Angles and Polygons - Short Problems

This is part of our collection of Short Problems.

You may also be interested in our longer problems on Angles and Polygons.

Right-angled Request

KS 2 & 3 Short Challenge Level:

Weekly Problem 26 - 2006
How many right angled triangles are formed by the points in this diagram?

Homely Angles

KS 2 & 3 Short Challenge Level:

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

Pentagon Ring

KS 3 Short Challenge Level:

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

Diagonal Division

KS 3 Short Challenge Level:

Weekly Problem 45 - 2008
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?

Angular Reflection

KS 3 Short Challenge Level:

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?

Fangs

KS 3 Short Challenge Level:

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

Isosceles Meld

KS 3 Short Challenge Level:

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

Overbearing

KS 3 Short Challenge Level:

Weekly Problem 33 - 2009
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?

Heptagon Has

KS 3 Short Challenge Level:

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

Two Exterior Triangles

KS 3 Short Challenge Level:

Weekly Problem 35 - 2009
Two equilateral triangles have been drawn on two adjacent sides of a square. What is the angle between the triangles?

Robo-turn

KS 3 Short Challenge Level:

Weekly Problem 16 - 2007
A robot, which is initially facing North, is programmed so that each move consists of moving 5m in a straight line and then turning clockwise through an angle which increases by 10 degrees each move. How far has it travelled by the time it is first facing due East at the end of a move?

Stellar Angles

KS 3 Short Challenge Level:

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

Distinct Diagonals

KS 3 Short Challenge Level:

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

KS 3 Short Challenge Level:

Weekly Problem 18 - 2007
A regular pentagon together with three sides of a regular hexagon form a cradle. What is the size of one of the angles?

Sharp Corners

KS 3 Short Challenge Level:

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

As Long as Possible

KS 3 Short Challenge Level:

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

Hexagon Cut Out

KS 3 Short Challenge Level:

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?

Angle Hunt

KS 3 Short Challenge Level:

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

Half Past Two

KS 3 Short Challenge Level:

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

KS 3 Short Challenge Level:

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

Outside the Nonagon

KS 3 Short Challenge Level:

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

Central Distance

KS 3 Short Challenge Level:

Weekly Problem 1 - 2006
The diagram shows two circles enclosed in a rectangle. What is the distance between the centres of the circles?

Handy Angles

KS 3 Short Challenge Level:

Weekly Problem 39 - 2008
How big is the angle between the hour hand and the minute hand of a clock at twenty to five?

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?

Square in a Triangle

KS 3 Short Challenge Level:

Weekly Problem 33 - 2006
A square is inscribed in an isoscles right angled triangle of area $x$. What is the area of the square?

Rectangle Dissection

KS 3 Short Challenge Level:

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?

Nonagon Angle

KS 3 Short Challenge Level:

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

Right Angled Octagon

KS 3 Short Challenge Level:

Weekly Problem 38 - 2008
A quadrilateral can have four right angles. What is the largest number of right angles an octagon can have?

Integral Polygons

KS 3 Short Challenge Level:

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?

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?

Parallelogram in the Middle

KS 3 Short Challenge Level:

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

Hexapentagon

KS 3 & 4 Short Challenge Level:

Weekly Problem 53 - 2007
The diagram shows a regular pentagon and regular hexagon which overlap. What is the value of x?

Triangle Split

KS 3 & 4 Short Challenge Level:

Weekly Problem 50 - 2008
The lengths SP, SQ and SR are equal and the angle SRQ is x degrees. What is the size of angle PQR?

Inscribed Hexagon

KS 3 & 4 Short Challenge Level:

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

Square Bisection

KS 3 & 4 Short Challenge Level:

Weekly Problem 8 - 2008
In how many ways can a square be cut in half using a single straight line cut?

Extended Parallelogram

KS 3 & 4 Short Challenge Level:

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

Outside the Boxes

KS 3 & 4 Short Challenge Level:

Weekly Problem 13 - 2008
The diagram shows three squares drawn on the sides of a triangle. What is the sum of the three marked angles?

Rhombus Diagonal

KS 3 & 4 Short Challenge Level:

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

U in a Pentagon

KS 3 & 4 Short Challenge Level:

Weekly Problem 18 - 2008
The diagram shows a regular pentagon. Can you work out the size of the marked angle?

Angle of Overlap

KS 3 & 4 Short Challenge Level:

Weekly Problem 26 - 2007
The diagram shows two equilateral triangles. What is the value of x?

Regular Vertex

KS 3 & 4 Short Challenge Level:

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

Trapezium Arch

KS 3 & 4 Short Challenge Level:

Weekly Problem 27 - 2007
Ten stones form an arch. What is the size of the smallest angles of the trapezoidal stones?

Six Minutes Past Eight

KS 3 & 4 Short Challenge Level:

Weekly Problem 45 - 2007
What is the obtuse angle between the hands of a clock at 6 minutes past 8 o'clock?

Parallel Base

KS 3 & 4 Short Challenge Level:

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

Other Side

KS 3 & 4 Short Challenge Level:

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 $\angle ABC$?

Equal Lengths

KS 3 & 4 Short Challenge Level:

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?

Tent Poles

KS 3 & 4 Short Challenge Level:

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