Circles and Nets

Circumference and Diameter #

Edge and width always keep the same ratio.

Every circle is the same shape, so its edge and its width always keep the same ratio

The line from the middle out to the edge is called the radius.

All the way across is the diameter. It is always twice the radius: d = 2r.

The edge — the circumference — is a little over 3 diameters. That number is pi.

So the circumference is exactly pi diameters: $C=\pi d$C = πd. And d is 2r, so $C=2\pi r$C = 2πr.

Area of a Circle #

Pi lots of the square built on the radius.

Cutting a circle into wedges and relaying them makes a rectangle of pi r by r

A circle is every point the same distance from the center. That distance is r.

Cut it into wedges. Nothing is added or thrown away, so the area is unchanged.

Lay them tip up, tip down. Cut more wedges and the top edge straightens out.

Half the arcs make the top: half of $2\pi r$2πr is $\pi r$πr wide, r tall. Area: $\pi r\times r=\pi r^2$πr × r = πr².

Nets #

The solid unfolded flat.

A net is the solid unfolded flat, so its faces are the solid faces

A cube has 6 faces, but from here you can only spot 3 of them.

Open it out. The gold lid swings clear, and all 6 lie flat to count.

A pyramid unfolds differently: 1 square base and 4 triangles.