paperclip

and this one?

concentric circular rings

the powerpoint is here

thanks to David Wells for the first question

lunes

Hippocrates of Chios made some achievements with lunes on route to seeing if they could square the circle

maybe tell students that the semicircles have a side of a triangle as a diameter
maybe wait for them to ask
maybe they will assume that this is the case anyway...

following on from areas of circles (and Pythagoras' theorem) you can set your students two problems that are almost 2500 years old

area for a fixed perimeter

you have a 32 unit long electro rope for enclosing animals
what areas can be enclosed?

oh, by the way, you need to know about a world famous right angled triangle

some suggest that the Ancient Egyptians may have used this triangle to construct right-angles
when building the pyramids

but
there seems little evidence for this view






students could explore the (given or discovered) areas of rectangles with a perimeter of 32

establish that the perimeters of these shapes are all 32

what are their areas?

question (8) needs a calculator (for the circle with a perimeter of 32)

what is a message from this work?

pi development

a history/development of ideas that seems probable

circle: introducing circumference

for a regular baked bean tin size of tin (you need to have one really), which do you think is longer:
twice the height or
the distance round the circumference?

paper rulers work quite well for practical measuring
this version may need scaling
or use graph paper...

how many times do you think the diameter will wrap around the outside of a circle (the circumference)?

a  geogebra applet that links diameter with circumference:

several of these questions are adapted from KS3 SATs questions (England)

'satellite' is from a KS3 SAT paper











a similar question to 'rope around the Earth' appeared in William Whiston's book on the Elements of Euclid, dated 1702
he was an English theologian, historian and mathematician