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

 

Placeholder: several colourful numbers
problem

Nested Squares

Age
11 to 14
Challenge level
filled star empty star empty star
What proportion of the diagram is shaded?
Mid-Point Area
problem

Mid-Point Area

Age
11 to 14
Challenge level
filled star empty star empty star
M is the midpoint of the side of the rectangle. What is the area (in square units) of the triangle PMR?
Six Circles
problem

Six Circles

Age
11 to 14
Challenge level
filled star empty star empty star
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?
Squares in a Square
problem

Squares in a Square

Age
11 to 14
Challenge level
filled star empty star empty star
In the diagram, the small squares are all the same size. What fraction of the large square is shaded?
Chequered Cuboid
problem

Chequered Cuboid

Age
11 to 14
Challenge level
filled star empty star empty star
A large cuboid is made from cubes of equal size. What fraction of the surface area of the large cuboid is black?
Star in a Hexagon
problem

Star in a Hexagon

Age
11 to 14
Challenge level
filled star empty star empty star
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?
Cuboid faces
problem

Cuboid faces

Age
11 to 14
Challenge level
filled star empty star empty star
Can you find the volume of a cuboid, given the areas of its faces?
Hexa-split
problem

Hexa-split

Age
11 to 14
Challenge level
filled star empty star empty star
Which of the areas shown in the hexagons are equal to each other?
Giant Rubik's Cube
problem

Giant Rubik's Cube

Age
11 to 14
Challenge level
filled star empty star empty star
How many cubes would be visible in a 12 by 12 by 12 Rubik’s cube?
3x3 Areas
problem

3x3 Areas

Age
11 to 14
Challenge level
filled star empty star empty star
Which of the following shaded regions has an area different from the other shaded regions?
Tangram Area
problem

Tangram Area

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Triangles' Triangle
problem

Triangles' Triangle

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Intersecting Squares
problem

Intersecting Squares

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Sideways Ratio
problem

Sideways Ratio

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Cubic Masterpiece
problem

Cubic Masterpiece

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Open the Box
problem

Open the Box

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Shaded End
problem

Shaded End

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Cubes on a Cube
problem

Cubes on a Cube

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Line of Squares
problem

Line of Squares

Age
11 to 14
Challenge level
filled star filled star empty star
Weekly Problem 35 - 2016
What is the total perimeter of the squares, if the line GH in the diagram is 24cm?
Strawberries and Peas
problem

Strawberries and Peas

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Corner Cut
problem

Corner Cut

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Leaning Over
problem

Leaning Over

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Christmas Cut-Out
problem

Christmas Cut-Out

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Placeholder: several colourful numbers
problem

Hawaiian Earring

Age
11 to 14
Challenge level
filled star filled star empty star
What fraction of the larger circle is outside the smaller circle?
Scalene area
problem

Scalene area

Age
11 to 14
Challenge level
filled star filled star empty star
Can you find the area of this scalene triangle?
Guillotine
problem

Guillotine

Age
11 to 14
Challenge level
filled star filled star empty star
Weekly Problem 9 - 2006
What fraction of the area of the rectangle is shaded?
L-emental
problem

L-emental

Age
11 to 14
Challenge level
filled star filled star empty star
Weekly Problem 28 - 2006
What can you say about the rectangles that form this L-shape?
Placeholder: several colourful numbers
problem

Tile border

Age
11 to 14
Challenge level
filled star filled star empty star
What is the largest possible number of yellow tiles in this pattern?
Tilted tank
problem

Tilted tank

Age
11 to 14
Challenge level
filled star filled star empty star
Can you find the height of the water in this tilted tank when it is flat?
Double Cover
problem

Double Cover

Age
11 to 14
Challenge level
filled star filled star empty star
Weekly Problem 3 - 2007
What is the ratio of the area of the table covered twice, to the uncovered area?
Dividing a square
problem

Dividing a square

Age
11 to 14
Challenge level
filled star filled star empty star
A square is divided into three shapes which all have equal areas. Can you find the length of this side?
Exactly Three-Quarters
problem

Exactly Three-Quarters

Age
11 to 14
Challenge level
filled star filled star empty star
Weekly Problem 30 - 2007
Three-quarters of the area of the rectangle has been shaded. What is the length of x?
Placeholder: several colourful numbers
problem

Semicircle Shape

Age
11 to 14
Challenge level
filled star filled star empty star
Find the shaded area of these shapes with perimeters made of semicircles.
Margins
problem

Margins

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Placeholder: several colourful numbers
problem

Packing small boxes

Age
11 to 14
Challenge level
filled star filled star empty star
How many small boxes will fit inside the big box?
Four Square
problem

Four Square

Age
11 to 14
Challenge level
filled star filled star filled star
Weekly Problem 17 - 2015
A square contains two overlapping squares. What is the total of the shaded regions?
Pile Driver
problem

Pile Driver

Age
11 to 14
Challenge level
filled star filled star filled star
Weekly Problem 38 - 2015
Where does the line through P that halves the figure shown meet the edge XY?
rectangle split
problem

rectangle split

Age
11 to 14
Challenge level
filled star filled star filled star
Draw another line through the centre of this rectangle to split it into 4 pieces of equal area.
Hexagon Slices
problem

Hexagon Slices

Age
11 to 14
Challenge level
filled star filled star filled star
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?
Triangle in a Hexagon
problem

Triangle in a Hexagon

Age
11 to 14
Challenge level
filled star filled star filled star
Weekly Problem 3 - 2009
What fraction of the area of this regular hexagon is the shaded triangle?
Smile
problem

Smile

Age
14 to 16
Challenge level
filled star empty star empty star
Four semicircles are drawn on a line to form a shape. What is the area of this shape?
Semicircular Design
problem

Semicircular Design

Age
14 to 16
Challenge level
filled star empty star empty star
Weekly Problem 9 - 2016
The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?
Four Parts
problem

Four Parts

Age
14 to 16
Challenge level
filled star empty star empty star
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?
Annulus Area
problem

Annulus Area

Age
14 to 16
Challenge level
filled star empty star empty star
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?
Bobbly perimeter
problem

Bobbly perimeter

Age
14 to 16
Challenge level
filled star empty star empty star
Find the perimeter of this shape made of semicircles
Semicircle Stack
problem

Semicircle Stack

Age
14 to 16
Challenge level
filled star empty star empty star
What is the total area enclosed by the three semicicles?
Placeholder: several colourful numbers
problem

Ratio of areas

Age
14 to 16
Challenge level
filled star empty star empty star
What is the ratio of the area of the hexagon to the area of the triangle?
Crazy Shading
problem

Crazy Shading

Age
14 to 16
Challenge level
filled star empty star empty star
Can you work out the fraction of the larger square that is covered by the shaded area?
Arc Area
problem

Arc Area

Age
14 to 16
Challenge level
filled star empty star empty star
What is the area of the shape enclosed by the line and arcs?
Penny Farthing
problem

Penny Farthing

Age
14 to 16
Challenge level
filled star empty star empty star
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?
Four Leaf Clover
problem

Four Leaf Clover

Age
14 to 16
Challenge level
filled star filled star empty star
The diagram shows four equal discs and a square. What is the perimeter of the figure?
Emptied Cube
problem

Emptied Cube

Age
14 to 16
Challenge level
filled star filled star empty star
Weekly Problem 26 - 2015
What are the volume and surface area of this 'Cubo Vazado' or 'Emptied Cube'?
Square Flower
problem

Square Flower

Age
14 to 16
Challenge level
filled star filled star empty star
The diagram shows 8 circles surrounding a region. What is the perimeter of the shaded region?
Trisected Triangle
problem

Trisected Triangle

Age
14 to 16
Challenge level
filled star filled star empty star
Weekly Problem 34 - 2015
Four tiles are given. For which of them can three be placed together to form an equilateral triangle?
Semicircle distance
problem

Semicircle distance

Age
14 to 16
Challenge level
filled star filled star empty star
Can you find the shortest distance between the semicircles given the area between them?
F'arc'tion
problem

F'arc'tion

Age
14 to 16
Challenge level
filled star filled star empty star
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.
Circle in a Semicircle
problem

Circle in a Semicircle

Age
14 to 16
Challenge level
filled star filled star empty star
Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?
Placeholder: several colourful numbers
problem

Half an Annulus

Age
14 to 16
Challenge level
filled star filled star empty star
Can you locate the point on an annulus that splits it into two areas?
Pencil Turning
problem

Pencil Turning

Age
14 to 16
Challenge level
filled star filled star empty star
Rotating a pencil twice about two different points gives surprising results...
Placeholder: several colourful numbers
problem

Two Paths

Age
14 to 16
Challenge level
filled star filled star empty star
Which of these two paths made of semicircles is shorter?
Maximised Area
problem

Maximised Area

Age
14 to 16
Challenge level
filled star filled star empty star
Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radius.
Cones and spheres
problem

Cones and spheres

Age
14 to 16
Challenge level
filled star filled star empty star
A solid metal cone is melted down and turned into spheres. How many spheres can be made?
Canny Fraction
problem

Canny Fraction

Age
14 to 16
Challenge level
filled star filled star empty star
What fraction of the volume of this can is filled with lemonade?
Sticky Tape
problem

Sticky Tape

Age
14 to 16
Challenge level
filled star filled star empty star
Work out the radius of a roll of adhesive tape.
Loo roll emergency
problem

Loo roll emergency

Age
14 to 16
Challenge level
filled star filled star empty star
When the roll of toilet paper is half as wide, what percentage of the paper is left?
Placeholder: several colourful numbers
problem

Candy floss

Age
14 to 16
Challenge level
filled star filled star empty star
What length of candy floss can Rita spin from her cylinder of sugar?
Placeholder: several colourful numbers
problem

Similar Cylinders

Age
14 to 16
Challenge level
filled star filled star empty star
Two similar cylinders are formed from a block of metal. What is the volume of the smaller cylinder?
Square Ratio
problem

Square Ratio

Age
14 to 16
Challenge level
filled star filled star empty star
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
problem

Rolling Inside

Age
14 to 16
Challenge level
filled star filled star empty star
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.
Rectangle Cutting
problem

Rectangle Cutting

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Circled Corners
problem

Circled Corners

Age
14 to 16
Challenge level
filled star filled star empty star
Three circles have been drawn at the vertices of this triangle. What is the area of the inner shaded area?
Wood Pile Perimeter
problem

Wood Pile Perimeter

Age
14 to 16
Challenge level
filled star filled star empty star
Weekly Problem 30 - 2011
Three touching circles have an interesting area between them...
Tadpoles
problem

Tadpoles

Age
14 to 16
Challenge level
filled star filled star empty star
The diagram shows a shaded shape bounded by circular arcs. What is the difference in area betweeen this and the equilateral triangle shown?
Cut-Up Square
problem

Cut-Up Square

Age
14 to 16
Challenge level
filled star filled star filled star
Weekly Problem 15 - 2015
In the diagram, two lines have been drawn in a square. What is the ratio of the areas marked?
Clown Hats
problem

Clown Hats

Age
14 to 16
Challenge level
filled star filled star filled star
Weekly Problem 51 - 2015
Charlie is making clown hats from a piece of cardboard. What is the maximum number he can make?
Sinking Feeling
problem

Sinking Feeling

Age
14 to 16
Challenge level
filled star filled star filled star
Two vases are cylindrical in shape. Can you work out the original depth of the water in the larger vase?
ratio cut
problem

ratio cut

Age
14 to 16
Challenge level
filled star filled star filled star
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
problem

Snake eyes

Age
14 to 16
Challenge level
filled star filled star filled star
Can you find the area of the yellow part of this snake's eye?
Roll On
problem

Roll On

Age
14 to 16
Challenge level
filled star filled star filled star
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?
Running Race
problem

Running Race

Age
14 to 16
Challenge level
filled star filled star filled star
Weekly Problem 13 - 2006
If three runners run at the same constant speed around the race tracks, in which order do they finish?
Eyelids
problem

Eyelids

Age
14 to 16
Challenge level
filled star filled star filled star
Can you work out the shaded area surrounded by these arcs?
Placeholder: several colourful numbers
problem

Tilted Aquarium

Age
14 to 16
Challenge level
filled star filled star filled star
Can you find the depth of water in this aquarium?
In or Out?
problem

In or Out?

Age
14 to 16
Challenge level
filled star filled star filled star
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?