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?

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

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.

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

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

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

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. . . .

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

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?

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. . . .

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

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. . . .

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. . . .

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

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

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. . . .

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.

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

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.

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. . . .

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?

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

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?

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?

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.

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

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?

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?

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

This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.

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

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

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

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?

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.

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

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.

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

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.

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

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. . . .

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).

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and. . . .