Circles
51. Inside, On, or Outside? · Exploring the regions of a circle
P is inside if dist(O,P) < r, on if dist(O,P) = r, outside if dist(O,P) > r.
A point P is inside the circle if its distance from the centre is less than the radius, on the circle if it equals the radius, and outside if it is greater than the radius. Drag P — the indicator switches as you cross the circle's boundary.
What this lesson covers
Try to break it
The circle line itself is the boundary. Points on it are neither inside nor outside.
How you build it
Place points inside, on, and outside the circle.
- Draw a circle.
- Click inside the circle to place a point.
- Click outside the circle to place a point.
- Click on the circle to place a point.
The proof, step by step
Since this is a definition, there's no formal proof. But by exploring the positions, you see why these terms are fundamental to all circle geometry.
- Every point on the circle lies exactly a distance r — the radius — from the centre O.
- For any point P, measure its distance OP from the centre.
- Compare the distance OP with the radius r.
- P is inside when OP < r, on the circle when OP = r, and outside when OP > r.
Worked example
O is the center of a circle with radius 8 cm. Point P is 6 cm from O. Which region contains P?
Since OP (6 cm) < radius (8 cm), P is in the interior.
- Interior — correct
- Exterior
- On the circle
- None