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.
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
How efficiently can you pack together disks?
Can you draw the height-time chart as this complicated vessel fills
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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. . . .
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. . . .
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. . . .
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
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
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Explore one of these five pictures.
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
Analyse these beautiful biological images and attempt to rank them in size order.
A follow-up activity to Tiles in the Garden.
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. . . .
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
Can you work out the area of the inner square and give an
explanation of how you did it?
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
Derive a formula for finding the area of any kite.
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
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?
Can you maximise the area available to a grazing goat?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
If I print this page which shape will require the more yellow ink?
A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?
What happens to the area and volume of 2D and 3D shapes when you
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?
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?
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.
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
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.
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.
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.
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?
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?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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!
What is the same and what is different about these circle
questions? What connections can you make?
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.
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 ?
Straight lines are drawn from each corner of a square to the mid
points of the opposite sides. Express the area of the octagon that
is formed at the centre as a fraction of the area of the square.