Angles, Polygons and Geometrical Proof Stage 3 - Short Problems

This is part of our collection of Short Problems.

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

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

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?

Triangle in a Corner

KS 3 Short Challenge Level:

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

Shared Vertex

KS 3 Short Challenge Level:

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

KS 3 Short Challenge Level:

Weekly Problem 10 - 2012
If you know how long Meg's shadow is, can you work out how long the shadow is when she stands on her brother's shoulders?

Distinct Diagonals

KS 3 Short Challenge Level:

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

Right-angled Request

KS 3 Short Challenge Level:

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

Isometric Rhombuses

KS 3 Short Challenge Level:

Weekly Problem 31 - 2016
The diagram shows a grid of $16$ identical equilateral triangles. How many rhombuses are there made up of two adjacent small triangles?

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?

Overlapping Beer Mats

KS 3 Short Challenge Level:

Can you find the area of the overlap when these two beer mats are placed on top of each other?

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?

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?

Equilateral Pair

KS 3 Short Challenge Level:

Weekly Problem 39 - 2016
In the diagram, VWX and XYZ are congruent equilateral triangles. What is the size of angle VWY?

Square Bisection

KS 3 Short Challenge Level:

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

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?

Robo-turn

KS 3 Short Challenge Level:

Weekly Problem 16 - 2007
Can you figure out how far the robot has travelled by the time it is first facing due East?

Stellar Angles

KS 3 Short Challenge Level:

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

Tent Poles

KS 3 Short Challenge Level:

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

Homely Angles

KS 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?

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?

Stacking Shapes

KS 3 Short Challenge Level:

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?

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?

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?

Parallel Base

KS 3 Short Challenge Level:

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

Angle of Overlap

KS 3 Short Challenge Level:

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

KS 3 Short Challenge Level:

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

KS 3 Short Challenge Level:

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

Other Side

KS 3 Short Challenge Level:

Weekly Problem 8 - 2016
Can you work out the size of the angles in a quadrilateral?

Isosceles Meld

KS 3 Short Challenge Level:

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

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?

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?

Extended Parallelogram

KS 3 Short Challenge Level:

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

Six Minutes Past Eight

KS 3 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?

Diagonal Side

KS 3 Short Challenge Level:

Can you work out the area of a square drawn on a diagonal?

Rhombus Diagonal

KS 3 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?

Hexapentagon

KS 3 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 in the Corner

KS 3 Short Challenge Level:

A triangle is shaded within a regular hexagon. Can you find its area?

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?

Triangle Split

KS 3 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?

KS 3 Short Challenge Level:

Weekly Problem 41 - 2014
Three straight lines divide an equilateral triangle into seven regions. What is the side length of the original triangle?

Funky Hexagon

KS 3 Short Challenge Level:

Can you find the value of x in this diagram?

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 ?

KS 3 Short Challenge Level:

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

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?

Two Triangles

KS 3 Short Challenge Level:

Prove that the angle marked $a$ is half the size of the angle marked $b$.

Outside the Boxes

KS 3 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?

Perimeter in a Hexagon

KS 3 Short Challenge Level:

Can you find the perimeter of this triangle inscribed in a hexagon?

U in a Pentagon

KS 3 Short Challenge Level:

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

Tricky Tessellations

KS 3 Short Challenge Level:

Can you work out the fraction of the tiles that are painted black in this pattern?

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?

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 $x$?

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?

Descending Angles

KS 3 Short Challenge Level:

Given four of the angles in two triangles, can you find the smallest angle overall?

Dodecagon Angles

KS 3 Short Challenge Level:

Weekly Problem 50 - 2012
The diagram shows a regular dodecagon. What is the size of the marked angle?

Inscribed Hexagon

KS 3 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?

Perimeter Puzzle

KS 3 Short Challenge Level:

If four copies of this triangle are joined together to form a parallelogram, what is the largest possible perimeter of the parallelogram?

Clock Face Angles

KS 3 Short Challenge Level:

The time is 20:14. What is the smaller angle between the hour hand and the minute hand on an accurate analogue clock?

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 $x$?

Heptagon Has

KS 3 Short Challenge Level:

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

KS 3 Short Challenge Level:

Find the sum of all of the angles denoted by letters in this diagram

Clock Angle

KS 3 Short Challenge Level:

What is the angle between the the hands of a clock at 8:24?

Equal Lengths

KS 3 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?

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?

Inner Rectangle

KS 3 Short Challenge Level:

If the shape on the inside is a rectangle, what can you say about the shape on the outside?

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?

Overbearing

KS 3 Short Challenge Level:

A village has a pub, church and school. What is the bearing of the school from the church?

No Rights

KS 3 Short Challenge Level:

Prove that the angle bisectors of a triangle can never meet at right angles.

Trapezium Arch

KS 3 Short Challenge Level:

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

Two Isosceles

KS 3 Short Challenge Level:

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