MathGrade 9ages 14–15
◐ Inscribed angle
An inscribed angle sitting on a circle's edge is always exactly half of the central angle that leans on the same arc — the opposite of what many students expect.
Try this lesson freeNo card needed. The first lesson is free.
Download on the App StoreWhat your child learns
- A central angle sits at the centre of the circle.
- An inscribed angle sits on the edge and leans on the same arc.
- The inscribed angle is half the central one, because its corner is further from the arc and so sees it through a narrower opening.
- All the inscribed angles on the same arc are equal to each other.
The mistake most kids make here
Flipping the relation and thinking the inscribed angle is the bigger one.
Try one question
On the same arc, which is bigger, the central angle or the inscribed one?
Central against inscribed
- the central one
- the inscribed one
- they are equal
Answer
the central one — That's it. The central angle is twice the inscribed one.
How Maya teaches it
Your child tells Maya what they did in class. She teaches this topic at her desk, asks questions with tappable answers, and when the answer is wrong she explains why that answer was tempting instead of saying “try again”. She remembers what did not land and comes back to it.
Your child has it when they can: Say the inscribed angle when the central one is 80.
How to help at home
A central angle has its corner at the middle of the circle; an inscribed angle has its corner somewhere on the circle's edge. Both can "look at" the same arc, but from very different distances — and that distance is exactly why their sizes differ.
Draw a circle with a compass or trace a round lid. Mark the center, then draw a central angle opening onto an arc. Draw a second angle from a point on the edge, opening onto that same arc. Measure both with a protractor if you have one.
When your child guesses the inscribed angle is bigger (maybe because it looks stretched out from the edge), have them actually measure both angles from the drawing. Seeing the numbers — one exactly half the other — is more convincing than being told the rule.
A sign they've got it: given a central angle's measure, they can immediately halve it to get any inscribed angle on the same arc, without needing to measure again.
Questions parents ask
- Why is the inscribed angle always half the central angle?
- The inscribed angle's corner sits farther from the arc than the center does, so it 'sees' the same arc through a narrower opening — exactly half as wide, every time.
- Are all inscribed angles on the same arc equal to each other?
- Yes. No matter where on the remaining edge of the circle you place the inscribed angle's corner, as long as it leans on the same arc, it measures the same.
- What if the central angle is 180 degrees?
- Then the arc is a full half-circle, and any inscribed angle on it will be 90 degrees — a fact used often to spot right angles inside circles.
- How does Maya teach inscribed and central angles?
- Maya has the child draw and measure both angles on the same arc before stating the half-relationship, so the rule follows from something they've seen, not just been told.
No card needed. The first lesson is free.
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