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 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?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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
Can you find the area of a parallelogram defined by two vectors?
What is the same and what is different about these circle
questions? What connections can you make?
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
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?
Determine the total shaded area of the 'kissing triangles'.
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.
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).
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.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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 task which depends on members of the group noticing the needs of
others and responding.
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?
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. . . .
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
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.
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?
Can you maximise the area available to a grazing goat?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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 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
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.
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
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you work out the area of the inner square and give an
explanation of how you did it?
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?
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?
How efficiently can you pack together disks?
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
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.
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. . . .
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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?
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Derive a formula for finding the area of any kite.
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!
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
A follow-up activity to Tiles in the Garden.
Can you draw the height-time chart as this complicated vessel fills
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.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.