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

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

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.

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

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?

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

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

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

A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?

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

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 ?

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

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?

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.

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 is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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.

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?

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

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 rectangles where the value of the area is the same as the value of the perimeter?

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. . . .

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?

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

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.

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

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.

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

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.

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?

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.

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

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?

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

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.

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

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

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

A follow-up activity to Tiles in the Garden.

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.

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

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

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. . . .