# Perimeter, Area and Volume - Stage 3

These resources are designed to encourage you to explore perimeter, area and volume of shapes and solids based on rectangles and triangles. For resources about area, perimeter and volume that include shapes and solids with curved edges and surfaces, see our collection Perimeter, Area and Volume - Stage 4.

Scroll down to see the complete collection, or explore our subcollections on Perimeter and Area in two dimensions, and Surface Area and Volume in three dimensions.

### Perimeter and Area

This selection of problems is a great starting point for learning about Perimeter and Area.

### Surface Area and Volume

This selection of problems is a great starting point for learning about Surface Area and Volume

### Pick's Theorem

##### KS 3 Challenge Level:

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.

### Painted Cube

##### KS 3 Challenge Level:

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?

### Cuboids

##### KS 3 Challenge Level:

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?

### On the Edge

##### KS 3 Challenge Level:

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.

### Sending a Parcel

##### KS 3 Challenge Level:

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

### Fence It

##### KS 3 Challenge Level:

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

### Isosceles Triangles

##### KS 3 Challenge Level:

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?

### Warmsnug Double Glazing

##### KS 3 Challenge Level:

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

### Can They Be Equal?

##### KS 3 Challenge Level:

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

### Cuboid Challenge

##### KS 3 Challenge Level:

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

### Changing Areas, Changing Perimeters

##### KS 3 Challenge Level:

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?

### Changing Areas, Changing Volumes

##### KS 4 Challenge Level:

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?

### Area Stage 3 - Short Problems

##### KS 3 Challenge Level:

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

### Perimeter Possibilities

##### KS 3 Challenge Level:

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