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.
Stage: 3 Challenge Level: Challenge Level:1

Perimeter and Area

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

Stage: 3 Challenge Level: Challenge Level:1

Surface Area and Volume

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



Blue and White

KS 3 Challenge Level:

Challenge Level:1

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

Pick's Theorem

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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.

An Unusual Shape

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

Can you maximise the area available to a grazing goat?

Painted Cube

KS 3 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

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:

Challenge Level:3 Challenge Level:3 Challenge Level:3

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:

Challenge Level:2 Challenge Level:2

If you move the tiles around, can you make squares with different coloured edges?

Sending a Parcel

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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:

Challenge Level:1

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

Efficient Cutting

KS 3 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

Isosceles Triangles

KS 3 Challenge Level:

Challenge Level:1

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:

Challenge Level:3 Challenge Level:3 Challenge Level:3

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

Cola Can

KS 3 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Can They Be Equal?

KS 3 Challenge Level:

Challenge Level:1

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:

Challenge Level:2 Challenge Level:2

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:

Challenge Level:1

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 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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?

Perimeter Possibilities

KS 3 Challenge Level:

Challenge Level:1

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

Triangles in a Square

KS 3 Challenge Level:

Challenge Level:1

What are the possible areas of triangles drawn in a square?

Perimeter Challenge

KS 3 Challenge Level:

Challenge Level:1

Can you deduce the perimeters of the shapes from the information given?

Colourful Cube

KS 3 Challenge Level:

Challenge Level:1

A colourful cube is made from little red and yellow cubes. But can you work out how many of each?

Isometric Areaslive

KS 3 Challenge Level:

Challenge Level:1

We usually use squares to measure area, but what if we use triangles instead?

More Isometric Areaslive

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

Isometric Areas explored areas of parallelograms in triangular units. Here we explore areas of triangles...

Cuboid Faces

KS 3 Short Challenge Level:

Challenge Level:1

Can you find the volume of a cuboid, given the areas of its faces?

Hawaiian Earring

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

What fraction of the larger circle is outside the smaller circle?

Scalene Area

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Can you find the area of this scalene triangle?

Tilted Tank

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Can you find the height of the water in this tilted tank when it is flat?

Giant Rubik's Cube

KS 3 Short Challenge Level:

Challenge Level:1

How many cubes would be visible in a 12 by 12 by 12 Rubik’s cube?

Dividing a Square

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

A square is divided into three shapes which all have equal areas. Can you find the length of this side?

Rectangle Split

KS 3 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Draw another line through the centre of this rectangle to split it into 4 pieces of equal area.

Semicircle Shape

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Find the shaded area of these shapes with perimeters made of semicircles.