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 Stage 3 ★★★ Sheet 1 Solutions Sheet 2 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?

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?

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?

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?

KS 3 Short Challenge Level:

Weekly Problem 23 - 2016
The diagram shows a cuboid in which the area of the shaded face is one quarter of the area of each of the two visible unshaded faces. What is the area of one of the unshaded faces?

Strawberries and Peas

KS 3 Short Challenge Level:

Weekly Problem 17 - 2017
Yasmin has beds for peas and strawberries in her rectangular garden. She 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

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?

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?

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?

Guillotine

KS 3 Short Challenge Level:

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

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?

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?

Cubic Masterpiece

KS 3 Short Challenge Level:

Weekly Problem 49 - 2014
A solid wooden cube is painted blue on the outside. The cube is then cut into 27 smaller cubes of equal size. What fraction of the total surface area of these new cubes is blue?

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?

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?

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?

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?

Chequered Cuboid

KS 3 & 4 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

KS 3 & 4 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?

Leaning Over

KS 3 & 4 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

KS 3 & 4 Short Challenge Level:

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

Soma Surface

KS 3 & 4 Short Challenge Level:

Weekly Problem 48 - 2017
What is the surface area of the solid shown?