The idea

Equation of a circle (x² + y² = r²)

Why every point of a circle centred at the origin obeys x² + y² = r² (it is Pythagoras), how to read the radius off an equation and describe the graph fully, how to test whether a point is inside, on or outside, how to find the tangent at a point and where it meets an axis, where a line crosses the circle, and how to sketch the circle after a translation.

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Geometry & measures · Circles, arcs & sectors

Equation of a circle (x² + y² = r²)

Why it works

Every point 10 km from the mast

A phone mast stands at the origin

O

of a map grid, with one unit for

1

km. Its signal reaches exactly

10

km in every direction. Where is the edge of the signal, and is the farm at

F(6, 8)

right on it?To get from

O

F

you go

6

across and

8

up. Those two moves are the short sides of a right-angled triangle, and

OF

is its hypotenuse. Pythagoras gives

OF^2 = 6^2 + 8^2 = 100

, so

OF = 10

: the farm is exactly on the edge. Every point of the edge is

10

km from

O

, and together those points make a circle.

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