This is part of our collection of Short Problems.
You may also be interested in our longer problems on Perimeter, Area and Volume Age 11-14 and Age 14-16.
Printable worksheets containing selections of these problems are available here:
| Stage 3 ★ | Sheet 1 | Solutions | Stage 3 ★★ | Sheet 1 | Solutions | |
| Sheet 2 | Solutions | Sheet 2 | Solutions | |||
| Sheet 3 | Solutions | Sheet 3 | Solutions | |||
| Sheet 4 | Solutions | |||||
| Stage 3 ★★★ | Sheet 1 | Solutions | ||||
| Stage 4 ★ | Sheet 1 | Solutions | Stage 4 ★★ | Sheet 1 | Solutions | |
| Sheet 2 | Solutions | |||||
| Sheet 3 | Solutions | Stage 4 ★★★ | Sheet 1 | Solutions |
problem
Six Circles
Age
11 to 14
Challenge level
In the diagram, six circles of equal size touch adjacent circles and the sides of the large rectangle. What is the perimeter of the large rectangle?
problem
Mid-Point Area
Age
11 to 14
Challenge level
M is the midpoint of the side of the rectangle. What is the area (in square units) of the triangle PMR?
problem
Squares in a Square
Age
11 to 14
Challenge level
In the diagram, the small squares are all the same size. What fraction of the large square is shaded?
problem
Chequered Cuboid
Age
11 to 14
Challenge level
A large cuboid is made from cubes of equal size. What fraction of the surface area of the large cuboid is black?
problem
Star in a Hexagon
Age
11 to 14
Challenge level
Weekly Problem 20 - 2017
The diagram shows a design formed by drawing six lines in a regular hexagon. What fraction of the hexagon is shaded?
The diagram shows a design formed by drawing six lines in a regular hexagon. What fraction of the hexagon is shaded?
problem
Cuboid faces
Age
11 to 14
Challenge level
Can you find the volume of a cuboid, given the areas of its faces?
problem
Giant Rubik's Cube
Age
11 to 14
Challenge level
How many cubes would be visible in a 12 by 12 by 12 Rubik’s cube?
problem
Hexa-split
Age
11 to 14
Challenge level
Which of the areas shown in the hexagons are equal to each other?
problem
3x3 Areas
Age
11 to 14
Challenge level
Which of the following shaded regions has an area different from the other shaded regions?
problem
Cubes on a Cube
Age
11 to 14
Challenge level
Weekly Problem 26 - 2016
A cube has each of its faces covered by one face of an identical cube, making a solid as shown. What is the surface area of the solid?
A cube has each of its faces covered by one face of an identical cube, making a solid as shown. What is the surface area of the solid?
problem
Tangram Area
Age
11 to 14
Challenge level
Weekly Problem 29 - 2008
The seven pieces in this 12 cm by 12 cm square make a Tangram set. What is the area of the shaded parallelogram?
The seven pieces in this 12 cm by 12 cm square make a Tangram set. What is the area of the shaded parallelogram?
problem
Triangles' Triangle
Age
11 to 14
Challenge level
Weekly Problem 27 - 2009
The perimeter of a large triangle is 24 cm. What is the total length of the black lines used to draw the figure?
The perimeter of a large triangle is 24 cm. What is the total length of the black lines used to draw the figure?
problem
Line of Squares
Age
11 to 14
Challenge level
Weekly Problem 35 - 2016
What is the total perimeter of the squares, if the line GH in the diagram is 24cm?
What is the total perimeter of the squares, if the line GH in the diagram is 24cm?
problem
Intersecting Squares
Age
11 to 14
Challenge level
Weekly Problem 32 - 2014
Three overlapping squares are shown. If you know the areas of the overlapping and non-overlapping parts, can you work out the side lengths of the squares?
Three overlapping squares are shown. If you know the areas of the overlapping and non-overlapping parts, can you work out the side lengths of the squares?
problem
Sideways Ratio
Age
11 to 14
Challenge level
Weekly Problem 33 - 2014
A rectangle with area $125\text{cm}^2$ has sides in the ratio $4:5$. What is the perimeter of the rectangle?
A rectangle with area $125\text{cm}^2$ has sides in the ratio $4:5$. What is the perimeter of the rectangle?
problem
Cubic Masterpiece
Age
11 to 14
Challenge level
Weekly Problem 49 - 2014
A blue cube is cut into 27 smaller cubes of equal size. What fraction of the total surface area of these cubes is blue?
A blue cube is cut into 27 smaller cubes of equal size. What fraction of the total surface area of these cubes is blue?
problem
Strawberries and Peas
Age
11 to 14
Challenge level
Weekly Problem 17 - 2017
Yasmin lengthened one side of her pea bed by 3m to make it a square. This reduced her strawberry patch by $15m^2$. What was the original area of her pea bed?
Yasmin lengthened one side of her pea bed by 3m to make it a square. This reduced her strawberry patch by $15m^2$. What was the original area of her pea bed?
problem
Open the Box
Age
11 to 14
Challenge level
Weekly Problem 37 - 2015
A piece of card is folded to make an open box. Given its surface area, can you work out its volume?
A piece of card is folded to make an open box. Given its surface area, can you work out its volume?
problem
Corner Cut
Age
11 to 14
Challenge level
Weekly Problem 23 - 2017
Three small equilateral triangles of the same size are cut from the corners of a larger equilateral triangle. What is the side length of the small triangles?
Three small equilateral triangles of the same size are cut from the corners of a larger equilateral triangle. What is the side length of the small triangles?
problem
Leaning Over
Age
11 to 14
Challenge level
Weekly Problem 31 - 2017
The triangle HIJ has the same area as the square FGHI. What is the distance from J to the line extended through F and G?
The triangle HIJ has the same area as the square FGHI. What is the distance from J to the line extended through F and G?
problem
Christmas Cut-Out
Age
11 to 14
Challenge level
Weekly Problem 52 - 2017
Sue cuts some squares from a piece of paper to make a Christmas decoration. What is the perimeter of the resulting shape?
Sue cuts some squares from a piece of paper to make a Christmas decoration. What is the perimeter of the resulting shape?
problem
Hawaiian Earring
Age
11 to 14
Challenge level
What fraction of the larger circle is outside the smaller circle?
problem
Tile border
Age
11 to 14
Challenge level
What is the largest possible number of yellow tiles in this pattern?
problem
Tilted tank
Age
11 to 14
Challenge level
Can you find the height of the water in this tilted tank when it is flat?
problem
Dividing a square
Age
11 to 14
Challenge level
A square is divided into three shapes which all have equal areas. Can you find the length of this side?
problem
Semicircle Shape
Age
11 to 14
Challenge level
Find the shaded area of these shapes with perimeters made of semicircles.
problem
Packing small boxes
Age
11 to 14
Challenge level
How many small boxes will fit inside the big box?
problem
Guillotine
Age
11 to 14
Challenge level
Weekly Problem 9 - 2006
What fraction of the area of the rectangle is shaded?
What fraction of the area of the rectangle is shaded?
problem
L-emental
Age
11 to 14
Challenge level
Weekly Problem 28 - 2006
What can you say about the rectangles that form this L-shape?
What can you say about the rectangles that form this L-shape?
problem
Double Cover
Age
11 to 14
Challenge level
Weekly Problem 3 - 2007
What is the ratio of the area of the table covered twice, to the uncovered area?
What is the ratio of the area of the table covered twice, to the uncovered area?
problem
Exactly Three-Quarters
Age
11 to 14
Challenge level
Weekly Problem 30 - 2007
Three-quarters of the area of the rectangle has been shaded. What is the length of x?
Three-quarters of the area of the rectangle has been shaded. What is the length of x?
problem
Margins
Age
11 to 14
Challenge level
Weekly Problem 16 - 2008
A 30cm x 40cm page of a book includes a 2cm margin on each side... What percentage of the page is occupied by the margins?
A 30cm x 40cm page of a book includes a 2cm margin on each side... What percentage of the page is occupied by the margins?
problem
Shaded End
Age
11 to 14
Challenge level
If the area of a face of a cuboid is one quarter of the area of each of the other two visible faces, what is the area of these faces?
problem
Four Square
Age
11 to 14
Challenge level
Weekly Problem 17 - 2015
A square contains two overlapping squares. What is the total of the shaded regions?
A square contains two overlapping squares. What is the total of the shaded regions?
problem
Pile Driver
Age
11 to 14
Challenge level
Weekly Problem 38 - 2015
Where does the line through P that halves the figure shown meet the edge XY?
Where does the line through P that halves the figure shown meet the edge XY?
problem
rectangle split
Age
11 to 14
Challenge level
Draw another line through the centre of this rectangle to split it into 4 pieces of equal area.
problem
Hexagon Slices
Age
11 to 14
Challenge level
Weekly Problem 37 - 2007
This regular hexagon has been divided into four trapezia and one hexagon.... what is the ratio of the lengths of sides p, q and r?
This regular hexagon has been divided into four trapezia and one hexagon.... what is the ratio of the lengths of sides p, q and r?
problem
Triangle in a Hexagon
Age
11 to 14
Challenge level
Weekly Problem 3 - 2009
What fraction of the area of this regular hexagon is the shaded triangle?
What fraction of the area of this regular hexagon is the shaded triangle?
problem
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?
problem
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?
The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?
problem
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?
problem
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?
Given three concentric circles, shade in the annulus formed by the smaller two. What percentage of the larger circle is now shaded?
problem
Bobbly perimeter
Age
14 to 16
Challenge level
Find the perimeter of this shape made of semicircles
problem
Semicircle Stack
Age
14 to 16
Challenge level
What is the total area enclosed by the three semicicles?
problem
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?
problem
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?
problem
Arc Area
Age
14 to 16
Challenge level
What is the area of the shape enclosed by the line and arcs?
problem
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?
problem
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?
problem
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'?
What are the volume and surface area of this 'Cubo Vazado' or 'Emptied Cube'?
problem
Square Flower
Age
14 to 16
Challenge level
The diagram shows 8 circles surrounding a region. What is the perimeter of the shaded region?
problem
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?
Four tiles are given. For which of them can three be placed together to form an equilateral triangle?
problem
Semicircle distance
Age
14 to 16
Challenge level
Can you find the shortest distance between the semicircles given the area between them?
problem
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?
problem
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.
problem
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?
problem
Half an Annulus
Age
14 to 16
Challenge level
Can you locate the point on an annulus that splits it into two areas?
problem
Pencil Turning
Age
14 to 16
Challenge level
Rotating a pencil twice about two different points gives surprising results...
problem
Two Paths
Age
14 to 16
Challenge level
Which of these two paths made of semicircles is shorter?
problem
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.
problem
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?
problem
Canny Fraction
Age
14 to 16
Challenge level
What fraction of the volume of this can is filled with lemonade?
problem
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?
problem
Candy floss
Age
14 to 16
Challenge level
What length of candy floss can Rita spin from her cylinder of sugar?
problem
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?
problem
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?
problem
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.
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.
problem
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?
problem
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?
problem
Wood Pile Perimeter
Age
14 to 16
Challenge level
Weekly Problem 30 - 2011
Three touching circles have an interesting area between them...
Three touching circles have an interesting area between them...
problem
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?
problem
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?
In the diagram, two lines have been drawn in a square. What is the ratio of the areas marked?
problem
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?
Charlie is making clown hats from a piece of cardboard. What is the maximum number he can make?
problem
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?
problem
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?
problem
Snake eyes
Age
14 to 16
Challenge level
Can you find the area of the yellow part of this snake's eye?
problem
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?
How many times does the inside disc have to roll around the inside of the ring to return to its initial position?
problem
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?
If three runners run at the same constant speed around the race tracks, in which order do they finish?
problem
Eyelids
Age
14 to 16
Challenge level
Can you work out the shaded area surrounded by these arcs?
problem
Tilted Aquarium
Age
14 to 16
Challenge level
Can you find the depth of water in this aquarium?
problem
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?
Four arcs are drawn in a circle to create a shaded area. What fraction of the area of the circle is shaded?