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

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

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?

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.

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?

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

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

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?

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

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

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. . . .

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

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. . . .

Prove that the area of a quadrilateral is given by half the product of the lengths of the diagonals multiplied by the sine of the angle between the diagonals.

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.

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

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

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

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?

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 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.

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.

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

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

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

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

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

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

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. . . .

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

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

A follow-up activity to Tiles in the Garden.

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

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).

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 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.

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?

Can you find the area of a parallelogram defined by two vectors?

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.

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

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

A task which depends on members of the group noticing the needs of others and responding.

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.

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Determine the total shaded area of the 'kissing triangles'.

What is the same and what is different about these circle questions? What connections can you make?