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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
Can you work out the area of the inner square and give an
explanation of how you did it?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
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. . . .
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?
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. . . .
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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. . . .
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
A follow-up activity to Tiles in the Garden.
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
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
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?
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.
What happens to the area of a square if you double the length of
the sides? Try the same thing with rectangles, diamonds and other
shapes. How do the four smaller ones fit into the larger one?
Explore one of these five pictures.
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
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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. . . .
What do these two triangles have in common? How are they related?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
Determine the total shaded area of the 'kissing triangles'.
Derive a formula for finding the area of any kite.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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?
What happens to the area and volume of 2D and 3D shapes when you
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. . . .
A task which depends on members of the group noticing the needs of
others and responding.
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
These practical challenges are all about making a 'tray' and covering it with paper.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
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
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
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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 fractions of the largest circle are the two shaded regions?
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. . . .
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).
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?
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
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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.