What fractions of the largest circle are the two shaded regions?
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
Explore one of these five pictures.
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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. . . .
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
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?
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 follow-up activity to Tiles in the Garden.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
What shape has Harry drawn on this clock face? Can you find its
area? What is the largest number of square tiles that could cover
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!
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.
Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?
Measure problems at primary level that require careful consideration.
Can you maximise the area available to a grazing goat?
Measure problems at primary level that may require determination.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
This article for teachers gives some food for thought when teaching
ideas about area.
Have a good look at these images. Can you describe what is happening? There are plenty more images like this on NRICH's Exploring Squares CD.
Measure problems for primary learners to work on with others.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
What do these two triangles have in common? How are they related?
An investigation that gives you the opportunity to make and justify
Use the information on these cards to draw the shape that is being described.
A simple visual exploration into halving and doubling.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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?
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?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
How many tiles do we need to tile these patios?
These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.
I cut this square into two different shapes. What can you say about
the relationship between them?
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.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
A task which depends on members of the group noticing the needs of
others and responding.
What happens to the area and volume of 2D and 3D shapes when you
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?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
What is the largest number of circles we can fit into the frame
without them overlapping? How do you know? What will happen if you
try the other shapes?
Measure problems for inquiring primary learners.
How would you move the bands on the pegboard to alter these shapes?