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

### Square Bisection

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

### Diagonal Division

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

### Overlapping beer mats

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

problem

### Half Past Two

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

### Long Shadows

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

### Two Exterior Triangles

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

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

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

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

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

### Angular Reflection

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

### As Long as Possible

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

### Regular Vertex

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

problem

### Bishop's Paradise

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

### Tent Poles

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

### Central Distance

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

### Parallel Base

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

### Right-Angled Request

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

### Other Side

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

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

problem

### Robo-Turn

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

### Isometric Rhombuses

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

### Polygon Cradle

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

### Equilateral Pair

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

### Angle of Overlap

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

### Stacking Shapes

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

### Outside the Boxes

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

### Triangle in the corner

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

problem

### U in a Pentagon

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

### Right Angled Octagon

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

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

### Tricky Tessellations

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

problem

### Descending Angles

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

problem

### Triangle Split

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

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

### Perimeter Puzzle

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

problem

### Nonagon Angle

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

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

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

### Fangs

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

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

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

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

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

### Square in a Triangle

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

### Six Minutes Past Eight

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

### Hexapentagon

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

### Two Isosceles

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

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

problem

### Pentagon Ring

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

### Overbearing

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

problem

### inner rectangle

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

problem

### Hexagon Cut Out

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

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

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

### Trapezium Arch

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

### Centred

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

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

### Diagonal Touch

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

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

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

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

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

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

### Internal - External

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

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

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

### Octagonal Ratio

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

problem

### Triangular intersection

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

problem

### Semicircle in a Triangle

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

problem

### Folded Square

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

problem

### Griddy Region

Weekly Problem 34 - 2008

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

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

problem

### Garden Fence

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?

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?

problem

### Altitude Inequalities

Weekly Problem 8 - 2010

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

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