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

Strawberries and Peas

Age 11 to 14 Short 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?

3x3 Areas

Age 11 to 14 Short Challenge Level:

Weekly Problem 24 - 2007
Which of the following shaded regions has an area different from the other shaded regions?

Mid-point Area

Age 11 to 14 Short Challenge Level:

Weekly Problem 30 - 2009
M is the midpoint of the side of the rectangle. What is the area (in square units) of the triangle PMR?

Hexa-split

Age 11 to 14 Short Challenge Level:

Weekly Problem 39 - 2013
Which of the areas shown in the hexagons are equal to each other?

Six Circles

Age 11 to 14 Short Challenge Level:

Weekly Problem 33 - 2016
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?

Rectangle Cutting

Age 11 to 14 Short Challenge Level:

Weekly Problem 50 - 2009
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?

Nested Squares

Age 11 to 14 Short Challenge Level:

What proportion of the diagram is shaded?

Giant Rubik's Cube

Age 11 to 14 Short Challenge Level:

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

Squares in a Square

Age 11 to 14 Short Challenge Level:

Weekly Problem 43 - 2016
In the diagram, the small squares are all the same size. What fraction of the large square is shaded?

Square Ratio

Age 11 to 14 Short Challenge Level:

Weekly Problem 52 - 2006
A square is divided into four rectangles and a square. Can you work out the ratio of the side lengths of the rectangles?

Cuboid Faces

Age 11 to 14 Short Challenge Level:

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

Chequered Cuboid

Age 11 to 14 Short Challenge Level:

Weekly Problem 50 - 2016
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

Age 11 to 14 Short 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?

Shaded End

Age 11 to 14 Short Challenge Level:

Weekly Problem 23 - 2016
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?

Open the Box

Age 11 to 14 Short 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?

Triangles' Triangle

Age 11 to 14 Short 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?

Tilted Tank

Age 11 to 14 Short Challenge Level:

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

Cubes on a Cube

Age 11 to 14 Short 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?

Packing Small Boxes

Age 11 to 14 Short Challenge Level:

How many small boxes will fit inside the big box?

Hawaiian Earring

Age 11 to 14 Short Challenge Level:

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

Corner Cut

Age 11 to 14 Short 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?

Exactly Three-quarters

Age 11 to 14 Short Challenge Level:

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

L-emental

Age 11 to 14 Short Challenge Level:

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

Leaning Over

Age 11 to 14 Short 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?

Line of Squares

Age 11 to 14 Short Challenge Level:

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

Intersecting Squares

Age 11 to 14 Short 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?

Christmas Cut-out

Age 11 to 14 Short 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?

Margins

Age 11 to 14 Short 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?

Sideways Ratio

Age 11 to 14 Short 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?

Guillotine

Age 11 to 14 Short Challenge Level:

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

Dividing a Square

Age 11 to 14 Short Challenge Level:

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

Cubic Masterpiece

Age 11 to 14 Short 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?

Scalene Area

Age 11 to 14 Short Challenge Level:

Can you find the area of this scalene triangle?

Double Cover

Age 11 to 14 Short Challenge Level:

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

Semicircle Shape

Age 11 to 14 Short Challenge Level:

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

Tangram Area

Age 11 to 14 Short 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?

Tile Border

Age 11 to 14 Short Challenge Level:

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

Rectangle Split

Age 11 to 14 Short Challenge Level:

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

Pile Driver

Age 11 to 14 Short Challenge Level:

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

Four Square

Age 11 to 14 Short Challenge Level:

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

Hexagon Slices

Age 11 to 14 Short 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?

Triangle in a Hexagon

Age 11 to 14 Short Challenge Level:

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

Arc Area

Age 14 to 16 Short Challenge Level:

Weekly Problem 49 - 2006
What is the area of the shape enclosed by the line and arcs?

Semicircle Stack

Age 14 to 16 Short Challenge Level:

What is the total area enclosed by the three semicicles?

Canny Fraction

Age 14 to 16 Short Challenge Level:

Weekly Problem 4 - 2012
What fraction of the volume of this can is filled with lemonade?

Four Leaf Clover

Age 14 to 16 Short Challenge Level:

Weekly Problem 5 - 2015
The diagram shows four equal discs and a square. What is the perimeter of the figure?

Tadpoles

Age 14 to 16 Short Challenge Level:

Weekly Problem 7 - 2014
The diagram shows a shaded shape bounded by circular arcs. What is the difference in area betweeen this and the equilateral triangle shown?

Pentagonal Area

Age 14 to 16 Short Challenge Level:

Weekly Problem 18 - 2006
What is the area of the pentagon?

Bobbly Perimeter

Age 14 to 16 Short Challenge Level:

Find the perimeter of this shape made of semicircles

Ratio of Areas

Age 14 to 16 Short Challenge Level:

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

Penny Farthing

Age 14 to 16 Short Challenge Level:

Weekly Problem 23 - 2014
Boris' bicycle has a bigger back wheel than front wheel. Can you work out how many revolutions the back wheel made if the front wheel did 120,000?

Cones and Spheres

Age 14 to 16 Short Challenge Level:

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

Circle in a Semicircle

Age 14 to 16 Short Challenge Level:

Weekly Problem 36 - 2011
Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?

Smile

Age 14 to 16 Short Challenge Level:

Weekly Problem 52 - 2015
Four semicircles are drawn on a line to form a shape. What is the area of this shape?

Crazy Shading

Age 14 to 16 Short Challenge Level:

Weekly Problem 34 - 2010
Can you work out the fraction of the larger square that is covered by the shaded area?

Pencil Turning

Age 14 to 16 Short Challenge Level:

Weekly Problem 37 - 2011
Rotating a pencil twice about two different points gives surprising results...

Semicircular Design

Age 14 to 16 Short 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?

Square Flower

Age 14 to 16 Short Challenge Level:

Weekly Problem 31 - 2015
The diagram shows 8 circles surrounding a region. What is the perimeter of the shaded region?

Annulus Area

Age 14 to 16 Short 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?

Four Parts

Age 14 to 16 Short Challenge Level:

Weekly Problem 13 - 2016
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?

Maximised Area

Age 14 to 16 Short Challenge Level:

Weekly Problem 39 - 2011
Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radius.

Square in a Circle in a Square

Age 14 to 16 Short Challenge Level:

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

F'arc'tion

Age 14 to 16 Short 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.

Loo Roll Emergency

Age 14 to 16 Short Challenge Level:

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

Sticky Tape

Age 14 to 16 Short Challenge Level:

Work out the radius of a roll of adhesive tape.

Half an Annulus

Age 14 to 16 Short Challenge Level:

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

Candy Floss

Age 14 to 16 Short Challenge Level:

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

Emptied Cube

Age 14 to 16 Short Challenge Level:

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

Two Paths

Age 14 to 16 Short Challenge Level:

Which of these two paths made of semicircles is shorter?

Similar Cylinders

Age 14 to 16 Short Challenge Level:

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

Circled Corners

Age 14 to 16 Short Challenge Level:

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

Wood Pile Perimeter

Age 14 to 16 Short Challenge Level:

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

Semicircle Distance

Age 14 to 16 Short Challenge Level:

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

Trisected Triangle

Age 14 to 16 Short Challenge Level:

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

Rolling Inside

Age 14 to 16 Short 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.

Snake Eyes

Age 14 to 16 Short Challenge Level:

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

Cut-up Square

Age 14 to 16 Short 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?

Sinking Feeling

Age 14 to 16 Short Challenge Level:

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

Clown Hats

Age 14 to 16 Short Challenge Level:

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

Eyelids

Age 14 to 16 Short Challenge Level:

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

Roll On

Age 14 to 16 Short 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?

Running Race

Age 14 to 16 Short 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?

Ratio Cut

Age 14 to 16 Short 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?

In or Out?

Age 14 to 16 Short 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?

Tilted Aquarium

Age 14 to 16 Short Challenge Level:

Can you find the depth of water in this aquarium?