# Perimeter, Area and Volume Stage 4 - Short Problems

This is part of our collection of Short Problems.

You may also be interested in our longer problems on Perimeter, Area and Volume - Stage 4.

Printable worksheets containing selections of these problems are available here:

 Stage 4 ★ Sheet 1 Solutions Stage 4 ★★ Sheet 1 Solutions Sheet 2 Solutions Sheet 3 Solutions Stage 4 ★★★ Sheet 1 Solutions

### Circle in a Semicircle

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 36 - 2011
Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?

### Bobbly Perimeter

##### Age 14 to 16 Short Challenge Level:

Find the perimeter of this shape made of semicircles

### Four Parts

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 13 - 2016
The circle of radius 4cm is divided into four congruent parts by arcs of radius 2cm as shown. What is the length of the perimeter of one of the parts, in cm?

### Pencil Turning

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 37 - 2011
Rotating a pencil twice about two different points gives surprising results...

### Cones and Spheres

##### Age 14 to 16 Short Challenge Level:

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

### Arc Area

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 49 - 2006
What is the area of the shape enclosed by the line and arcs?

### Semicircle Stack

##### Age 14 to 16 Short Challenge Level:

What is the total area enclosed by the three semicicles?

### Square Flower

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 31 - 2015
The diagram shows 8 circles surrounding a region. What is the perimeter of the shaded region?

### Annulus Area

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 38 - 2011
Given three concentric circles, shade in the annulus formed by the smaller two. What percentage of the larger circle is now shaded?

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 7 - 2014
The diagram shows a shaded shape bounded by circular arcs. What is the difference in area betweeen this and the equilateral triangle shown?

### Pentagonal Area

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 18 - 2006
What is the area of the pentagon?

### Maximised Area

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 39 - 2011
Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radius.

### Penny Farthing

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 23 - 2014
Boris' bicycle has a bigger back wheel than front wheel. Can you work out how many revolutions the back wheel made if the front wheel did 120,000?

### Canny Fraction

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 4 - 2012
What fraction of the volume of this can is filled with lemonade?

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 34 - 2010
Can you work out the fraction of the larger square that is covered by the shaded area?

### Smile

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 52 - 2015
Four semicircles are drawn on a line to form a shape. What is the area of this shape?

### Four Leaf Clover

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 5 - 2015
The diagram shows four equal discs and a square. What is the perimeter of the figure?

### Semicircular Design

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 9 - 2016
The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?

### Wood Pile Perimeter

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 30 - 2011
Three touching circles have an interesting area between them...

### Emptied Cube

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 26 - 2015
What are the volume and surface area of this 'Cubo Vazado' or 'Emptied Cube'?

### Semicircle Distance

##### Age 14 to 16 Short Challenge Level:

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

### Square in a Circle in a Square

##### Age 14 to 16 Short Challenge Level:

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

### Trisected Triangle

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 34 - 2015
Four tiles are given. For which of them can three be placed together to form an equilateral triangle?

### Rolling Inside

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 11 - 2007
A circle of radius 1 rolls without slipping round the inside of a square of side length 4. Find an expression for the number of revolutions the circle makes.

### Loo Roll Emergency

##### Age 14 to 16 Short Challenge Level:

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

### Half an Annulus

##### Age 14 to 16 Short Challenge Level:

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

### Candy Floss

##### Age 14 to 16 Short Challenge Level:

What length of candy floss can Rita spin from her cylinder of sugar?

### F'arc'tion

##### Age 14 to 16 Short Challenge Level:

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

### Two Paths

##### Age 14 to 16 Short Challenge Level:

Which of these two paths made of semicircles is shorter?

### Circled Corners

##### Age 14 to 16 Short Challenge Level:

Three circles have been drawn at the vertices of this triangle. What is the area of the inner shaded area?

### Similar Cylinders

##### Age 14 to 16 Short Challenge Level:

Two similar cylinders are formed from a block of metal. What is the volume of the smaller cylinder?

### Cut-up Square

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 15 - 2015
In the diagram, two lines have been drawn in a square. What is the ratio of the areas marked?

### Running Race

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 13 - 2006
If three runners run at the same constant speed around the race tracks, in which order do they finish?

### Ratio Cut

##### Age 14 to 16 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

##### Age 14 to 16 Short Challenge Level:

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

### Sinking Feeling

##### Age 14 to 16 Short Challenge Level:

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

### Clown Hats

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 51 - 2015
Charlie is making clown hats from a piece of cardboard. What is the maximum number he can make?

### In or Out?

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 52 - 2014
Four arcs are drawn in a circle to create a shaded area. What fraction of the area of the circle is shaded?

### Roll On

##### Age 14 to 16 Short Challenge Level:

Weekly Problem 5 - 2006
How many times does the inside disc have to roll around the inside of the ring to return to its initial position?