Can you work out the area of the inner square and give an explanation of how you did it?
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. . . .
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?
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. . . .
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?
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.
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
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?
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. . . .
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. . . .
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. . . .
Look at the mathematics that is all around us - this circular window is a wonderful example.
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
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?
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.
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.
How many centimetres of rope will I need to make another mat just like the one I have here?
Determine the total shaded area of the 'kissing triangles'.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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. . . .
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?
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!
What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?
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?
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.
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
An investigation that gives you the opportunity to make and justify predictions.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
A simple visual exploration into halving and doubling.
Use the information on these cards to draw the shape that is being described.
Are these statements always true, sometimes true or never true?
A follow-up activity to Tiles in the Garden.
Explore one of these five pictures.
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 enlarge them?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?
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.
This article for teachers gives some food for thought when teaching ideas about area.
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.
Derive a formula for finding the area of any kite.
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. . . .