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.

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?

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. . . .

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

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.

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

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?

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. . . .

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?

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

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

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?

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

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.

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?

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

A trapezium is divided into four triangles by its diagonals. Suppose the two triangles containing the parallel sides have areas a and b, what is the area of the trapezium?

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.

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

If the base of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) is the width decreased by ?

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .

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

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

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

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

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?

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?

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

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

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?

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two. . . .

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

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.

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

Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the. . . .

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

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).

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.

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

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

You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in the middle. Cut the carpet into two pieces to make a 10 by 10 foot square carpet.

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 rectangles where the value of the area is the same as the value of the perimeter?

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

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.

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?

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