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

### 3x3 Areas

Which of the following shaded regions has an area different from the other shaded regions?

problem

### Mid-Point Area

M is the midpoint of the side of the rectangle. What is the area (in square units) of the triangle PMR?

problem

### Six Circles

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

### Squares in a Square

In the diagram, the small squares are all the same size. What fraction of the large square is shaded?

problem

### Chequered Cuboid

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

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

### Leaning Over

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

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

### Dividing a square

A square is divided into three shapes which all have equal areas. Can you find the length of this side?

problem

### L-emental

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

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

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

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

### Tangram Area

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

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

### Intersecting Squares

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

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

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

### Open the Box

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

### Shaded End

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

### Cubes on a Cube

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

### Line of Squares

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

### Strawberries and Peas

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

### Corner Cut

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

### rectangle split

Draw another line through the centre of this rectangle to split it into 4 pieces of equal area.

problem

### Hexagon Slices

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

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

### Four Square

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

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

### Crazy Shading

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

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?

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?

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?

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?

problem

### Loo roll emergency

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

problem

### Similar Cylinders

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

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?

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.

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?

problem

### Circled Corners

Three circles have been drawn at the vertices of this triangle. What is the area of the inner shaded area?

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...

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?

problem

### Four Leaf Clover

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

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'?

problem

### Square Flower

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

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?

problem

### Semicircle distance

Can you find the shortest distance between the semicircles given the area between them?

problem

### Square in a circle in a square

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

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.

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?

problem

### Pencil Turning

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

problem

### Maximised Area

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

problem

### Cones and spheres

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

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?

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?

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?

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?

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?

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?

problem

### Sinking Feeling

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