# 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

##### KS 3 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

##### KS 3 Short Challenge Level:

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

### Cuboid Faces

##### KS 3 Short Challenge Level:

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

### Mid-point Area

##### KS 3 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

##### KS 3 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

##### KS 3 Short Challenge Level:

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

### Rectangle Cutting

##### KS 3 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

##### KS 3 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?

##### KS 3 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

##### KS 3 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

##### KS 3 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

##### KS 3 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

##### KS 3 Short Challenge Level:

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

### Dividing a Square

##### KS 3 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

##### KS 3 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

##### KS 3 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

##### KS 3 Short Challenge Level:

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

### Semicircle Shape

##### KS 3 Short Challenge Level:

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

### Guillotine

##### KS 3 Short Challenge Level:

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

### Scalene Area

##### KS 3 Short Challenge Level:

Can you find the area of this scalene triangle?

### Intersecting Squares

##### KS 3 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

##### KS 3 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?

### Cubes on a Cube

##### KS 3 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

##### KS 3 Short Challenge Level:

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

### Tile Border

##### KS 3 Short Challenge Level:

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

### Sideways Ratio

##### KS 3 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

##### KS 3 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

##### KS 3 Short Challenge Level:

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

### Cubic Masterpiece

##### KS 3 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

##### KS 3 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

##### KS 3 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

##### KS 3 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

##### KS 3 Short Challenge Level:

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

### Christmas Cut-out

##### KS 3 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

##### KS 3 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?

### Rectangle Split

##### KS 3 Short Challenge Level:

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

### Pile Driver

##### KS 3 Short Challenge Level:

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

### Hexagon Slices

##### KS 3 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

##### KS 3 Short Challenge Level:

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

### Four Square

##### KS 3 Short Challenge Level:

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