This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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?
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 task which depends on members of the group noticing the needs of others and responding.
Determine the total shaded area of the 'kissing triangles'.
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?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
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.
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).
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?
Derive a formula for finding the area of any kite.
What happens to the area and volume of 2D and 3D shapes when you enlarge them?
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?
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. . . .
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 are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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!
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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?
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. . . .
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
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.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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.
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?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
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?
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. . . .
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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.
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. . . .
Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
Triangle ABC is right angled at A and semi circles are drawn on all three sides producing two 'crescents'. Show that the sum of the areas of the two crescents equals the area of triangle ABC.
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?
What is the same and what is different about these circle questions? What connections can you make?
Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you find the area of a parallelogram defined by two vectors?