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.

A tower of squares is built inside a right angled isosceles triangle. The largest square stands on the hypotenuse. What fraction of the area of the triangle is covered by the series of squares?

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?

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

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

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?

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

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

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. . . .

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .

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. . . .

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?

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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

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

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.

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?

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?

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.

You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.

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.

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.

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

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

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

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?

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.

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.

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

Can you work out the area of the inner square and give an explanation of how you did it?

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?

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?

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. . . .

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

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. . . .

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

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

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).

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.

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. . . .

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

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?

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

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

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

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

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