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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Favourite

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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Favourite

Triangles and petals

Age

14 to 16

Challenge level

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

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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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Favourite

Salinon

Age

14 to 16

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?

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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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Favourite

Gutter

Age

14 to 16

Challenge level

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

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Favourite

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 ?

problem

Favourite

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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Favourite

Curvy areas

Age

14 to 16

Challenge level

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

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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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Efficient cutting poster

Age

14 to 16

Challenge level

Efficient Cutting Poster

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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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On the edge poster

Age

11 to 14

Challenge level

On the Edge Poster

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Favourite

Fill me up too

Age

14 to 16

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?

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Curvy areas poster

Age

14 to 16

Challenge level

Curvy Areas Poster

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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?