Perimeter, Area and Volume Short Problems

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?

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

A large cuboid is made from cubes of equal size. What fraction of the surface area of the large cuboid is black?

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?

Can you find the volume of a cuboid, given the areas of its faces?

Which of the areas shown in the hexagons are equal to each other?

How many cubes would be visible in a 12 by 12 by 12 Rubik's cube?

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

What proportion of the diagram is shaded?

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

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?

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?

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?

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?

Weekly Problem 35 - 2016
What is the total perimeter of the squares, if the line GH in the diagram is 24cm?

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?

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?

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?

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?

What fraction of the larger circle is outside the smaller circle?

Can you find the area of this scalene triangle?

What is the largest possible number of yellow tiles in this pattern?

Can you find the height of the water in this tilted tank when it is flat?

Weekly Problem 9 - 2006
What fraction of the area of the rectangle is shaded?

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

Weekly Problem 28 - 2006
What can you say about the rectangles that form this L-shape?

Weekly Problem 3 - 2007
What is the ratio of the area of the table covered twice, to the uncovered area?

Find the shaded area of these shapes with perimeters made of semicircles.

How many small boxes will fit inside the big box?

Weekly Problem 30 - 2007
Three-quarters of the area of the rectangle has been shaded. What is the length of x?

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?

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?

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?

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?

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?

Weekly Problem 38 - 2015
Where does the line through P that halves the figure shown meet the edge XY?

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

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?

Weekly Problem 3 - 2009
What fraction of the area of this regular hexagon is the shaded triangle?

Weekly Problem 17 - 2015
A square contains two overlapping squares. What is the total of the shaded regions?

Four semicircles are drawn on a line to form a shape. What is the area of this shape?

Weekly Problem 9 - 2016
The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two trapeziums each of equal area. How could he do this?

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?

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?

Find the perimeter of this shape made of semicircles

What is the total area enclosed by the three semicicles?

What is the ratio of the area of the hexagon to the area of the triangle?

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

What is the area of the shape enclosed by the line and arcs?

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?

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

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.

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

Three fences of different lengths form three sides of an enclosure. What arrangement maximises the area?

Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?

Can you locate the point on an annulus that splits it into two areas?

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

Which of these two paths made of semicircles is shorter?

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

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

What fraction of the volume of this can is filled with lemonade?

Work out the radius of a roll of adhesive tape.

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

What length of candy floss can Rita spin from her cylinder of sugar?

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

A square is divided into four rectangles and a square. Can you work out the ratio of the side lengths of the rectangles?

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.

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?

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

Weekly Problem 30 - 2011
Three touching circles have an interesting area between them...

The diagram shows a shaded shape bounded by circular arcs. What is the difference in area betweeen this and the equilateral triangle shown?

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

Weekly Problem 26 - 2015
What are the volume and surface area of this 'Cubo Vazado' or 'Emptied Cube'?

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

Weekly Problem 34 - 2015
Four tiles are given. For which of them can three be placed together to form an equilateral triangle?

Weekly Problem 51 - 2015
Charlie is making clown hats from a piece of cardboard. What is the maximum number he can make?

Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?

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

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?

Can you find the area of the yellow part of this snake's eye?

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?

Weekly Problem 13 - 2006
If three runners run at the same constant speed around the race tracks, in which order do they finish?

Can you work out the shaded area surrounded by these arcs?

Can you find the depth of water in this aquarium?

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?

Weekly Problem 15 - 2015
In the diagram, two lines have been drawn in a square. What is the ratio of the areas marked?