Four quadrants are drawn centred at the vertices of a square . Find the area of the central region bounded by the four arcs.

Can you prove this formula for finding the area of a quadrilateral from its diagonals?

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. . . .

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?

If I print this page which shape will require the more yellow ink?

If the base of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) is the width decreased by ?

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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. . . .

Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?

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.

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?

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?

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.

Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area

What fractions of the largest circle are the two shaded regions?

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?

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

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?

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

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. . . .

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?

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 draw the height-time chart as this complicated vessel fills with water?

Analyse these beautiful biological images and attempt to rank them in size order.

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?

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

What happens to the area and volume of 2D and 3D shapes when you enlarge them?

An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.

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?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

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).

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. . . .

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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?

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. . . .

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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.

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

Can you find the areas of the trapezia in this sequence?