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.

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

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

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

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?

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. . . .

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.

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?

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. . . .

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?

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. . . .

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?

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

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.

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

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?

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?

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.

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?

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

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. . . .

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?

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 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?

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?

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

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

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.

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

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

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.

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. . . .

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

Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?

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!

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

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

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

A follow-up activity to Tiles in the Garden.

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