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?

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.

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

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.

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

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. . . .

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

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.

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.

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

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.

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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

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

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 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.

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!

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.

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 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?

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?

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.

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. . . .

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

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

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

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

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?

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?

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

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?

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

A follow-up activity to Tiles in the Garden.

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

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

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

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. . . .

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. . . .

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

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?

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

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