of these five pictures.
Prove that the area of a quadrilateral is given by half the product of the lengths of the diagonals multiplied by the sine of the angle between the diagonals.
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?