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.
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?
Can you find the areas of the trapezia in this sequence?
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
Can you find the area of a parallelogram defined by two vectors?
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.
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
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. . . .
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
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.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
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?
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 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.
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?
Can you maximise the area available to a grazing goat?
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
Determine the total shaded area of the 'kissing triangles'.
Can you work out the area of the inner square and give an explanation of how you did it?
What is the same and what is different about these circle questions? What connections can you make?
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
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?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
What happens to the area and volume of 2D and 3D shapes when you enlarge them?
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?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?
A task which depends on members of the group noticing the needs of others and responding.
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.
How efficiently can you pack together disks?
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!
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.
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?
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
If the base of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) is the width decreased by ?
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?
Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?
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.
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).
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. . . .
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. . . .