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A quadrilateral ABCD is inscribed within a circle. AD crosses the centre of the circle ∠BAC = 26o ∠ DBC = 50o What are the internal angles of the quadrilateral ABCD?

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If θ = δ - 10o find the angles. (assume red triangles are similar, larger triangle sides are tangents to the circle and yellow shape is a rectangle)

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a tessellation of a single symmetric pentagon,
sometimes with (several) equal sides,
two (opposite) right angles
comes in various forms

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 remaining 3 of them 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 tessellations is created by the different angle 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)

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 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 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

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a tessellation of a single symmetric pentagon, sometimes with (several) equal sides, two (opposite) right angles comes in various forms 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?

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ideas and resources for mathematics teachers of 10 to 16 year olds

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Someone tweeted me this puzzler a couple of days ago (I forget who it was, apologies):

"If the square has side 2cm, what is the radius of the circle?"

I had a stab at it today and thought I'd share my process, which may well be awful, but as Bob Hoskins famously didn't quite say, "it's good to share"

My first thought was "it's a circle, I need a radius, where can I get one". So I drew in a few lines:

I know that where the dotted diagonal lines cross is the centre of the circle, and I drew in the bold black line because I know I can calculate it, which might come in handy.

I thought I better have a visual reminder of what lengths are equal, in the hope it might prompt a new thought:

Then I just stared at it for a while.

I couldn't visualise how calculating the black diagonal line would help me, even though I could do it. So I drew in another line as I thought about it:

Which helped me spot the blue arrow thing which kicked in a thought about circle theorems. At this point, note I didn't have a direction I was going in, I was just trying stuff out. Testing the waters to see where it's shallow. I thought about the circle theorem thing, and eventually decided against it, and ditched the idea. However, looking at the diagram with the two angles on it made me realise the direction I wanted to take:

I want to work with this triangle, because it has the radius (twice, bonus) as its side length, and I can use the length that I know I can calculate (the longest side of the green triangle, which was mentioned in the first diagram). So now all I need is the angles inside the green triangle and off I go. And to get those, I can use the big red bottom left triangle:

So now I have some ugly trig to work through:

[/insert elevator music whilst ugly trig is being worked through]

and out pops r = 1.25 cm.

The nice answer makes me suspect there's a far nicer approach to this. I'll ponder it some more.

*Update: A few people have sent me their approaches, which are predictably simpler than mine. Interestingly, some are asking why I 'chose to solve it in this complicated way'. I didn't choose to! This is where my first thoughts took me. Anyway, here are some alternative approaches:

And from the comments section:

"If you let the height of the square be r + h, where r is the radius of the circle and the width of the square to be 2w (w = 1), you can form a triangle with sides w and h and hypotenuse r. From here do a little Pythagoras and the answer of 1.25 drops out quite quickly. "

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Someone tweeted me this puzzler a couple of days ago (I forget who it was, apologies): "If the square has side 2cm, what is the radius of the circle?" I had a stab at it today and thought I'd share my process, which may well be awful, but as Bob Hoskins famously didn't quite say,...

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Too often is the labelling of a circle limited to this:

I suspect this plays a part in students struggling to solve geometry problems further down the line. They expect lines in circles to follow this format.

So I made a new activity to work alongside the above picture:
File is here : Radius Diameter
This entry was posted in Circles, Properties of Shape, Resources, Shape. Bookmark the permalink.

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Too often is the labelling of a circle limited to this: I suspect this plays a part in students struggling to solve geometry problems further down the line. They expect lines in circles to follow this format.

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DrFrostMaths.com is a huge bank of free educational resources for teaching mathematics, with full sets of slides, worksheets, games and assessments that span Year 7 to Further Maths and enrichment resources with a Maths Challenge/Olympiad focus. We are working hard on a new platform for setting, building and monitoring homework.

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Angles

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This page lists recommended resources for teaching geometry and measure at Key Stage 3/4 organised by topic.

Huge thanks to all individuals and organisations who share teaching resources.

Quick links: Loci | Constructions | Compound Measures | Similarity | Congruence | Angle Basics | Angles in Parallel Lines | Bearings | Angles in Polygons | Quadrilaterals | Circle Theorems | Perimeter & Area | Circles | Volume & Surface Area | Real-Life Graphs | Scales and Units | 3D Drawing | Coordinates | Symmetry | Shape Transformations | Enlargement | Pythagoras | Trigonometry | Bounds | Vectors | Geometric Proof | Revision
Loci

Quadrilateral & Polygon Properties

Circles - Circumference and Area

Non right angled

Graphs and exact values

Geometric Proof and Reasoning

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This page lists recommended resources for teaching geometry and measure at Key Stage 3/4 organised by topic. Huge thanks to all individuals and organisations who share teaching resources.

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Circle Theorems

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Maths Teaching Professional Development Web Site