# 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.

### Perimeter and Area

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

### 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:

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:

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

### Arclets

##### KS 4 Challenge Level:

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:

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:

Can you maximise the area available to a grazing goat?

### Salinon

##### KS 4 Challenge Level:

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:

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:

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:

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:

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:

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

### Curvy Areas

##### KS 4 Challenge Level:

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:

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:

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?

### Perimeter, Area and Volume Stage 4 - Short Problems

##### KS 4

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

### Semicircle Distance

##### KS 4 Short Challenge Level:

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:

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

### Sinking Feeling

##### KS 4 Short Challenge Level:

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

### Bobbly Perimeter

##### KS 4 Short Challenge Level:

Find the perimeter of this shape made of semicircles

### Cones and Spheres

##### KS 4 Short Challenge Level:

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

### Ratio Cut

##### KS 4 Short Challenge Level:

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:

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

### Loo Roll Emergency

##### KS 4 Short Challenge Level:

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