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.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to colour sixteen tiles so that four 4 by 4 squares can be made - one with a green edge, one with a blue edge, one with a yellow edge and one with a red edge.

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

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

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?

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

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

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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?

How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?

A collection of short Stage 3 problems on area and volume.

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