Shape and Space Problems Year 9
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 
169, 183, 193 
Paper
Folding 
Similarity,
Gradients 
A
square is simply folded over and then opened out  what does the crease look like ? 
Similar,
Slope, Gradient 
185, 188, 209 
Overlapping
Figures 
Angles, Symmetry,
Similar triangles 
How much overlap
between 2 congruent regular polygons ? 
Congruent,
Corresponding, Interior angle, Isometric 
185 
P
Where You Like 
Properties
of Triangles 
Prove that the sum of distances from any point in an equilateral triangle to the sides is constant 
Proof,
Triangle 
185, 187 
LShape 
Area, Proof 
How to divide any
2rectangle 'composite' in half? 
Symmetry,
Proof, Bisect 
123, 185, 247 
Overlapping
Squares 2 
2D Shape, Proof,
Trigonometry,
Surds 
Rotate 2 squares
 how much of the lower one can you see? Simple to state, scope to
explore... 
Congruent,
Symmetry, Pythagoras, Surds, Proof 
187,
199 
A
Conic Journey 
Pythagoras,
2D and 3D Shape 
Find
the shortest route across a rather special cone. 
Pythagoras,
Crosssection, Chord 
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 
127,
189 
Truncated
Square 
Pythagoras 
Another
'notenoughinformationsurely?' problem that comes out nicely when
Pythagoras is liberally applied ... 
Pythagoras,
Subject of the formula 
189 
Double
Squares 
Pythagoras 
A
surprisingly simple trick to writing numbers as the sums of squares  a
mustsee ! 
Pythagoras 
189, 219 
Quickest
Route 
Coordinate
distances, Pythagoras 
Jeeves
needs to escape across a swimming pool  but what's his quickest escape
route ? 
Pythagoras 
191, 193, 217 
Triangle
Dissections 
Congruence,
Similarity, Enlargement 
A
series of puzzles concerned with dissecting equilateral triangles into
smaller parts 
Congruent,
Similar 
191 
The
Radius 
Congruence,
Similarity 
A
simple application of Similar triangles 
Congruent,
Similar 
193,61, 81,
215 
Pendants 
Similarity,
Enlargement 
'Area Factors'
under enlargement give quick answers to this problem 
Similar,
Enlargement, 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,
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 
200, 207 
TetraCubes 
3D
Shape 
Identify
shapes, then use to build mini
Soma Cubes 
Plan,
View, Symmetry 
201 
Sliced
Cube 
3D
Shape, Pythagoras 
Visualize,
then prove, a result about a bisected cube. 
CrossSection,
Plane, Properties 
161, 165, 203 
Graphic
Convergence 
Coordinates,
Mappings, Combination of Transformations 
A
pair of mappings for x and y lead to a convergent sequence of points,
which can also be viewed as a combination of transformations  quite
pretty. 
Map,
Invariant 
213 
Square
in a Triangle 
Transformations,
Symmetries 
How
to squeeze the largest possible square inside any triangle? 
Enlargement 
215, 221 
Forensic
Triangles 
Constructions,
Dynamic Geometry 
Reconstruct the
original triangle from the sides' midpoints 
Medians,
Similar, Parallel 
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 
Eggs 
Constructions,
Compasses 
First
take
an egg... 
Compasses, Perpendicular, Arc, Tangent 
221 
Triangle
in a Square 
Constructions,
Dynamic Geometry 
How
to squeeze the largest possible equilateral triangle inside a square? 
Compasses,
Rotation 
223, 245 
Height
of the Tower 
Similar Triangles,
Scale Drawing 
There's a tower,
see, across this river, and what you've got to do is... 
Similarity,
Elevation 
133, 227 
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 
81, 91, 233 
Average
Speed 
Speed, Ratio
+ Proportionality 
A nice
introduction to ratio methods for combining average speeds over 2 sections
of a journey 
Speed,
Average 
91,
233 
A
Walk In The Bush 
Fractions,
Measurements 
An
'average speed' problem that comes out very sweetly  involving some up
and down hills 
Speed,
Average 
235 
How
Deep Is The Well 
Circles 
Fairly
straightforward circumference calculations 
Pi,
Circumference 
237 
Pie
Free Circles 
Circles,
Area 
A
section of a circle turns out to have an area independent of Pi 
Pi,
Arc, Pythagoras 
237 
Snake
Eyes 
Circles,
Pythagoras 
A circular area
that turns out not to involve Pi ! 
Pi,
Hypotenuse, Pythagoras 
237 
Target
Practice 
Circles 
The
middle ring of a circular target has a simple area... 
Pi,
Proportion 
237 
Loo
Roll Emergency 
Circles 
When
the loo roll looks halfsize, how much is really left? 
Pi,
Circle, Area 
35, 237 
Wiggly
Paths 
Circles 
Surprisingly
pretty result about the area of winding pathways...(cf P.35 Garden Path) 
Pi,
Radius, Arc 
237 
Square
Peg, Round Hole 
Circles 
Which
fits bettera square peg in a round hole, or a round peg in a...? 
Pi 
155,
237 
Pick
A Shape 
Generate
sequence, Find nth term 
Pick's
Theorem for Areas on a dotty grid 
Formula,
Generate Tn, Proof 
