Perfect Round Shapes

Algebra 2 · Axiom Academy

What makes a circle perfectly round? One rule — every point the same distance from a center — and the equation (x-h)^2+(y-k)^2=r^2 that captures it. A circle isn't just a round blob — it's every point that sits exactly the same distance from one fixed center. That single rule, the one a compass obeys, is the seed of every conic section you're about to meet. Watch the compass arm turn around the center. Its length never changes, so every point it leaves behind is exactly r units from the center — and together they trace a perfect circle. Keep an eye on the distance to center readout: it never budges off r . A circle is the set of all points at distance r from the center — the compass, written as algebra. Move the center, stretch the radius Drag the center dot, drag the radius handle, or use the sliders. The circle and its equation move together — h and k are the center's coordinates, r is the radius. The algebra isn't separate from the picture; it is the picture, written down. Change h , k or r and the equation rewrites itself in lockstep — the same three numbers live in both. A point lies on the circle exactly when its distance from the center equals r — which is exactly when it makes the equation true. Drag the test point around the circle (x-2)^2+(y-1)^2=9 : watch its distance d , and whether (x-2)^2+(y-1)^2 lands on 9 . Let go near the ring and it snaps on.

This is the written version of the interactive lesson above. See the full Algebra 2 course.