![Perimeter, Area and Volume Short Problems](/sites/default/files/styles/medium/public/thumbnails/content-id-9374-icon.png?itok=MO1DSTf8)
![F'arc'tion](/sites/default/files/styles/medium/public/thumbnails/content-00-05-six4-icon.jpg?itok=Y4Pe0yOh)
problem
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.
![Arclets](/sites/default/files/styles/medium/public/thumbnails/arclets.gif?itok=OyLIqv2Q)
problem
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".
![Triangles and petals](/sites/default/files/styles/medium/public/thumbnails/content-02-11-six3-icon.gif?itok=K4YCjM-x)
problem
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?
![Circle in a Semicircle](/sites/default/files/styles/medium/public/thumbnails/content-04-weekly-prob6-icon.png?itok=KkZtmTpS)
problem
Circle in a Semicircle
Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?
![Pencil Turning](/sites/default/files/styles/medium/public/thumbnails/content-04-weekly-prob7-icon.png?itok=m-RQXMA4)
problem
Pencil Turning
Rotating a pencil twice about two different points gives surprising results...
![Annulus Area](/sites/default/files/styles/medium/public/thumbnails/content-04-weekly-prob8-icon.png?itok=NosiqEqy)
problem
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?
![Maximised Area](/sites/default/files/styles/medium/public/thumbnails/content-04-weekly-prob9-icon.png?itok=1pRpakZu)
problem
Maximised Area
Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radius.
![Canny Fraction](/sites/default/files/styles/medium/public/thumbnails/content-id-2406-icon.png?itok=m7ONYjl2)
![Salinon](/sites/default/files/styles/medium/public/thumbnails/content-id-2425-icon.png?itok=09ypcPQx)
problem
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?
![Sticky Tape](/sites/default/files/styles/medium/public/thumbnails/content-id-4736-icon.png?itok=AE6SGdOt)
![Roll On](/sites/default/files/styles/medium/public/thumbnails/content-id-4737-icon.png?itok=lKsEcSem)
problem
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?
![Running Race](/sites/default/files/styles/medium/public/thumbnails/content-id-4875-icon.png?itok=wJTPy3zT)
problem
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?
![Semicircle Stack](/sites/default/files/styles/medium/public/thumbnails/content-id-4881-icon.png?itok=MF7HrG_S)
![Eyelids](/sites/default/files/styles/medium/public/thumbnails/content-id-4987-icon.jpg?itok=kEGJo9pW)
![Square Ratio](/sites/default/files/styles/medium/public/thumbnails/content-id-5003-icon.png?itok=IX04FLGf)
problem
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?
![Rolling Inside](/sites/default/files/styles/medium/public/thumbnails/content-id-5019-icon.png?itok=DY9HAkW2)
problem
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.
![Gutter](/sites/default/files/styles/medium/public/thumbnails/content-id-5673-icon.jpg?itok=OF1ia8c1)
problem
Gutter
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
![Funnel](/sites/default/files/styles/medium/public/thumbnails/content-id-5890-icon.png?itok=RyYhWbEa)
problem
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 ?
![Immersion](/sites/default/files/styles/medium/public/thumbnails/content-id-6439-icon.gif?itok=1Ef_dxXY)
problem
Immersion
Various solids are lowered into a beaker of water. How does the
water level rise in each case?
![Curvy areas](/sites/default/files/styles/medium/public/thumbnails/content-id-6468-icon.png?itok=BhJjQwuY)
problem
Curvy areas
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
![Rectangle Cutting](/sites/default/files/styles/medium/public/thumbnails/content-id-6751-icon.png?itok=KFcc-IRe)
problem
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?
![Crazy Shading](/sites/default/files/styles/medium/public/thumbnails/content-id-6798-icon.png?itok=2tFMN9yY)
problem
Crazy Shading
Can you work out the fraction of the larger square that is covered by the shaded area?
![Efficient Cutting Poster](/sites/default/files/styles/medium/public/thumbnails/content-id-6831-icon.png?itok=B4B-17Rl)
![Circled Corners](/sites/default/files/styles/medium/public/thumbnails/content-id-7146-icon.png?itok=G2Vj0LfF)
problem
Circled Corners
Three circles have been drawn at the vertices of this triangle. What is the area of the inner shaded area?
![Wood Pile Perimeter](/sites/default/files/styles/medium/public/thumbnails/content-id-7181-icon.png?itok=4gFDtpHR)
problem
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...
![On The Edge Poster](/sites/default/files/styles/medium/public/thumbnails/content-id-7195-icon.png?itok=FDxWw7nC)
![Track design](/sites/default/files/styles/medium/public/thumbnails/content-id-7359-icon.png?itok=SoM9Ko59)
problem
Track design
Where should runners start the 200m race so that they have all run the same distance by the finish?
![Fill Me Up Too](/sites/default/files/styles/medium/public/thumbnails/content-id-7499-icon.jpg?itok=tBCBnfqy)
problem
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?
![Curvy Areas Poster](/sites/default/files/styles/medium/public/thumbnails/content-id-8001-icon.png?itok=QJqMnssC)
![Arc Area](/sites/default/files/styles/medium/public/thumbnails/content-id-8679-icon.png?itok=tI-ik3XA)
![Tadpoles](/sites/default/files/styles/medium/public/thumbnails/content-id-10103-icon.png?itok=hL3_alMT)
problem
Tadpoles
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](/sites/default/files/styles/medium/public/thumbnails/content-id-10119-icon.png?itok=G3PmEMWh)
problem
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?
![In or Out?](/sites/default/files/styles/medium/public/thumbnails/content-id-10148-icon.png?itok=hJPrb7mO)
problem
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?
![Four Leaf Clover](/sites/default/files/styles/medium/public/thumbnails/content-id-10153-icon.png?itok=ihpN955z)
problem
Four Leaf Clover
The diagram shows four equal discs and a square. What is the perimeter of the figure?
![Cut-Up Square](/sites/default/files/styles/medium/public/thumbnails/content-id-10163-icon.png?itok=hE1zi5pN)
problem
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?
![Emptied Cube](/sites/default/files/styles/medium/public/thumbnails/content-id-10174-icon.png?itok=MwR25E8k)
problem
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'?
![Square Flower](/sites/default/files/styles/medium/public/thumbnails/content-id-10179-icon.png?itok=zf80sNju)
problem
Square Flower
The diagram shows 8 circles surrounding a region. What is the perimeter of the shaded region?
![Trisected Triangle](/sites/default/files/styles/medium/public/thumbnails/content-id-10182-icon.png?itok=egHGlu9P)
problem
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?
![Clown Hats](/sites/default/files/styles/medium/public/thumbnails/content-id-10199-icon.png?itok=I_C9ouv8)
problem
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?
![Smile](/sites/default/files/styles/medium/public/thumbnails/content-id-10200-icon.png?itok=A4CRyahF)
![Semicircular Design](/sites/default/files/styles/medium/public/thumbnails/content-id-11615-icon.png?itok=oY7Z4fV2)
problem
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?
![Four Parts](/sites/default/files/styles/medium/public/thumbnails/content-id-11623-icon.png?itok=xOtg89Hy)
problem
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?
![Semicircle distance](/sites/default/files/styles/medium/public/thumbnails/content-id-12519-icon.png?itok=YEdFxtC_)
problem
Semicircle distance
Can you find the shortest distance between the semicircles given the area between them?
![Square in a circle in a square](/sites/default/files/styles/medium/public/thumbnails/content-id-12521-icon.png?itok=T6Il19Z8)
problem
Square in a circle in a square
What is the ratio of the areas of the squares in the diagram?
![Sinking Feeling](/sites/default/files/styles/medium/public/thumbnails/content-id-12540-icon.png?itok=kJ_zlkhg)
problem
Sinking Feeling
Two vases are cylindrical in shape. Can you work out the original depth of the water in the larger vase?
![Placeholder: several colourful numbers](/themes/nrich/images/nrich_small_placeholder.png)
![Placeholder: several colourful numbers](/themes/nrich/images/nrich_small_placeholder.png)
![Bobbly perimeter](/sites/default/files/styles/medium/public/thumbnails/content-id-12791-icon.png?itok=6zJOkTUy)
![Cones and spheres](/sites/default/files/styles/medium/public/thumbnails/content-id-12837-icon.png?itok=tAa7ymNC)
problem
Cones and spheres
A solid metal cone is melted down and turned into spheres. How many spheres can be made?
![ratio cut](/sites/default/files/styles/medium/public/thumbnails/content-id-12914-icon.png?itok=hpmq9Rev)
problem
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?
![Snake eyes](/sites/default/files/styles/medium/public/thumbnails/content-id-12997-icon.png?itok=cqkQaOek)
![Loo roll emergency](/sites/default/files/styles/medium/public/thumbnails/content-id-12998-icon.png?itok=hwf_0Anu)
problem
Loo roll emergency
When the roll of toilet paper is half as wide, what percentage of the paper is left?
![Placeholder: several colourful numbers](/themes/nrich/images/nrich_small_placeholder.png)
![Placeholder: several colourful numbers](/themes/nrich/images/nrich_small_placeholder.png)
problem
Similar Cylinders
Two similar cylinders are formed from a block of metal. What is the volume of the smaller cylinder?
![Placeholder: several colourful numbers](/themes/nrich/images/nrich_small_placeholder.png)
![Placeholder: several colourful numbers](/themes/nrich/images/nrich_small_placeholder.png)