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 find the areas of the trapezia in this sequence?

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

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.

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).

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. . . .

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.

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

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?

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

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

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

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

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 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!

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?

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?

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?

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

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?

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

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

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. . . .

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

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

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?

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?

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.

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.

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.

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

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?

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

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

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

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

A follow-up activity to Tiles in the Garden.

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

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

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

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?