Analyse these beautiful biological images and attempt to rank them in size order.
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
How efficiently can you pack together disks?
Can you draw the height-time chart as this complicated vessel fills
Explore one of these five pictures.
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
A follow-up activity to Tiles in the Garden.
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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?
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.
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. . . .
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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. . . .
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?
Derive a formula for finding the area of any kite.
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?
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.
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.
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?
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
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?
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?
This article, written for teachers, discusses the merits of
different kinds of resources: those which involve exploration and
those which centre on calculation.
Can you find the area of a parallelogram defined by two vectors?
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.
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
A task which depends on members of the group noticing the needs of
others and responding.
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
What is the same and what is different about these circle
questions? What connections can you make?
What happens to the area and volume of 2D and 3D shapes when you
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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?
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?
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
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?
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!
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
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?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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.
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
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