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.

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. . . .

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

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

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.

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

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?

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

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 the area of a parallelogram defined by two vectors?

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?

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. . . .

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

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?

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.

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.

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. . . .

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?

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?

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

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.

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

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 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.

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.

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.

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

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?

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.

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

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?

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

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

A follow-up activity to Tiles in the Garden.

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

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

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

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

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

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?

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.