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 squares?
Make an eight by eight square, the layout is the same as a chessboard. You can print out and use the square below. What is the area of the square? Divide the square in the way shown by the red dashed. . . .
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.
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
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. . . .
Explore one of these five pictures.
Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .
A follow-up activity to Tiles in the Garden.
Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .
This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.
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?
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?
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 maximise the area available to a grazing goat?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
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.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.
What happens to the area and volume of 2D and 3D shapes when you enlarge them?
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.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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 into?
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 square carpet.
Can you work out the area of the inner square and give an explanation of how you did it?
Determine the total shaded area of the 'kissing triangles'.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
What fractions of the largest circle are the two shaded regions?
Derive a formula for finding the area of any kite.
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).
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. . . .
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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. . . .
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?
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
Look at the mathematics that is all around us - this circular window is a wonderful example.
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?
Read about David Hilbert who proved that any polygon could be cut up into a certain number of pieces that could be put back together to form any other polygon of equal area.
You have pitched your tent (the red triangle) on an island. Can you move it to the position shown by the purple triangle making sure you obey the rules?
This article for teachers gives some food for thought when teaching ideas about area.
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
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.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
In how many ways can you halve a piece of A4 paper? How do you know they are halves?
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?
Are these statements always true, sometimes true or never true?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?