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?
Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Four quadrants are drawn centred at the vertices of a square . Find the area of the central region bounded by the four arcs.
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.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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 maximise the area available to a grazing goat?
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
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. . . .
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?
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?
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
What fractions of the largest circle are the two shaded regions?
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?
If I print this page which shape will require the more yellow ink?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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. . . .
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
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.
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Explore one of these five pictures.
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
A follow-up activity to Tiles in the Garden.
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. . . .
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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 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.
Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.
What is the same and what is different about these circle questions? What connections can you make?
Can you find the area of a parallelogram defined by two vectors?
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?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
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.
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!
If the base of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) is the width decreased by ?
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 are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
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.
A trapezium is divided into four triangles by its diagonals. Suppose the two triangles containing the parallel sides have areas a and b, what is the area of the trapezium?
A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two. . . .