Perimeter, Area and Volume Stage 3 - 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 - Stage 3.

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        

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?

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?

Cuboid Faces

Age 11 to 14 Short Challenge Level:

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

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

Hawaiian Earring

Age 11 to 14 Short Challenge Level:

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

Semicircle Shape

Age 11 to 14 Short Challenge Level:

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

Guillotine

Age 11 to 14 Short Challenge Level:

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

Scalene Area

Age 11 to 14 Short Challenge Level:

Can you find the area of this scalene triangle?

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?

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?

Packing Small Boxes

Age 11 to 14 Short Challenge Level:

How many small boxes will fit inside the big box?

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?

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?

Tile Border

Age 11 to 14 Short Challenge Level:

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

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?