TwiddleLab / Math / Shapes and Proof
Circles and Angles
What you will learn
- An inscribed angle (corner on the circle) is half the central angle on the same arc.
- So every point on the same arc sees the chord at the same angle.
- On a diameter the arc is 180°, so the angle at the rim is 90°.
How the lesson goes
A player sees the goal at 50°. Where else can they run and still see exactly 50°?
Draw the path with your finger or mouse.
Hunt for 50° spots.
Drag the player. Each new spot at exactly 50° leaves a green dot. Your path is dotted orange.
Why a circle? Compare the angle at the rim with the angle at the center.
Slide the arc, drag P anywhere on the circle, and Record three different arcs.
Two angles on one arc.
The central angle has its corner at the center O. The inscribed angle has its corner on the circle.
On the same arc, inscribed = half of central. That's why every rim point sees the arc at the same angle.
Your player ran on such an arc: the goal is the chord, and every spot sees it at 50°.
Make the angle at P exactly 90°.
Drag B around the circle.
Game: penalty spots.
Place three defenders so each sees the goal at exactly 40°. The angle shows only when you let go. Fewest measures wins.
Find the center of a round plate with just a sheet of paper.
Slide the paper's corner along the rim, draw the line between the two dots, then do it again somewhere else.
The central angle is 130°. What is the inscribed angle on the same arc?
An inscribed angle is half the central angle on the same arc: 130° ÷ 2 = 65°.
One more use: cameras around a stage.
The stage is a diameter. Tap three spots on the half circle to mount cameras.
Previous lesson: Proof as a Chain · Next lesson: Area by Cutting and Sliding · Part of the course: Shapes and Proof
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