List

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.
Perimeter and Area
list

Perimeter and Area

Age
14 to 16
Challenge level
filled star empty star empty star
This selection of problems challenges you to use your understanding of perimeter and area of shapes made from circles.
Surface Area and Volume
list

Surface Area and Volume

Age
14 to 16
Challenge level
filled star empty star empty star
This selection of problems challenges you to use your understanding of surface area and volume of three dimensional shapes based on circles.


F'arc'tion
problem

F'arc'tion

Age
14 to 16
Challenge level
filled star filled star empty star
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.
Arclets
problem
Favourite

Arclets

Age
14 to 16
Challenge level
filled star filled star empty star
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
problem
Favourite

Triangles and petals

Age
14 to 16
Challenge level
filled star filled star empty star
An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?
Circle in a Semicircle
problem

Circle in a Semicircle

Age
14 to 16
Challenge level
filled star filled star empty star
Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?
Pencil Turning
problem

Pencil Turning

Age
14 to 16
Challenge level
filled star filled star empty star
Rotating a pencil twice about two different points gives surprising results...
Annulus Area
problem

Annulus Area

Age
14 to 16
Challenge level
filled star empty star empty star
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?
Maximised Area
problem

Maximised Area

Age
14 to 16
Challenge level
filled star filled star empty star
Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radius.
Canny Fraction
problem

Canny Fraction

Age
14 to 16
Challenge level
filled star filled star empty star
What fraction of the volume of this can is filled with lemonade?
Salinon
problem
Favourite

Salinon

Age
14 to 16
Challenge level
filled star empty star empty star
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?
Sticky Tape
problem

Sticky Tape

Age
14 to 16
Challenge level
filled star filled star empty star
Work out the radius of a roll of adhesive tape.
Roll On
problem

Roll On

Age
14 to 16
Challenge level
filled star filled star filled star
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?
Running Race
problem

Running Race

Age
14 to 16
Challenge level
filled star filled star filled star
Weekly Problem 13 - 2006
If three runners run at the same constant speed around the race tracks, in which order do they finish?
Semicircle Stack
problem

Semicircle Stack

Age
14 to 16
Challenge level
filled star empty star empty star
What is the total area enclosed by the three semicicles?
Eyelids
problem

Eyelids

Age
14 to 16
Challenge level
filled star filled star filled star
Can you work out the shaded area surrounded by these arcs?
Square Ratio
problem

Square Ratio

Age
14 to 16
Challenge level
filled star filled star empty star
A square is divided into four rectangles and a square. Can you work out the ratio of the side lengths of the rectangles?
Rolling Inside
problem

Rolling Inside

Age
14 to 16
Challenge level
filled star filled star empty star
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.
Gutter
problem
Favourite

Gutter

Age
14 to 16
Challenge level
filled star filled star empty star
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Funnel
problem
Favourite

Funnel

Age
14 to 16
Challenge level
filled star filled star filled star
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
problem
Favourite

Immersion

Age
14 to 16
Challenge level
filled star filled star empty star
Various solids are lowered into a beaker of water. How does the water level rise in each case?
Curvy areas
problem
Favourite

Curvy areas

Age
14 to 16
Challenge level
filled star empty star empty star
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Rectangle Cutting
problem

Rectangle Cutting

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Crazy Shading
problem

Crazy Shading

Age
14 to 16
Challenge level
filled star empty star empty star
Can you work out the fraction of the larger square that is covered by the shaded area?
Circled Corners
problem

Circled Corners

Age
14 to 16
Challenge level
filled star filled star empty star
Three circles have been drawn at the vertices of this triangle. What is the area of the inner shaded area?
Wood Pile Perimeter
problem

Wood Pile Perimeter

Age
14 to 16
Challenge level
filled star filled star empty star
Weekly Problem 30 - 2011
Three touching circles have an interesting area between them...
Track design
problem
Favourite

Track design

Age
14 to 16
Challenge level
filled star empty star empty star
Where should runners start the 200m race so that they have all run the same distance by the finish?
Fill Me Up Too
problem
Favourite

Fill Me Up Too

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Arc Area
problem

Arc Area

Age
14 to 16
Challenge level
filled star empty star empty star
What is the area of the shape enclosed by the line and arcs?
Tadpoles
problem

Tadpoles

Age
14 to 16
Challenge level
filled star filled star empty star
The diagram shows a shaded shape bounded by circular arcs. What is the difference in area betweeen this and the equilateral triangle shown?
Penny Farthing
problem

Penny Farthing

Age
14 to 16
Challenge level
filled star empty star empty star
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?
In or Out?
problem

In or Out?

Age
14 to 16
Challenge level
filled star filled star filled star
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 Leaf Clover
problem

Four Leaf Clover

Age
14 to 16
Challenge level
filled star filled star empty star
The diagram shows four equal discs and a square. What is the perimeter of the figure?
Cut-Up Square
problem

Cut-Up Square

Age
14 to 16
Challenge level
filled star filled star filled star
Weekly Problem 15 - 2015
In the diagram, two lines have been drawn in a square. What is the ratio of the areas marked?
Emptied Cube
problem

Emptied Cube

Age
14 to 16
Challenge level
filled star filled star empty star
Weekly Problem 26 - 2015
What are the volume and surface area of this 'Cubo Vazado' or 'Emptied Cube'?
Square Flower
problem

Square Flower

Age
14 to 16
Challenge level
filled star filled star empty star
The diagram shows 8 circles surrounding a region. What is the perimeter of the shaded region?
Trisected Triangle
problem

Trisected Triangle

Age
14 to 16
Challenge level
filled star filled star empty star
Weekly Problem 34 - 2015
Four tiles are given. For which of them can three be placed together to form an equilateral triangle?
Clown Hats
problem

Clown Hats

Age
14 to 16
Challenge level
filled star filled star filled star
Weekly Problem 51 - 2015
Charlie is making clown hats from a piece of cardboard. What is the maximum number he can make?
Smile
problem

Smile

Age
14 to 16
Challenge level
filled star empty star empty star
Four semicircles are drawn on a line to form a shape. What is the area of this shape?
Semicircular Design
problem

Semicircular Design

Age
14 to 16
Challenge level
filled star empty star empty star
Weekly Problem 9 - 2016
The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?
Four Parts
problem

Four Parts

Age
14 to 16
Challenge level
filled star empty star empty star
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?
Semicircle distance
problem

Semicircle distance

Age
14 to 16
Challenge level
filled star filled star empty star
Can you find the shortest distance between the semicircles given the area between them?
Sinking Feeling
problem

Sinking Feeling

Age
14 to 16
Challenge level
filled star filled star filled star
Two vases are cylindrical in shape. Can you work out the original depth of the water in the larger vase?
Placeholder: several colourful numbers
problem

Half an Annulus

Age
14 to 16
Challenge level
filled star filled star empty star
Can you locate the point on an annulus that splits it into two areas?
Placeholder: several colourful numbers
problem

Two Paths

Age
14 to 16
Challenge level
filled star filled star empty star
Which of these two paths made of semicircles is shorter?
Bobbly perimeter
problem

Bobbly perimeter

Age
14 to 16
Challenge level
filled star empty star empty star
Find the perimeter of this shape made of semicircles
Cones and spheres
problem

Cones and spheres

Age
14 to 16
Challenge level
filled star filled star empty star
A solid metal cone is melted down and turned into spheres. How many spheres can be made?
ratio cut
problem

ratio cut

Age
14 to 16
Challenge level
filled star filled star filled star
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
problem

Snake eyes

Age
14 to 16
Challenge level
filled star filled star filled star
Can you find the area of the yellow part of this snake's eye?
Loo roll emergency
problem

Loo roll emergency

Age
14 to 16
Challenge level
filled star filled star empty star
When the roll of toilet paper is half as wide, what percentage of the paper is left?
Placeholder: several colourful numbers
problem

Candy floss

Age
14 to 16
Challenge level
filled star filled star empty star
What length of candy floss can Rita spin from her cylinder of sugar?
Placeholder: several colourful numbers
problem

Similar Cylinders

Age
14 to 16
Challenge level
filled star filled star empty star
Two similar cylinders are formed from a block of metal. What is the volume of the smaller cylinder?
Placeholder: several colourful numbers
problem

Tilted Aquarium

Age
14 to 16
Challenge level
filled star filled star filled star
Can you find the depth of water in this aquarium?
Placeholder: several colourful numbers
problem

Ratio of areas

Age
14 to 16
Challenge level
filled star empty star empty star
What is the ratio of the area of the hexagon to the area of the triangle?