Same Math › Geometry · grades 9–10 › Circles: arcs, angles and equations
How do angles and arcs in a circle work?
In a circle, a central angle (vertex at the center) equals the arc it cuts off. An inscribed angle (vertex on the circle) is half the arc it cuts off. So an 80° arc makes an 80° central angle and a 40° inscribed angle. On a coordinate grid, a circle with center (h, k) and radius r has the equation (x − h)² + (y − k)² = r².
- Arc: measured in degrees, like an angle.
- Central angle: equals the arc.
- Inscribed angle: half the arc.
Why schools teach it
The rules that unlock circle problems
Geometry students learn the relationships among central, inscribed and circumscribed angles (HSG.C.A.2) and derive the equation of a circle from the Pythagorean theorem (HSG.GPE.A.1).
Most circle questions are one of these two angle rules wearing a costume. One consequence kids love: an angle inscribed in a semicircle is always 90°.
At the kitchen table
What to say, and what to skip
Say this
Where's the vertex of the angle?
At the center or on the circle?Which arc does the angle cut off?
Trace it with your finger.Equal or half?
In the equation, what are h, k and r?
Watch the signs: (x + 3) means h = −3.
Skip this
- Treating an inscribed angle like a central one.On the circle means half.
- Reading the center's signs wrong.(x − 2)² + (y + 5)² = 9 has center (2, −5).
- Using r² as the radius.If r² = 9, the radius is 3.
See it in action
An inscribed angle cuts off a 130° arc
Vertex is on the circle
inscribed angleTake half the arc
130° ÷ 2Answer
65°
A central angle cutting off the same arc would be 130°.
Free practice sheet
Make a circles sheet
Loading practice sheet…
Fridge cheat sheets
Math Fridge Cheat Sheets
Every K–8 math method in one printable book (159 pages): a cheat sheet page like this guide, plus a practice sheet and its answer key for each, in Classic and handwritten Notebook styles. $9 one time, or all 7 subjects for $29.99. See all prices.
Teacher? This method has a classroom pack on TeachersPayTeachers: an anchor chart, a parent letter and practice sheets with answer keys, $4. See the pack
Parents: tell your child's teacher about this guide
Parents also ask
How many degrees are in a whole circle?
360°. A semicircle is 180°.
What's a major arc?
The long way around, more than 180°. It's named with three letters, like arc ACB.
What if the equation isn't in that form?
Complete the square on x and on y to get it there.
Is there a rule for two chords crossing?
Yes: the angle is half the sum of the two arcs. It's usually taught right after these.