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.



Arclets

KS 4 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?

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?

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?

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?

Semicircle Distance

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Can you find the shortest distance between the semicircles given the area between them?

Square in a Circle in a Square

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

What is the ratio of the areas of the squares in the diagram?

Sinking Feeling

KS 4 Short Challenge Level:

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

Two vases are cylindrical in shape. Can you work out the original depth of the water in the larger vase?

Half an Annulus

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Can you locate the point on an annulus that splits it into two areas?

Two Paths

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Which of these two paths made of semicircles is shorter?

Bobbly Perimeter

KS 4 Short Challenge Level:

Challenge Level:1

Find the perimeter of this shape made of semicircles

Cones and Spheres

KS 4 Short Challenge Level:

Challenge Level:1

A solid metal cone is melted down and turned into spheres. How many spheres can be made?

Ratio Cut

KS 4 Short Challenge Level:

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

Cutting a rectangle from a corner to a point on the opposite side splits its area in the ratio 1:2. What is the ratio of a:b?

Snake Eyes

KS 4 Short Challenge Level:

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

Can you find the area of the yellow part of this snake's eye?

Loo Roll Emergency

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

When the roll of toilet paper is half as wide, what percentage of the paper is left?