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

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

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

Can you find rectangles where the value of the area is the same as the value of the perimeter?

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

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

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

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

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

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

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?

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

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?

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

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?

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.

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.

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

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

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

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 area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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?

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?

A follow-up activity to Tiles in the Garden.

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?

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

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.

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.

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

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

If I print this page which shape will require the more yellow ink?

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

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

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?

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

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. . . .

Four quadrants are drawn centred at the vertices of a square . Find the area of the central region bounded by the four arcs.

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. . . .

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. . . .

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. . . .

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.