Angles and Polygons - Short Problems

A collection of short Stage 3 problems on Angles and Polygons



Pentagon Ring

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 47 - 2011
Place equal, regular pentagons together to form a ring. How many pentagons will be needed?

Isosceles Meld

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 9 - 2012
What is the angle QPT in this diagram?

Heptagon Has

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 15 - 2012
How many of the five properties can a heptagon have?

Nonagon Angle

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 53 - 2012
ABCDEFGHI is a regular nine-sided polygon (called a 'nonagon' or 'enneagon'). What is the size of the angle FAE ?

Hexagon Cut Out

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 52 - 2012
An irregular hexagon can be made by cutting the corners off an equilateral triangle. How can an identical hexagon be made by cutting the corners off a different equilateral triangle?

Integral Polygons

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 51 - 2012
Each interior angle of a particular polygon is an obtuse angle which is a whole number of degrees. What is the greatest number of sides the polygon could have?

Parallelogram in the Middle

Stage: 3 Short Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Weekly Problem 27 - 2013
The diagram shows a parallelogram inside a triangle. What is the value of $x$?

Angular Reflection

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 28 - 2013
Two lines meet at a point. Another line through this point is reflected in both of these lines. What is the angle between the image lines?

Equal Lengths

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 29 - 2013
An equilateral triangle is drawn inside a rhombus, both with equal side lengths. What is one of the angles of the rhombus?

Stellar Angles

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 30 - 2013
What is the angle $x$ in the star shape shown?

As Long as Possible

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 40 - 2013
Given three sides of a quadrilateral, what is the longest that the fourth side can be?

Bishop's Paradise

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 37 - 2013
Which of the statements about diagonals of polygons is false?

Weekly Problem 8 - 2008

Stage: 3 and 4 Challenge Level: Challenge Level:1

In how many ways can a square be cut in half using a single straight line cut?

Weekly Problem 13 - 2008

Stage: 3 and 4 Challenge Level: Challenge Level:1

The diagram shows three squares drawn on the sides of a triangle. What is the sum of the three marked angles?

Weekly Problem 18 - 2008

Stage: 3 and 4 Challenge Level: Challenge Level:1

The diagram shows a regular pentagon. Can you work out the size of the marked angle?

Weekly Problem 38 - 2008

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

A quadrilateral can have four right angles. What is the largest number of right angles an octagon can have?

Weekly Problem 39 - 2008

Stage: 3 Challenge Level: Challenge Level:1

How big is the angle between the hour hand and the minute hand of a clock at twenty to five?

Weekly Problem 45 - 2008

Stage: 3 Challenge Level: Challenge Level:1

The diagram shows a regular pentagon with two of its diagonals. If all the diagonals are drawn in, into how many areas will the pentagon be divided?

Weekly Problem 50 - 2008

Stage: 3 Short Challenge Level: Challenge Level:1

The lengths SP, SQ and SR are equal and the angle SRQ is x degrees. What is the size of angle PQR?

Rectangle Dissection

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 2 - 2009
The 16 by 9 rectangle is cut as shown. Rearrange the pieces to form a square. What is the perimeter of the square?

Sharp Corners

Stage: 3 Challenge Level: Challenge Level:1

Weekly Problem 6 - 2009
In a triangle the smallest angle is 20 degrees. What is the largest possible angle in the triangle?

Half Past Two

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 21 - 2009
What is the angle between the two hands of a clock at 2.30?

Weekly Problem 33 - 2009

Stage: 3 Short Challenge Level: Challenge Level:1

In a village the pub, church and school are at different lengths and bearings from each other. What is the bearing of the school from the church?

Weekly Problem 35 - 2009

Stage: 3 Short Challenge Level: Challenge Level:1

Two equilateral triangles have been drawn on two adjacent sides of a square. What is the angle between the triangles?

Distinct Diagonals

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 21 - 2010
How many diagonals can you draw on this square...

Angle Hunt

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 39 - 2010
If you know three lengths and an angle in this diagram, can you find another angle by calculation?

Outside the Nonagon

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 44 - 2010
Extend two of the sides of a nonagon to form an angle. How large is this acute angle?

Homely Angles

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Weekly Problem 18 - 2011
Draw an equilateral triangle onto one side of a square. Can you work out one particular angle?

Fangs

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 7 - 2013
Three of the angles in this diagram all have size $x$. What is the value of $x$?

Inscribed Hexagon

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 1 - 2014
The diagram shows a regular hexagon inside a rectangle. What is the sum of the four marked angles?

Extended Parallelogram

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 11 - 2014
The diagram shows a parallelogram and an isosceles triangle. What is the size of angle TQR?

Rhombus Diagonal

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 19 - 2014
The diagram shows a rhombus and an isosceles triangle. Can you work out the size of the angle JFI?

Regular Vertex

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 51 - 2014
A square, regular pentagon and equilateral triangle share a vertex. What is the size of the other angle there?

Tent Poles

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 40 - 2015
In the diagram, $PT = QT = TS$ and $QS = SR$. What is the value of $x$?

Parallel Base

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 46 - 2015
The diagram shows two parallel lines and two angles. What is the value of x?

Other Side

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Weekly Problem 8 - 2016
The diagram shows a quadrilateral $ABCD$, in which $AD=BC$, $\angle CAD=50^\circ$, $\angle ACD=65^\circ$ and $\angle ACB=70^\circ$. What is the size of $\angle ABC$?

Triangle in a Corner

Stage: 4 Short Challenge Level: Challenge Level:1

Weekly Problem 17 - 2016
The diagram shows an equilateral triangle touching two straight lines. What is the sum of the four marked angles?

Six Circles

Stage: 3 Short Challenge Level: Challenge Level:1

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?

Adding Angles

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 47 - 2016
What is the sum of the six marked angles?

Stacking Shapes

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 28 - 2017
The diagram on the right shows an equilateral triangle, a square and a regular pentagon. What is the sum of the interior angles of the resulting polygon?

Angle Please

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 19 - 2017
In the figure, what is the value of x?

Two Isosceles

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

Weekly Problem 37 - 2017
A quadrilateral is divided into two isosceles triangles. Can you work out the perimeter of the quadrilateral?

Shared Vertex

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 38 - 2017
In the diagram, what is the value of $x$?

Geometric Quadrilateral

Stage: 3 Short Challenge Level: Challenge Level:1

Weekly Problem 49 - 2017
Each interior angle in a quadrilateral (apart from the smallest) is twice the previous one. What is the size of the smallest interior angle?