What fractions of the largest circle are the two shaded regions?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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. . . .
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?
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.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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.
Explore one of these five pictures.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
What happens to the area and volume of 2D and 3D shapes when you
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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.
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?
A task which depends on members of the group noticing the needs of
others and responding.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Can you maximise the area available to a grazing goat?
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?
Can you work out the area of the inner square and give an
explanation of how you did it?
Derive a formula for finding the area of any kite.
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!
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. . . .
Determine the total shaded area of the 'kissing triangles'.
You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in
the middle. Cut the carpet into two pieces to make a 10 by 10 foot
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).
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. . . .
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
A follow-up activity to Tiles in the Garden.
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
What do these two triangles have in common? How are they related?
Measure problems for primary learners to work on with others.
Measure problems at primary level that require careful consideration.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Measure problems at primary level that may require determination.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
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 can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?
Use the information on these cards to draw the shape that is being described.
A simple visual exploration into halving and doubling.
My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?
These practical challenges are all about making a 'tray' and covering it with paper.
Are these statements always true, sometimes true or never true?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
How many tiles do we need to tile these patios?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?