# Perimeter, Area and Volume - Stage 4

### 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?