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?

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.

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

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

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. . . .

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

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

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

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

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

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

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 red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the 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. . . .

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

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

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?

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

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

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

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. . . .

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?

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

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.

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!

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

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?

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?

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.

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.

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?

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?

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

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

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?

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

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

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

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make. . . .

A follow-up activity to Tiles in the Garden.

You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in the middle. Cut the carpet into two pieces to make a 10 by 10 foot square carpet.

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

What fractions of the largest circle are the two shaded regions?