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 problem-and-solution 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 |
2-D 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 |
2-D 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,
2-D and 3-D Shape |
Find
the shortest route across a rather special cone. |
Pythagoras,
Cross-section, Chord |
187,
189, 199 |
A
Taut Rope |
3-D
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 |
Co-ordinate
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 integer-area 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 |
2-D
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 Semi-Circle |
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 3-D 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 3-D 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 |
2-D
Representation of 3-D 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 |
Co-ordinates |
Finding
mid-points and points of trisection, using 'weighted average'
co-ordinates, and hence finding the 'centroid' |
Co-ordinates,
Line, Mid-point, 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
Out-of-Town 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, 2-D
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
Simpsons 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 |
|