Perimeter, Area and Volume - Stage 4

These resources are designed to encourage you to explore perimeter, area and volume of shapes and solids with curved edges and surfaces. For resources about perimeter, area and volume of shapes with straight edges, see our collection Perimeter, Area and Volume - Stage 3.

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: 4 Challenge Level: Challenge Level:1

Perimeter and Area

This selection of problems challenges you to use your understanding of perimeter and area of shapes made from circles.

Stage: 4 Challenge Level: Challenge Level:1

Surface Area and Volume

This selection of problems challenges you to use your understanding of surface area and volume of three dimensional shapes based on circles.



Of All the Areas

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Blue and White

KS 3 Challenge Level:

Challenge Level:1

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

Arclets

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

Triangles and Petals

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

An Unusual Shape

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

Can you maximise the area available to a grazing goat?

Salinon

KS 4 Challenge Level:

Challenge Level:1

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?

Efficient Cutting

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

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.

Gutter

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Cola Can

KS 4 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?

Funnel

KS 4 Challenge Level:

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

A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?

Immersion

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

Various solids are lowered into a beaker of water. How does the water level rise in each case?

Curvy Areas

KS 4 Challenge Level:

Challenge Level:1

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

Track Design

KS 4 Challenge Level:

Challenge Level:1

Where should runners start the 200m race so that they have all run the same distance by the finish?

Fill Me up Too

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

In Fill Me Up we invited you to sketch graphs as vessels are filled with water. Can you work out the equations of the graphs?

Area Stage 4 - Short Problems

KS 4 Challenge Level:

Challenge Level:1

A collection of short Stage 4 problems on area and volume.