Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
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?
You can move the 4 pieces of the jigsaw and fit them into both
outlines. Explain what has happened to the missing one unit of
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you maximise the area available to a grazing goat?
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.
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. . . .
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?
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?
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?
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
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 the area of a parallelogram defined by two vectors?
If I print this page which shape will require the more yellow ink?
Can you work out the area of the inner square and give an
explanation of how you did it?
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
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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
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?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.
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. . . .
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
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.
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
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.
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?
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. . . .
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. . . .
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 prove this formula for finding the area of a quadrilateral from its diagonals?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
How efficiently can you pack together disks?
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?
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.
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.
What is the same and what is different about these circle
questions? What connections can you make?
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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. . . .
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?