The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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. . . .
Can you maximise the area available to a grazing goat?
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 rectangles where the value of the area is the same as the value of the perimeter?
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?
If I print this page which shape will require the more yellow ink?
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.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
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.
What fractions of the largest circle are the two shaded regions?
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
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.
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?
What is the same and what is different about these circle
questions? What connections can you make?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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 properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
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?
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
Can you work out the area of the inner square and give an
explanation of how you did it?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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
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
Explore one of these five pictures.
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.
A task which depends on members of the group noticing the needs of
others and responding.
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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
Can you find the area of a parallelogram defined by two vectors?
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.
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?
What happens to the area and volume of 2D and 3D shapes when you
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Determine the total shaded area of the 'kissing triangles'.
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 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.
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?
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. . . .