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?

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

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

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

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.

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

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?

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 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. . . .

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

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

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. . . .

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?

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?

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. . . .

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

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?

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

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

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.

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. . . .

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.

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?

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.

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.

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?

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?

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

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

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?

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

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

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.

A follow-up activity to Tiles in the Garden.

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

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

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

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

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

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?

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