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

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.

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

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. . . .

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

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

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. . . .

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?

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

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

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.

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

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

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?

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 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?

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

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

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?

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

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.

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?

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. . . .

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

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.

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?

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

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.

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?

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

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?

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.

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.

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?

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

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?

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.

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. . . .

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

Can you find the areas of the trapezia in this sequence?

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. . . .

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