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?
Analyse these beautiful biological images and attempt to rank them in size order.
How efficiently can you pack together disks?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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.
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?
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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. . . .
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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.
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!
Can you find the area of a parallelogram defined by two vectors?
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
What happens to the area and volume of 2D and 3D shapes when you
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?
Derive a formula for finding the area of any kite.
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
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.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
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. . . .
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 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.
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?
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
A task which depends on members of the group noticing the needs of
others and responding.
What is the same and what is different about these circle
questions? What connections can you make?
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?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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
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?
Can you maximise the area available to a grazing goat?
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?
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
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
Can you work out the area of the inner square and give an
explanation of how you did it?