A central angle has its vertex at the centre, and it equals the arc it cuts off.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
An inscribed angle is half
An angle with its vertex on the circle is half the central angle on the same arc.
A useful consequence
An angle inscribed in a semicircle is 90°, because the arc is 180°.
Arc and sector are fractions
A 90° slice is a quarter of the circle, so its arc is a quarter of the circumference.
Everything is a fraction of the whole
An arc is a fraction of the circumference and a sector is the same fraction of the area, with the fraction given by the central angle over 360°. One idea covers both formulas, so neither needs separate memorising.
Radians make the fractions vanish
Measuring the angle in radians gives arc length = rθ and sector area = ½r²θ, with no fraction at all. That simplification is the reason radians exist, and it is why calculus uses them exclusively.
Arc length and arc measure differ
The measure of an arc is its central angle in degrees; its length is a distance. Two circles of different sizes can have arcs of the same measure and very different lengths. Keeping the two apart avoids a persistent confusion.
Segments need a subtraction
A segment is the region between a chord and its arc, found by taking the sector and subtracting the triangle. It is the standard composite-area strategy, applied to a curved figure.
Step 2: Try It Yourself
Tap and try it out.
- Diameter10 cm
- Circumference31.42 cm
- Area78.54 cm²
- Circumference ÷ diameter3.14
Circumference is a length, measured in cm. Area is a coverage, measured in cm². The units tell them apart.
Step 3: Watch an Example
One step at a time.
Watch Sam Find a Sector Area
A circle of radius 6 has a 60° sector. Sam finds its area.
- Step 1
The whole circle has area π × 36, about 113.1.
Step 4: Your Turn
Practice makes it stick.
The Inscribed Angle
Problem 1 of 2
A central angle is 80°. What is the inscribed angle on the same arc?
degrees
The Semicircle
Problem 2 of 2
An angle inscribed in a semicircle is how many degrees?
degrees
Fractions of a Circle
1 of 4
Central angle 100°. Inscribed angle on the same arc?
2 of 4
A 90° sector is what fraction of the circle? Give the bottom number.
3 of 4
Inscribed angle 35°. The central angle on the same arc?
4 of 4
A 120° sector is what fraction? Give the bottom number.
Step 5: Quick Check
Show what you know.
Question 1 of 2
Central angle 140°. Inscribed angle on the same arc?
Question 2 of 2
How does an inscribed angle compare with the central angle on the same arc?
What You Learned
- A central angle equals its arc; an inscribed angle is half.
- An angle in a semicircle is a right angle.
- Arc length and sector area are fractions of the whole.