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# Angles, Polygons and Geometrical Proof Short Problems

### Equilateral Pair

### Angular Reflection

### Angle Hunt

### Tent Poles

### Shared Vertex

### Outside the Nonagon

### Parallel Base

### Two Exterior Triangles

### Robo-turn

### Stellar Angles

### Square Bisection

### Other Side

### Square Within a Square Within...

### Polygon Cradle

### Isosceles Meld

### Diagonal Division

### Stacking Shapes

### As Long as Possible

### Homely Angles

### Angle of Overlap

### Overlapping Beer Mats

### Long Shadows

### Angle Please

### Bishop's Paradise

### Triangle in a Corner

### Distinct Diagonals

### Regular Vertex

### Central Distance

### Isometric Rhombuses

### Half Past Two

### Right-angled Request

### Right Angled Octagon

### Hexapentagon

### Two Triangles

### Square in a Triangle

### Rhombus Diagonal

### Handy Angles

### Perimeter Puzzle

### Perimeter in a Hexagon

### Nonagon Angle

### Triangle in the Corner

### Adding Angles

### Diagonal Side

### Tricky Tessellations

### Descending Angles

### Outside the Boxes

### Integral Polygons

### Radioactive Triangle

### Triangle Split

### Fangs

### U in a Pentagon

### Dodecagon Angles

### Rectangle Dissection

### Inscribed Hexagon

### Six Minutes Past Eight

### Extended Parallelogram

### Overbearing

### Equal Lengths

### Pentagon Ring

### Inner Rectangle

### Hexagon Cut Out

### Add All the Angles

### No Rights

### Clock Face Angles

### Trapezium Arch

### Clock Angle

### Two Isosceles

### Parallelogram in the Middle

### Heptagon Has

### Eulerian

### Centred

### Octagonal Ratio

### Diagonal Touch

### Shaded Square

### Two Right Angles

### Quarters

### 3-4-5 Circle

### Incentre Angle

### Internal - External

### Doubly Isosceles

### Angle to Chord

### Isosceles Reduction

### Overlapping Semicircles

### Inscribed Semicircle

### Circular Inscription

### Griddy Region

### Garden Fence

### Overlapping Annuli

### Altitude Inequalities

### Semicircle in a Triangle

Links to the University of Cambridge website
Links to the NRICH website Home page

Nurturing young mathematicians: teacher webinars

30 April (Primary), 1 May (Secondary)

30 April (Primary), 1 May (Secondary)

Or search by topic

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 |

Age 11 to 14

ShortChallenge Level

Weekly Problem 39 - 2016

In the diagram, VWX and XYZ are congruent equilateral triangles. What is the size of angle VWY?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Weekly Problem 39 - 2010

If you know three lengths and an angle in this diagram, can you find another angle by calculation?

Age 11 to 14

ShortChallenge Level

Weekly Problem 40 - 2015

In the diagram, $PT = QT = TS$ and $QS = SR$. What is the value of $x$?

Age 11 to 14

ShortChallenge Level

Weekly Problem 38 - 2017

In the diagram, what is the value of $x$?

Age 11 to 14

ShortChallenge Level

Weekly Problem 44 - 2010

Extend two of the sides of a nonagon to form an angle. How large is this acute angle?

Age 11 to 14

ShortChallenge Level

Weekly Problem 46 - 2015

The diagram shows two parallel lines and two angles. What is the value of x?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Weekly Problem 16 - 2007

Can you figure out how far the robot has travelled by the time it is first facing due East?

Age 11 to 14

ShortChallenge Level

Weekly Problem 30 - 2013

What is the angle $x$ in the star shape shown?

Age 11 to 14

ShortChallenge Level

Weekly Problem 8 - 2008

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

Age 11 to 14

ShortChallenge Level

Weekly Problem 8 - 2016

Can you work out the size of the angles in a quadrilateral?

Age 11 to 14

ShortChallenge Level

What fraction of this square is shaded?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Weekly Problem 9 - 2012

What is the angle QPT in this diagram?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Weekly Problem 40 - 2013

Given three sides of a quadrilateral, what is the longest that the fourth side can be?

Age 11 to 14

ShortChallenge Level

Weekly Problem 18 - 2011

Draw an equilateral triangle onto one side of a square. Can you work out one particular angle?

Age 11 to 14

ShortChallenge Level

Weekly Problem 26 - 2007

The diagram shows two equilateral triangles. What is the value of x?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Weekly Problem 19 - 2017

In the figure, what is the value of x?

Age 11 to 14

ShortChallenge Level

Weekly Problem 37 - 2013

Which of the statements about diagonals of polygons is false?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

Weekly Problem 21 - 2010

How many diagonals can you draw on this square...

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

Weekly Problem 1 - 2006

The diagram shows two circles enclosed in a rectangle. What is the distance between the centres of the circles?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Weekly Problem 21 - 2009

What is the angle between the two hands of a clock at 2.30?

Age 11 to 14

ShortChallenge Level

Weekly Problem 26 - 2006

How many right angled triangles are formed by the points in this diagram?

Age 11 to 14

ShortChallenge Level

Weekly Problem 38 - 2008

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

Age 11 to 14

ShortChallenge Level

Weekly Problem 53 - 2007

The diagram shows a regular pentagon and regular hexagon which overlap. What is the value of x?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Weekly Problem 19 - 2014

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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 ?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

Weekly Problem 47 - 2016

What is the sum of the six marked angles?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

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?

Age 11 to 14

ShortChallenge Level

Weekly Problem 41 - 2014

Three straight lines divide an equilateral triangle into seven regions. What is the side length of the original triangle?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Weekly Problem 7 - 2013

Three of the angles in this diagram all have size $x$. What is the value of $x$?

Age 11 to 14

ShortChallenge Level

Weekly Problem 18 - 2008

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

Age 11 to 14

ShortChallenge Level

Weekly Problem 50 - 2012

The diagram shows a regular dodecagon. What is the size of the marked angle?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Weekly Problem 1 - 2014

The diagram shows a regular hexagon inside a rectangle. What is the sum of the four marked angles?

Age 11 to 14

ShortChallenge Level

Weekly Problem 45 - 2007

What is the obtuse angle between the hands of a clock at 6 minutes past 8 o'clock?

Age 11 to 14

ShortChallenge Level

Weekly Problem 11 - 2014

The diagram shows a parallelogram and an isosceles triangle. What is the size of angle TQR?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

Weekly Problem 47 - 2011

Place equal, regular pentagons together to form a ring. How many pentagons will be needed?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

Weekly Problem 27 - 2007

Ten stones form an arch. What is the size of the smallest angles of the trapezoidal stones?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

Weekly Problem 37 - 2017

A quadrilateral is divided into two isosceles triangles. Can you work out the perimeter of the quadrilateral?

Age 11 to 14

ShortChallenge Level

Weekly Problem 27 - 2013

The diagram shows a parallelogram inside a triangle. What is the value of $x$?

Age 11 to 14

ShortChallenge Level

Weekly Problem 15 - 2012

How many of the five properties can a heptagon have?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

Weekly Problem 13 - 2012

The diagram shows contains some equal lengths. Can you work out one of the angles?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

Weekly Problem 21 - 2012

Two rectangles are drawn in a rectangle. What fraction of the rectangle is shaded?

Age 14 to 16

ShortChallenge Level

Weekly Problem 41 - 2016

The diagram shows a square, with lines drawn from its centre. What is the shaded area?

Age 14 to 16

ShortChallenge Level

Weekly Problem 4 - 2008

In the figure given in the problem, calculate the length of an edge.

Age 14 to 16

ShortChallenge Level

Weekly Problem 27 - 2014

Four congruent isosceles trapezia are placed in a square. What fraction of the square is shaded?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

Weekly Problem 1 - 2011

Use facts about the angle bisectors of this triangle to work out another internal angle.

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

Find the missing distance in this diagram with two isosceles triangles

Age 14 to 16

ShortChallenge Level

Weekly Problem 23 - 2008

A triangle has been drawn inside this circle. Can you find the length of the chord it forms?

Age 14 to 16

ShortChallenge Level

Weekly Problem 29 - 2010

An isosceles triangle is drawn inside another triangle. Can you work out the length of its base?

Age 14 to 16

ShortChallenge Level

Two semicircles overlap, can you find this length?

Age 14 to 16

ShortChallenge Level

Weekly Problem 43 - 2017

The diagram shows a semicircle inscribed in a right angled triangle. What is the radius of the semicircle?

Age 14 to 16

ShortChallenge Level

In the diagram, the radius of the circle is equal to the length AB. Can you find the size of angle ACB?

Age 14 to 16

ShortChallenge Level

Weekly Problem 34 - 2008

What is the area of the region common to this triangle and square?

Age 14 to 16

ShortChallenge Level

Weekly Problem 44 - 2009

A garden has the shape of a right-angled triangle. A fence goes from the corner with the right-angle to a point on the opposite side. How long is the fence?

Age 14 to 16

ShortChallenge Level

Just from the diagram, can you work out the radius of the smaller circles?

Age 14 to 16

ShortChallenge Level

Weekly Problem 8 - 2010

Are you able to find triangles such that these five statements are true?

Age 14 to 16

ShortChallenge Level

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