It's easy to work out the areas of most squares that we meet, but what if they were tilted?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
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.
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. . . .
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. . . .
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
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.
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 have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Determine the total shaded area of the 'kissing triangles'.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
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?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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.
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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. . . .
Can you find the area of a parallelogram defined by two vectors?
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?
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.
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!
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.
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. . . .
Can you maximise the area available to a grazing goat?
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?
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.
Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the. . . .
Explore one of these five pictures.
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 follow-up activity to Tiles in the Garden.
Derive a formula for finding the area of any kite.
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. . . .
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.
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?
Can you work out the area of the inner square and give an explanation of how you did it?
How efficiently can you pack together disks?
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).
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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 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.