rings of circles

an application of trigonometry (in right angled triangles)

a ppt is here

the task is to relate the radius of the black circles (in the ring) to the radius of the red (centre) radius

it might be simpler to put the radius of the red circle, k = 1, and find the associated value of r
or vice versus

an intention is to work towards establishing a general rule
so that e.g. if you wanted to find the relationship between the radii for a 50 circle ring you could do so with a formula

surds can be involved for some of the cases

what relationships are there between the radii of the three different sized circles?

prove that r = k

two of Damien Hirst's pictures

slopes of hills

gradients of steep hills
there was a Guinness record update in July 2019

a powerpoint is here

there are some links (in 'hidden' slides) to youtube clips

there's some difference of opinion about whether it is sinA or tanA (where A is the angle of inclination) that is used as a gradient (slope) measure but it seems (appropriately) to be the tangent of the angle: the elevation distance / horizontal distance

Guinness World Records sets out a definition for the steepest street: the maximum gradient over a ten metre span, comparing the vertical rise to the horizontal distance

see the wikipedia entry on grade(slopes) here

the quoted slope on road signs appears to be the steepest section of the road rather than an average slope over the length of it

gradient contexts

slides about the gradients of

• qanads (water channels)
• wheelchair ramps
• roof pitches
• pyramid inclinations
• stadium seating rake

mainly useful to introduce arc tan (rise / run) to obtain the angle of inclination

stairs steepness

the powerpoint is here

this task is based on an idea from Malcolm Swan and Jim Ridgway

Fawn Nguyen writes about using these resources here








some scale drawings of staircases are shown

students write down the letter of the staircases
in order of their steepness (most down to least)

without using any form of measuring device, initially anyway

they then consider how they might measure the steepness to check whether their order is reasonable or not

what could they do if protractors are not available?

perpendicular to the hypotenuse

thanks to 'five triangles' on twitter  (on jan 7th 2016)

'Cairo' pentagon tilings

a tessellation of a single symmetric pentagon,
can have (several) equal sides,
two (opposite) right angles

these come in various forms (have degrees of freedom)

found in Cairo as paving tiles
(not that old)

what relationship is there between the angles if two are right angles and two of the other (obtuse) angles are equal?

what if the 3 obtuse angles are equal?

David Bailey (from Grimsby, England, with a keen interest in recreational mathematics) has undertaken a very thorough analysis of the variety and dates of 'in situ' tiles
so far the oldest dated version he has been able to establish is 1956

due to the two 90 degree angles in the pentagon
the tessellation(s) have a 'skeleton' (as David Wells calls them) of squares

so the variety of this type of tessellation is created by the different angles in the rhombus:

the pentagon tessellation can be viewed with other square 'skeletons'
















or another way
are these squares?

or with isosceles (at least) triangles


what relationship is there for the apex and two equal base angles?














or with trapeziums
what is the relationship here?

















a version of the tiling can be created from the 4, 3, 3, 4, 3 semi-regular tessellation, as a dual (corners of the original tessellation become centres of the related one - and vice versus)

what are the angles in the tiles?

are the tiles congruent?

a tessellation of Cairo-like tiles can be drawn on isometric paper

but...  there are two different types of (non-congruent) pentagon here
establish that their angles are the same

David Bailey quotes Robert H Macmillan's claim that collinearity is a feature of some of the 'Cairo' tessellations and explores this

four of the sides are the same length

what are the angles in each pentagon tile for this arrangement?

use trigonometry to establish the angles in each pentagon for this arrangement

what collinearity is there? 

it is possible to have
all the sides the same length


















what are the angles for this equilateral pentagon tile?















if the triangle formed by joining the apex of the pentagon to the two bottom corners is equilateral

there is a collinearity

the triangle shown is equilateral

find angle 'a'
and establish the (seeming) collinearity property

roof steepness

flat roofs

normal roofs

steep roofs

what are the ranges of base angles?

roof pitches

what are the angles?

pyramid steepness

the Rhind mathematical papyrus