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.

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

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

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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### F'arc'tion

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

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

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

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

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

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### Annulus Area

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?

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

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

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Favourite

### Salinon

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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### Roll On

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?

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

Weekly Problem 13 - 2006

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

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

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### Square Ratio

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

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.

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

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

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 ?

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Favourite

### Immersion

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

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

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

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

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### Circled Corners

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

Weekly Problem 30 - 2011

Three touching circles have an interesting area between them...

Three touching circles have an interesting area between them...

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Favourite

### Track design

Where should runners start the 200m race so that they have all run the same distance by the finish?

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Favourite

### Fill Me Up Too

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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### Tadpoles

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

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?

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?

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

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

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### Cut-Up Square

Weekly Problem 15 - 2015

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

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

Weekly Problem 26 - 2015

What are the volume and surface area of this 'Cubo Vazado' or 'Emptied Cube'?

What are the volume and surface area of this 'Cubo Vazado' or 'Emptied Cube'?

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### Square Flower

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

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### Trisected Triangle

Weekly Problem 34 - 2015

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

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

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### Clown Hats

Weekly Problem 51 - 2015

Charlie is making clown hats from a piece of cardboard. What is the maximum number he can make?

Charlie is making clown hats from a piece of cardboard. What is the maximum number he can make?

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### Semicircular Design

Weekly Problem 9 - 2016

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

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

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

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

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

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### Sinking Feeling

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

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### Cones and spheres

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

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### ratio cut

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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### Loo roll emergency

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

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### Similar Cylinders

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