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.

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Perimeter and area

Age

14 to 16

Challenge level

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

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Surface area and volume

Age

14 to 16

Challenge level

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

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Perimeter, area and volume short problems

Age

11 to 16

A collection of short problems on area and volume.

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F'arc'tion

Age

14 to 16

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.

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Arclets

Age

14 to 16

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".

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Circle in a semicircle

Age

14 to 16

Challenge level

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

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Pencil turning

Age

14 to 16

Challenge level

Rotating a pencil twice about two different points gives surprising results...

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Annulus area

Age

14 to 16

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?

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Maximised area

Age

14 to 16

Challenge level

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

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Canny fraction

Age

14 to 16

Challenge level

What fraction of the volume of this can is filled with lemonade?

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Sticky tape

Age

14 to 16

Challenge level

Work out the radius of a roll of adhesive tape.

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Roll on

Age

14 to 16

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?

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Running race

Age

14 to 16

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?

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Semicircle stack

Age

14 to 16

Challenge level

What is the total area enclosed by the three semicicles?

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Eyelids

Age

14 to 16

Challenge level

Can you work out the shaded area surrounded by these arcs?

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Square ratio

Age

14 to 16

Challenge level

A square is divided into four rectangles and a square. Can you work out the ratio of the side lengths of the rectangles?

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Rolling inside

Age

14 to 16

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.

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Rectangle cutting

Age

14 to 16

Challenge level

Tom and Jerry start with identical sheets of paper. Each one cuts his sheet in a different way. Can you find the perimeter of the original sheet?

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Crazy shading

Age

14 to 16

Challenge level

Can you work out the fraction of the larger square that is covered by the shaded area?

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Circled corners

Age

14 to 16

Challenge level

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

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Wood pile perimeter

Age

14 to 16

Challenge level

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

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Arc area

Age

14 to 16

Challenge level

What is the area of the shape enclosed by the line and arcs?

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Tadpoles

Age

14 to 16

Challenge level

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

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Penny farthing

Age

14 to 16

Challenge level

Boris' bicycle has a smaller back wheel than front wheel. Can you work out how many revolutions the front wheel made if the back wheel did 120,000?

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In or out?

Age

14 to 16

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?

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Four leaf clover

Age

14 to 16

Challenge level

The diagram shows four equal discs and a square. What is the perimeter of the figure?

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Cut-up square

Age

14 to 16

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?

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Emptied cube

Age

14 to 16

Challenge level

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

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Square flower

Age

14 to 16

Challenge level

The diagram shows 8 circles surrounding a region. What is the perimeter of the shaded region?

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Trisected triangle

Age

14 to 16

Challenge level

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

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Clown hats

Age

14 to 16

Challenge level

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

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Smile

Age

14 to 16

Challenge level

Four semicircles are drawn on a line to form a shape. What is the area of this shape?

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Semicircular design

Age

14 to 16

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?

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Four parts

Age

14 to 16

Challenge level

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?

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Semicircle distance

Age

14 to 16

Challenge level

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

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Square in a circle in a square

Age

14 to 16

Challenge level

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

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Sinking feeling

Age

14 to 16

Challenge level

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

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Half an annulus

Age

14 to 16

Challenge level

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

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Two paths

Age

14 to 16

Challenge level

Which of these two paths made of semicircles is shorter?

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Bobbly perimeter

Age

14 to 16

Challenge level

Find the perimeter of this shape made of semicircles

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Cones and spheres

Age

14 to 16

Challenge level

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

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Ratio cut

Age

14 to 16

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?

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Snake eyes

Age

14 to 16

Challenge level

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

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Loo roll emergency

Age

14 to 16

Challenge level

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

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Candy floss

Age

14 to 16

Challenge level

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

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Similar cylinders

Age

14 to 16

Challenge level

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

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Tilted aquarium

Age

14 to 16

Challenge level

Can you find the depth of water in this aquarium?

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Ratio of areas

Age

14 to 16

Challenge level

What is the ratio of the area of the hexagon to the area of the triangle?

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Immersion

Age

14 to 16

Challenge level

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

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Funnel

Age

14 to 16

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 ?