Angles, Polygons and Geometrical Proof Short Problems: Age 11-14

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.

We have put together a selection of these short problems as printable worksheets:

Printable worksheets: Age 11-14 ⭐   Printable worksheets: Age 11-14 ⭐⭐   Printable worksheet solutions


  • Small pepper seedlings in turquoise pots.
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    Isosceles Meld

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Long Shadows

    Age
    11 to 14
    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?

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    Angular Reflection

    Age
    11 to 14
    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?

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    Stellar Angles

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    As Long as Possible

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Bishop's Paradise

    Age
    11 to 14
    Challenge level

    Which of the statements about diagonals of polygons is false?

  • Small pepper seedlings in turquoise pots.
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    Central Distance

    Age
    11 to 14
    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?

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    Right-Angled Request

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Robo-Turn

    Age
    11 to 14
    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?

  • Small pepper seedlings in turquoise pots.
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    Polygon Cradle

    Age
    11 to 14
    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?

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    Angle of Overlap

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Square Bisection

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Diagonal Division

    Age
    11 to 14
    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?

  • Small pepper seedlings in turquoise pots.
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    Half Past Two

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Two Exterior Triangles

    Age
    11 to 14
    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?

  • Small pepper seedlings in turquoise pots.
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    Distinct Diagonals

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Angle Hunt

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Outside the Nonagon

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Homely Angles

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Regular Vertex

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Tent Poles

    Age
    11 to 14
    Challenge level

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

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    Parallel Base

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Other Side

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Triangle in a Corner

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Isometric Rhombuses

    Age
    11 to 14
    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?

  • Small pepper seedlings in turquoise pots.
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    Equilateral Pair

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Stacking Shapes

    Age
    11 to 14
    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?

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    Angle Please

    Age
    11 to 14
    Challenge level

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

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    Shared Vertex

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Overlapping Beer Mats

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Fraction of a Square

    Age
    11 to 14
    Challenge level

    What fraction of this square is shaded?

  • Small pepper seedlings in turquoise pots.
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    Nonagon Angle

    Age
    11 to 14
    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 ?

  • Small pepper seedlings in turquoise pots.
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    Integral Polygons

    Age
    11 to 14
    Challenge 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?

  • Small pepper seedlings in turquoise pots.
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    Dodecagon Angles

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Square in a Triangle

    Age
    11 to 14
    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?

  • Small pepper seedlings in turquoise pots.
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    Six Minutes Past Eight

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Hexapentagon

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Outside the Boxes

    Age
    11 to 14
    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?

  • Small pepper seedlings in turquoise pots.
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    U in a Pentagon

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Right Angled Octagon

    Age
    11 to 14
    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?

  • Small pepper seedlings in turquoise pots.
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    Handy Angles

    Age
    11 to 14
    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?

  • Small pepper seedlings in turquoise pots.
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    Triangle Split

    Age
    11 to 14
    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?

  • Small pepper seedlings in turquoise pots.
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    Rectangle Dissection

    Age
    11 to 14
    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?

  • Small pepper seedlings in turquoise pots.
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    Fangs

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Inscribed Hexagon

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Extended Parallelogram

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Rhombus Diagonal

    Age
    11 to 14
    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?

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    Radioactive Triangle

    Age
    11 to 14
    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?

  • Small pepper seedlings in turquoise pots.
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    Adding Angles

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Triangle in the Corner

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Funky Hexagon

    Age
    11 to 14
    Challenge level

    Can you find the value of x in this diagram?

  • Small pepper seedlings in turquoise pots.
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    Perimeter in a Hexagon

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Tricky Tessellations

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Descending Angles

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Two Triangles

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Perimeter Puzzle

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Diagonal Side

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Pentagon Ring

    Age
    11 to 14
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Heptagon Has

    Age
    11 to 14
    Challenge level

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

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