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?

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?

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?

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

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.

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 fractions of the largest circle are the two shaded regions?

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. . . .

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

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. . . .

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

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?

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.

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.

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 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.

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.

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?

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. . . .

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

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

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.

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.

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

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

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?

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?

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

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

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

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

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

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. . . .

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!

A follow-up activity to Tiles in the Garden.

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

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

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

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

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?