Angles, Polygons and Geometrical Proof 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 Age 11-14 and Age 14-16.

Printable worksheets containing selections of these problems are available here:

Stage 3 ★ Sheet 1 Solutions
  Sheet 2 Solutions
  Sheet 3 Solutions
  Sheet 4 Solutions
     
Stage 3 ★★ Sheet 1 Solutions
  Sheet 2 Solutions
  Sheet 3 Solutions
     
Stage 4 ★★ Sheet 1 Solutions
  Sheet 2 Solutions
     
Stage 4 ★★★ Sheet 1 Solutions


Isosceles Meld

Age 11 to 14 Short Challenge Level:

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

Diagonal Division

Age 11 to 14 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?

Stacking Shapes

Age 11 to 14 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?

As Long as Possible

Age 11 to 14 Short Challenge Level:

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

Homely Angles

Age 11 to 14 Short Challenge Level:

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

Angle of Overlap

Age 11 to 14 Short Challenge Level:

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

Overlapping Beer Mats

Age 11 to 14 Short Challenge Level:

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

Long Shadows

Age 11 to 14 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?

Angle Please

Age 11 to 14 Short Challenge Level:

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

Bishop's Paradise

Age 11 to 14 Short Challenge Level:

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

Triangle in a Corner

Age 11 to 14 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?

Distinct Diagonals

Age 11 to 14 Short Challenge Level:

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

Regular Vertex

Age 11 to 14 Short Challenge Level:

A square, regular pentagon and equilateral triangle share a vertex. What is the size of the other angle?

Central Distance

Age 11 to 14 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?

Isometric Rhombuses

Age 11 to 14 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?

Half Past Two

Age 11 to 14 Short Challenge Level:

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

Fraction of a Square

Age 11 to 14 Short Challenge Level:

What fraction of this square is shaded?

Right-angled Request

Age 11 to 14 Short Challenge Level:

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

Equilateral Pair

Age 11 to 14 Short Challenge Level:

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

Angular Reflection

Age 11 to 14 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

Age 11 to 14 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?

Tent Poles

Age 11 to 14 Short Challenge Level:

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

Shared Vertex

Age 11 to 14 Short Challenge Level:

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

Outside the Nonagon

Age 11 to 14 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?

Parallel Base

Age 11 to 14 Short Challenge Level:

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

Two Exterior Triangles

Age 11 to 14 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

Age 11 to 14 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

Age 11 to 14 Short Challenge Level:

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

Square Bisection

Age 11 to 14 Short Challenge Level:

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

Other Side

Age 11 to 14 Short Challenge Level:

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

Square Within a Square Within...

Age 11 to 14 Short Challenge Level:

What fraction of this square is shaded?

Polygon Cradle

Age 11 to 14 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?

Descending Angles

Age 11 to 14 Short Challenge Level:

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

Outside the Boxes

Age 11 to 14 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?

Integral Polygons

Age 11 to 14 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?

Funky Hexagon

Age 11 to 14 Short Challenge Level:

Can you find the value of x in this diagram?

Radioactive Triangle

Age 11 to 14 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?

Triangle Split

Age 11 to 14 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?

Fangs

Age 11 to 14 Short Challenge Level:

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

U in a Pentagon

Age 11 to 14 Short Challenge Level:

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

Dodecagon Angles

Age 11 to 14 Short Challenge Level:

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

Rectangle Dissection

Age 11 to 14 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?

Inscribed Hexagon

Age 11 to 14 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?

Six Minutes Past Eight

Age 11 to 14 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?

Extended Parallelogram

Age 11 to 14 Short Challenge Level:

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

Right Angled Octagon

Age 11 to 14 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?

Hexapentagon

Age 11 to 14 Short Challenge Level:

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

Two Triangles

Age 11 to 14 Short Challenge Level:

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

Square in a Triangle

Age 11 to 14 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?

Rhombus Diagonal

Age 11 to 14 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?

Handy Angles

Age 11 to 14 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?

Perimeter Puzzle

Age 11 to 14 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?

Perimeter in a Hexagon

Age 11 to 14 Short Challenge Level:

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

Nonagon Angle

Age 11 to 14 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 ?

Triangle in the Corner

Age 11 to 14 Short Challenge Level:

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

Adding Angles

Age 11 to 14 Short Challenge Level:

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

Diagonal Side

Age 11 to 14 Short Challenge Level:

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

Tricky Tessellations

Age 11 to 14 Short Challenge Level:

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

Add All the Angles

Age 11 to 14 Short Challenge Level:

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

No Rights

Age 11 to 14 Short Challenge Level:

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

Clock Face Angles

Age 11 to 14 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?

Trapezium Arch

Age 11 to 14 Short Challenge Level:

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

Clock Angle

Age 11 to 14 Short Challenge Level:

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

Two Isosceles

Age 11 to 14 Short Challenge Level:

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

Parallelogram in the Middle

Age 11 to 14 Short Challenge Level:

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

Heptagon Has

Age 11 to 14 Short Challenge Level:

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

Overbearing

Age 11 to 14 Short Challenge Level:

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

Equal Lengths

Age 11 to 14 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?

Pentagon Ring

Age 11 to 14 Short Challenge Level:

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

Inner Rectangle

Age 11 to 14 Short Challenge Level:

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

Hexagon Cut Out

Age 11 to 14 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?

Centred

Age 14 to 16 Short Challenge Level:

Weekly Problem 13 - 2012
The diagram shows contains some equal lengths. Can you work out one of the angles?

Cat on a Wall

Age 14 to 16 Short Challenge Level:

How high is the wall that this cat is lying on?

Eulerian

Age 14 to 16 Short Challenge Level:

Weekly Problem 37 - 2014
Which of the five diagrams below could be drawn without taking the pen off the page and without drawing along a line already drawn?

Internal - External

Age 14 to 16 Short Challenge Level:

Weekly Problem 12 - 2016
The diagram shows a square PQRS and two equilateral triangles RSU and PST. PQ has length 1. What is the length of TU?

Square and Triangle

Age 14 to 16 Short Challenge Level:

Can you find the area of the yellow square?

Doubly Isosceles

Age 14 to 16 Short Challenge Level:

Find the missing distance in this diagram with two isosceles triangles

Block Challenge

Age 14 to 16 Short Challenge Level:

Can you work out the shaded area in this shape?

Angle to Chord

Age 14 to 16 Short Challenge Level:

Weekly Problem 23 - 2008
A triangle has been drawn inside this circle. Can you find the length of the chord it forms?

Isosceles Reduction

Age 14 to 16 Short Challenge Level:

Weekly Problem 29 - 2010
An isosceles triangle is drawn inside another triangle. Can you work out the length of its base?

Overlapping Semicircles

Age 14 to 16 Short Challenge Level:

Two semicircles overlap, can you find this length?

Inscribed Semicircle

Age 14 to 16 Short Challenge Level:

Weekly Problem 43 - 2017
The diagram shows a semicircle inscribed in a right angled triangle. What is the radius of the semicircle?

Tunnel Vision

Age 14 to 16 Short Challenge Level:

How wide is this tunnel?

Octagonal Ratio

Age 14 to 16 Short Challenge Level:

Can you find the ratio of the area shaded in this regular octagon to the unshaded area?

Diagonal Touch

Age 14 to 16 Short Challenge Level:

Weekly Problem 21 - 2012
Two rectangles are drawn in a rectangle. What fraction of the rectangle is shaded?

Circles on a Triangle

Age 14 to 16 Short Challenge Level:

Find the missing length on this diagram.

Tied Up

Age 14 to 16 Short Challenge Level:

How much of the field can the animals graze?

Shaded Square

Age 14 to 16 Short Challenge Level:

Weekly Problem 41 - 2016
The diagram shows a square, with lines drawn from its centre. What is the shaded area?

Two Right Angles

Age 14 to 16 Short Challenge Level:

Weekly Problem 4 - 2008
In the figure given in the problem, calculate the length of an edge.

Quarters

Age 14 to 16 Short Challenge Level:

Weekly Problem 27 - 2014
Four congruent isosceles trapezia are placed in a square. What fraction of the square is shaded?

3-4-5 Circle

Age 14 to 16 Short Challenge Level:

Can you find the radius of the circle inscribed inside a '3-4-5 triangle'?

Incentre Angle

Age 14 to 16 Short Challenge Level:

Weekly Problem 1 - 2011
Use facts about the angle bisectors of this triangle to work out another internal angle.

Slide Height

Age 14 to 16 Short Challenge Level:

How high is the top of the slide?

Altitude Inequalities

Age 14 to 16 Short Challenge Level:

Weekly Problem 8 - 2010
Are you able to find triangles such that these five statements are true?

Semicircle in a Triangle

Age 14 to 16 Short Challenge Level:

A semicircle is drawn inside a right-angled triangle. Find the distance marked on the diagram.

Circled Square

Age 14 to 16 Short Challenge Level:

Can you find the area of this square inside a circle?

Folded Square

Age 14 to 16 Short Challenge Level:

This square piece of paper has been folded and creased. Where does the crease meet the side AD?

Cut-up Square

Age 14 to 16 Short Challenge Level:

Weekly Problem 15 - 2015
In the diagram, two lines have been drawn in a square. What is the ratio of the areas marked?

Triangular Intersection

Age 14 to 16 Short Challenge Level:

What is the largest number of intersection points that a triangle and a quadrilateral can have?

Height and Sides

Age 14 to 16 Short Challenge Level:

Can you find the area of the triangle from its height and two sides?