List

Perimeter, Area and Volume Short Problems



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

 

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Cuboid Faces

Age
11 to 14
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Hexa-Split

Age
11 to 14
Challenge level

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

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

3×3 Areas

Age
11 to 14
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Nested Squares

Age
11 to 14
Challenge level

What proportion of the diagram is shaded?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Hawaiian Earring

Age
11 to 14
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Scalene Area

Age
11 to 14
Challenge level

Can you find the area of this scalene triangle?

Small pepper seedlings in turquoise pots.
problem

Tile Border

Age
11 to 14
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Guillotine

Age
11 to 14
Challenge level

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

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Semicircle Shape

Age
11 to 14
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Packing Small Boxes

Age
11 to 14
Challenge level

How many small boxes will fit inside the big box?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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.

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Bobbly Perimeter

Age
14 to 16
Challenge level

Find the perimeter of this shape made of semicircles

Small pepper seedlings in turquoise pots.
problem

Semicircle Stack

Age
14 to 16
Challenge level

What is the total area enclosed by the three semicicles?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Arc Area

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Dividing the Field

Age
14 to 16
Challenge level

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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Pencil Turning

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Two Paths

Age
14 to 16
Challenge level

Which of these two paths made of semicircles is shorter?

Small pepper seedlings in turquoise pots.
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.

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Canny Fraction

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Sticky Tape

Age
14 to 16
Challenge level

Work out the radius of a roll of adhesive tape.

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Candy Floss

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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.

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Wood Pile Perimeter

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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'?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Semicircle Distance

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
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.

Small pepper seedlings in turquoise pots.
problem

Biggest Enclosure

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Half an Annulus

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Snake Eyes

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Eyelids

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Tilted Aquarium

Age
14 to 16
Challenge level

Can you find the depth of water in this aquarium?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
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?

Small pepper seedlings in turquoise pots.
problem

Two Circles

Age
14 to 16
Challenge level

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?

Small pepper seedlings in turquoise pots.
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?