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 |
problem
Stellar angles
Age
11 to 14
Challenge level
Weekly Problem 30 - 2013
What is the angle $x$ in the star shape shown?
What is the angle $x$ in the star shape shown?
problem
Shared vertex
Age
11 to 14
Challenge level
Weekly Problem 38 - 2017
In the diagram, what is the value of $x$?
In the diagram, what is the value of $x$?
problem
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?
Given three sides of a quadrilateral, what is the longest that the fourth side can be?
problem
Bishop's paradise
Age
11 to 14
Challenge level
Weekly Problem 37 - 2013
Which of the statements about diagonals of polygons is false?
Which of the statements about diagonals of polygons is false?
problem
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?
The diagram shows two circles enclosed in a rectangle. What is the distance between the centres of the circles?
problem
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?
How many right angled triangles are formed by the points in this diagram?
problem
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?
Can you figure out how far the robot has travelled by the time it is first facing due East?
problem
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?
A regular pentagon together with three sides of a regular hexagon form a cradle. What is the size of one of the angles?
problem
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?
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?
problem
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?
The diagram shows two equilateral triangles. What is the value of x?
problem
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?
problem
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?
What is the angle between the two hands of a clock at 2.30?
problem
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?
In how many ways can a square be cut in half using a single straight line cut?
problem
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?
Two equilateral triangles have been drawn on two adjacent sides of a square. What is the angle between the triangles?
problem
Distinct diagonals
Age
11 to 14
Challenge level
Weekly Problem 21 - 2010
How many diagonals can you draw on this square...
How many diagonals can you draw on this square...
problem
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?
If you know three lengths and an angle in this diagram, can you find another angle by calculation?
problem
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?
Extend two of the sides of a nonagon to form an angle. How large is this acute angle?
problem
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?
Draw an equilateral triangle onto one side of a square. Can you work out one particular angle?
problem
Square within a square within...
Age
11 to 14
Challenge level
What fraction of this square is shaded?
problem
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?
problem
Isosceles meld
Age
11 to 14
Challenge level
Weekly Problem 9 - 2012
What is the angle QPT in this diagram?
What is the angle QPT in this diagram?
problem
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 $x$?
In the diagram, $PT = QT = TS$ and $QS = SR$. What is the value of $x$?
problem
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?
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?
problem
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?
The diagram shows two parallel lines and two angles. What is the value of x?
problem
Other side
Age
11 to 14
Challenge level
Weekly Problem 8 - 2016
Can you work out the size of the angles in a quadrilateral?
Can you work out the size of the angles in a quadrilateral?
problem
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?
problem
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?
The diagram shows a grid of $16$ identical equilateral triangles. How many rhombuses are there made up of two adjacent small triangles?
problem
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?
In the diagram, VWX and XYZ are congruent equilateral triangles. What is the size of angle VWY?
problem
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?
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?
problem
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?
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?
problem
Angle please
Age
11 to 14
Challenge level
Weekly Problem 19 - 2017
In the figure, what is the value of x?
In the figure, what is the value of x?
problem
Triangle in the corner
Age
11 to 14
Challenge level
A triangle is shaded within a regular hexagon. Can you find its area?
problem
Perimeter in a hexagon
Age
11 to 14
Challenge level
Can you find the perimeter of this triangle inscribed in a hexagon?
problem
Tricky tessellations
Age
11 to 14
Challenge level
Can you work out the fraction of the tiles that are painted black in this pattern?
problem
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?
A square is inscribed in an isoscles right angled triangle of area $x$. What is the area of the square?
problem
Descending angles
Age
11 to 14
Challenge level
Given four of the angles in two triangles, can you find the smallest angle overall?
problem
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?
The lengths SP, SQ and SR are equal and the angle SRQ is x degrees. What is the size of angle PQR?
problem
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?
The 16 by 9 rectangle is cut as shown. Rearrange the pieces to form a square. What is the perimeter of the square?
problem
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?
What is the obtuse angle between the hands of a clock at 6 minutes past 8 o'clock?
problem
Two triangles
Age
11 to 14
Challenge level
Prove that the angle marked $a$ is half the size of the angle marked $b$.
problem
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?
The diagram shows a regular pentagon and regular hexagon which overlap. What is the value of x?
problem
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?
The diagram shows three squares drawn on the sides of a triangle. What is the sum of the three marked angles?
problem
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?
problem
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?
The diagram shows a regular pentagon. Can you work out the size of the marked angle?
problem
Diagonal side
Age
11 to 14
Challenge level
Can you work out the area of a square drawn on a diagonal?
problem
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?
A quadrilateral can have four right angles. What is the largest number of right angles an octagon can have?
problem
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?
How big is the angle between the hour hand and the minute hand of a clock at twenty to five?
problem
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 $x$?
Three of the angles in this diagram all have size $x$. What is the value of $x$?
problem
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?
The diagram shows a regular hexagon inside a rectangle. What is the sum of the four marked angles?
problem
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?
The diagram shows a parallelogram and an isosceles triangle. What is the size of angle TQR?
problem
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?
The diagram shows a rhombus and an isosceles triangle. Can you work out the size of the angle JFI?
problem
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?
Three straight lines divide an equilateral triangle into seven regions. What is the side length of the original triangle?
problem
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 ?
ABCDEFGHI is a regular nine-sided polygon (called a 'nonagon' or 'enneagon'). What is the size of the angle FAE ?
problem
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?
problem
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?
The diagram shows a regular dodecagon. What is the size of the marked angle?
problem
Adding angles
Age
11 to 14
Challenge level
Weekly Problem 47 - 2016
What is the sum of the six marked angles?
What is the sum of the six marked angles?
problem
Two isosceles
Age
11 to 14
Challenge level
Weekly Problem 37 - 2017
A quadrilateral is divided into two isosceles triangles. Can you work out the perimeter of the quadrilateral?
A quadrilateral is divided into two isosceles triangles. Can you work out the perimeter of the quadrilateral?
problem
Clock face angles
Age
11 to 14
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?
problem
Trapezium arch
Age
11 to 14
Challenge level
Weekly Problem 27 - 2007
Ten stones form an arch. What is the size of the smallest angles of the trapezoidal stones?
Ten stones form an arch. What is the size of the smallest angles of the trapezoidal stones?
problem
Clock angle
Age
11 to 14
Challenge level
What is the angle between the the hands of a clock at 8:24?
problem
Overbearing
Age
11 to 14
Challenge level
A village has a pub, church and school. What is the bearing of the school from the church?
problem
Inner rectangle
Age
11 to 14
Challenge level
If the shape on the inside is a rectangle, what can you say about the shape on the outside?
problem
No rights
Age
11 to 14
Challenge level
Prove that the angle bisectors of a triangle can never meet at right angles.
problem
Add all the angles
Age
11 to 14
Challenge level
Find the sum of all of the angles denoted by letters in this diagram
problem
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?
Place equal, regular pentagons together to form a ring. How many pentagons will be needed?
problem
Heptagon has
Age
11 to 14
Challenge level
Weekly Problem 15 - 2012
How many of the five properties can a heptagon have?
How many of the five properties can a heptagon have?
problem
Hexagon cut out
Age
11 to 14
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?
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?
problem
Parallelogram in the middle
Age
11 to 14
Challenge level
Weekly Problem 27 - 2013
The diagram shows a parallelogram inside a triangle. What is the value of $x$?
The diagram shows a parallelogram inside a triangle. What is the value of $x$?
problem
Equal lengths
Age
11 to 14
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?
An equilateral triangle is drawn inside a rhombus, both with equal side lengths. What is one of the angles of the rhombus?
problem
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?
The diagram shows contains some equal lengths. Can you work out one of the angles?
problem
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?
Which of the five diagrams below could be drawn without taking the pen off the page and without drawing along a line already drawn?
problem
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?
The diagram shows a square PQRS and two equilateral triangles RSU and PST. PQ has length 1. What is the length of TU?
problem
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?
The diagram shows a square, with lines drawn from its centre. What is the shaded area?
problem
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?
The diagram shows a semicircle inscribed in a right angled triangle. What is the radius of the semicircle?
problem
Overlapping semicircles
Age
14 to 16
Challenge level
Two semicircles overlap, can you find this length?
problem
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?
problem
Block challenge
Age
14 to 16
Challenge level
Can you work out the shaded area in this shape?
problem
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'?
problem
Square and triangle
Age
14 to 16
Challenge level
Can you find the area of the yellow square?
problem
Doubly isosceles
Age
14 to 16
Challenge level
Find the missing distance in this diagram with two isosceles triangles
problem
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?
Two rectangles are drawn in a rectangle. What fraction of the rectangle is shaded?
problem
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.
In the figure given in the problem, calculate the length of an edge.
problem
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?
A triangle has been drawn inside this circle. Can you find the length of the chord it forms?
problem
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?
An isosceles triangle is drawn inside another triangle. Can you work out the length of its base?
problem
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.
Use facts about the angle bisectors of this triangle to work out another internal angle.
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?
Four congruent isosceles trapezia are placed in a square. What fraction of the square is shaded?
problem
Overlapping annuli
Age
14 to 16
Challenge level
Just from the diagram, can you work out the radius of the smaller circles?
problem
Circled square
Age
14 to 16
Challenge level
Can you find the area of this square inside a circle?
problem
Height and sides
Age
14 to 16
Challenge level
Can you find the area of the triangle from its height and two sides?
problem
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?
problem
Triangular intersection
Age
14 to 16
Challenge level
What is the largest number of intersection points that a triangle and a quadrilateral can have?
problem
Two equilateral triangles
Age
14 to 16
Challenge level
Prove that these two lengths are equal.
problem
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.