Shape and Space Problems Extension
To view the problem statement, double click on
the problem title. This will open up the problem in a new window. From there,
you can download the full problemandsolution Word document.
Examples
Page Ref 
Problem
Title 
Objectives
Ref 
Description 
Key Words 
185, 188, 209 
Overlapping
Figures 
Angles, Symmetry,
Similar triangles 
How much overlap
between 2 congruent regular polygons ? 
Congruent,
Corresponding, Interior angle 
123, 185, 247 
Overlapping
Squares 2 
2D Shape, Proof,
Pythagoras, Trigonometry 
Rotate 2 squares
 how much of the lower one can you see? Simple to state, scope to
explore... 
Congruent,
Symmetry, Pythagoras, Surds, Proof 
187,
189 
Parallel
Squares 
2D Shape, Proof, Symmetry,
Congruence 
Draw
squares on the sides of any parallelogram, and join their centres... what
have you got? 
Congruent,
Symmetry, Proof 
187,
199 
A
Conic Journey 
Pythagoras,
2D and 3D Shape 
Find
the shortest route across a rather special cone. 
Pythagoras,
Crosssection, Chord 
187,
189, 199 
A
Taut Rope 
3D
Shape, Pythagoras 
The
old Fly walking around the cuboid problem, redressed (Alternative Nets). 
Pythagoras,
Net 
189 
The
Right Plot 
Pythagoras 
A
rather special plot of land  quite easy. 
Pythagoras 
189 
Construct
A Root 
Pythagoras,
Squares 
Use
ruler and compasses to construct the square root of 29  several ways... 
Pythagoras,
Constructions, Surds 
91, 127, 189 
The
Magic of Pythagoras 
Pythagoras 
This Problem is a variation on the Well in the
Courtyard problem... and has a lovely, surprising answer as unwanted
terms 'cancel out' ! 
Pythagoras,
Subject of the formula 
189, 219 
Quickest
Route 
Coordinate
distances, Pythagoras 
Jeeves
needs to escape across a swimming pool  but what's his quickest escape
route ? 
Pythagoras 
189 
Hero
Triangles 
Pythagoras 
Putting
two Pythagorean triples together to form an integerarea Hero triangle 
Pythagoras 
193 
Lego
Triangles 
Similarity 
Making
similar triangles from 6 pieces of Lego 
Ratio,
Similar 
193 
Quad
Parks 
Similarity,
Proof 
Show
that the midpoints of any quadrilateral form a parallelogram 
Proof,
Similar 
193 
Diagonal
Cuts 
Similarity,
Proof 
Similar
triangles show how a diagonal line is trisected 
Proof,
Similar 
193 
Inner
Triangle 
Similarity,
Proof 
Using
similar triangles to find the largest area of a triangle drawn within a
triangle 
Proof,
Similar 
193 
Diamond
Ring 
Similar
Triangles 
Find the size of a circle inside a Rhombus 
Similar 
193,
235, 237 
Shortest
Half 
2D
Shape,
Dynamic Geometry 
What
is the most efficient way to divide an equilateral triangle into two equal
areas ? 
Arcs,
Sectors, Scale Factor 
193, 215, 234 
Sculpt
Big 
Ratio,
Enlargement, Similarity, Cylinders 
A sculptor
chooses between similar large and small designs, in order to maximize
profits... 
Scale
factor, Volume, Enlarge, Proportional 
193 
Intersecting
Chords 
Similarity,
Angles in a Circle 
The
simplest of theorems using 'equal angles on an arc' 
Similar,
Corresponding, Arc, Chord 
193 
Tunnel
Vision 
Similar
Triangles 
Find the width of a tunnel given the size of the truck going through it 
Similar 
193 
Paper
Crease 
Similar
Triangles 
Find the length of the crease when you fold a piece of paper over 
Similar 
193 
Circle
Transversals 
Similarity,
Angles in a Circle 
An
interesting property of a circle and two lines, with an unexpected degree
of freedom in the construction... 
Arc,
Similar, Construction 
195, 235, 247 
Polygon Pi
Approximations 
Polygons,
Circles, Trigonometry 
Using Archimedes' method of Regular Polygons to find
increasingly good estimates of Pi  plus some Trial and
Improvement practice.

Limit,
Sin, Tan, Opposite etc Arc, Chord, Pi 
197 
The
Security Cameras 
Angles
In A Circle 
Angles
at the centre of a circular Art gallery room  an application of the
circle Theorems! 
Arc,
Circles 
197 
Four
Corners 
Angle
in a SemiCircle 
Two
rectangles overlap  can you see which groups of points lie on common
circles? 
Semicircle,
Circles 
197 
CircumCircle 
Angles
In A Circle 
Finding the radius of a circumcircle turns out to be a treat 
Arc,
Circles 
197, 221 
Altitudes
and Orthocentres 
Constructions,
Loci, Dynamic Geometry 
A 'dynamic
geometry' investigation 
Constructions,
Locus, Perpendicular 
201 
Facing Up
To Football 1 
Properties of 3D shape 
Use the unfamiliar idea of angle sums at a
vertex to deduce how many vertices an Icosahedron has, and extend...! 
Vertex, Angle 
201 
Facing Up
To Football 2 
Properties of 3D shape 
Use the now familiar idea of angle sums at a
vertex to deduce how many faces a Truncated Icosahedron (football) has, and extend...! 
Vertex, Angle 
155, 201, 281 
Ringing
The Changes 
2D
Representation of 3D Shape 
Working through
permutations of 4 'bells', using systematic sequencing. Has a very
beautiful solution, modelled as the vertices of a truncated Octahedron ! 
Plane
projection, Vertex, Edge 
213 
Paper
Sizes 
Ratio,
Enlargement 
A0, A1 etc, then
into 3D for a 'Golden Cuboid' 
Scale
factor, Enlarge, Proportional 
219 
Points
In Between 
Coordinates 
Finding
midpoints and points of trisection, using 'weighted average'
coordinates, and hence finding the 'centroid' 
Coordinates,
Line, Midpoint, Average, Graphs 
221, 227 
Centroids 
Triangles, Constructions,
Loci, Dynamic Geometry 
A 'dynamic
geometry' investigation 
Constructions,
Locus, Median 
221, 227 
Angle
Puzzle 
Find
Locus, Constructions 
As the angles of a triangle vary, find the minimum area 
Constructions,
Locus, Proof 
221 
Circle
Tangents 
Circles, Dynamic
Geometry, Proof 
An unexpected
property of circles and their tangents 
Constructions,
Tangents, Inscribed circle 
223, 245 
Height
of the Tower 
Similar Triangles,
Scale Drawing, Trigonometry 
There's a tower,
see, across this river, and what you've got to do is... 
Trigonometry,
Similarity, Elevation 
225 
The
OutofTown Store 
Loci,
Constructions 
Find
the point within any triangle which minimises the total distance from each
vertex...( the Fermat point) 
Constructions,
Proof, Locus 
227,
197 
The
Down and Out Sponge 
Loci,
Constructions 
What is the area that a semicircular sponge wipes in the corner of a window ? 
Locus,
Angles In A Circle, Region 
227, 133, 235 
Overlapping
Squares 1 
Loci, Graphs, 2D
Shape 
This problem offers an
element of surprise in that the locus of possible solutions isn’t the
straight line that pupils may well expect 
Region,
Proof, Locus 
233, 91 
A
Walk In The Bush 
Fractions,
Measurements 
An
'average speed' problem that comes out very sweetly  involving some up
and down hills 
Speed,
Average 
239, 120 
Simpson's Rule and
the Volume of a Sphere 
Volumes 
This exercise leads the pupil through applications of
Simpson’s rule to the formula for the volume of a
sphere. It is intriguing that all these volumes are
given exactly by the rule. 
Right
prisms, Volume, 
237 
Target
Practice 
Circles 
The
middle ring of a circular target has a simple area... 
Pi,
Proportion 
237 
What
Size are the Cylinders 
Volumes
and Enlargement 
Two similar cylinders come from a block of material  how big is each? 
Ratio, Enlargement, Volume, Scale 
239 
Candy
Floss 
Volume
of Cylinders 
How long is a thread of Candy Floss spun from a block of sugar? 
Volume 
247 
Square
in a Triangle 
Trigonometry 
What's the
largest square that can be drawn inside a regular triangle ? This
problem would benefit from the use of the Sine Rule. 
Sin,
Hypotenuse etc 
