Angles, Polygons and Geometrical Proof Short Problems: Age 14-16

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 14-16.

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

Printable worksheets: Age 14-16 ⭐⭐   Printable worksheet: Age 14-16 ⭐⭐⭐   Printable worksheet solutions


  • Small pepper seedlings in turquoise pots.
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    Centred

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Eulerian

    Age
    14 to 16
    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?

  • Small pepper seedlings in turquoise pots.
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    Cat on a Wall

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Diagonal Touch

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Two Right Angles

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Angle to Chord

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Isosceles Reduction

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Incentre Angle

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
    problem

    Quarters

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Internal - External

    Age
    14 to 16
    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?

  • Small pepper seedlings in turquoise pots.
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    Shaded Square

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Inscribed Semicircle

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
    problem

    Overlapping Semicircles

    Age
    14 to 16
    Challenge level

    Two semicircles overlap, can you find this length?

  • Small pepper seedlings in turquoise pots.
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    Octagonal Ratio

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Block Challenge

    Age
    14 to 16
    Challenge level

    Can you work out the shaded area in this shape?

  • Small pepper seedlings in turquoise pots.
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    3-4-5 Circle

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Slide Height

    Age
    14 to 16
    Challenge level

    How high is the top of the slide?

  • Small pepper seedlings in turquoise pots.
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    Tunnel Vision

    Age
    14 to 16
    Challenge level

    How wide is this tunnel?

  • Small pepper seedlings in turquoise pots.
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    Square and Triangle

    Age
    14 to 16
    Challenge level

    Can you find the area of the yellow square?

  • Small pepper seedlings in turquoise pots.
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    Doubly Isosceles

    Age
    14 to 16
    Challenge level

    Find the missing distance in this diagram with two isosceles triangles

  • Small pepper seedlings in turquoise pots.
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    Circles on a Triangle

    Age
    14 to 16
    Challenge level

    Find the missing length on this diagram.

  • Small pepper seedlings in turquoise pots.
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    Griddy Region

    Age
    14 to 16
    Challenge level

    Weekly Problem 34 - 2008
    What is the area of the region common to this triangle and square?

  • Small pepper seedlings in turquoise pots.
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    Garden Fence

    Age
    14 to 16
    Challenge 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?

  • Small pepper seedlings in turquoise pots.
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    Altitude Inequalities

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Cut-Up Square

    Age
    14 to 16
    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?

  • Small pepper seedlings in turquoise pots.
    problem

    Overlapping Annuli

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
    problem

    Circled Square

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Height and Sides

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Circular Inscription

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Triangular Intersection

    Age
    14 to 16
    Challenge level

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

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

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Folded Square

    Age
    14 to 16
    Challenge level

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

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