Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
What fractions of the largest circle are the two shaded regions?
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. . . .
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?
Measure problems at primary level that require careful consideration.
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 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?
Derive a formula for finding the area of any kite.
Measure problems for primary learners to work on with others.
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?
My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.
Measure problems for inquiring primary learners.
Measure problems at primary level that may require determination.
Explore this interactivity and see if you can work out what it does. Could you use it to estimate the area of a shape?
What do these two triangles have in common? How are they related?
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?
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?
Can you maximise the area available to a grazing goat?
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 this area?
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.
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.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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.
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?
I cut this square into two different shapes. What can you say about the relationship between them?
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.
A task which depends on members of the group noticing the needs of others and responding.
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?
An investigation that gives you the opportunity to make and justify predictions.
What happens to the area and volume of 2D and 3D shapes when you enlarge them?
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?
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.
Look at the mathematics that is all around us - this circular window is a wonderful example.