What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you maximise the area available to a grazing goat?
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.
Can you work out the area of the inner square and give an
explanation of how you did it?
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. . . .
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. . . .
The area of a square inscribed in a circle with a unit radius is,
satisfyingly, 2. What is the area of a regular hexagon inscribed in
a circle with a unit radius?
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
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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 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?
Can you draw a square in which the perimeter is numerically equal
to the area?
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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. . . .
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?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
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.
An investigation that gives you the opportunity to make and justify
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 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.
How many centimetres of rope will I need to make another mat just
like the one I have here?
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 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?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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. . . .
These practical challenges are all about making a 'tray' and covering it with paper.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
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?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
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?
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. . . .
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Determine the total shaded area of the 'kissing triangles'.
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.
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Explore one of these five pictures.
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?