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?

These pieces of wallpaper need to be ordered from smallest to largest. Can you find a way to do it?

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

Can you put these shapes in order of size? Start with the smallest.

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

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

Draw three equal line segments in a unit circle to divide the circle into four parts of equal 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?

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation. . . .

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.

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

How would you move the bands on the pegboard to alter these shapes?

Can you draw a square in which the perimeter is numerically equal to the area?

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. . . .

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

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.

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Use the information on these cards to draw the shape that is being described.

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

These practical challenges are all about making a 'tray' and covering it with paper.

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

A simple visual exploration into halving and doubling.

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.

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.