Central angles
Key Notes :
🔵 What is a Central Angle?
A central angle is an angle whose vertex is at the center of a circle.
Its rays (arms) point to the circumference of the circle.
➡️ Example: Angle ∠AOB in circle O.
🔴 Central Angle and Arc Measure
A central angle cuts an arc on the circle.
📏 The measure of a central angle = the measure of its intercepted arc.
👉 If arc AB = 60°, then ∠AOB = 60°.
🟣 Central Angle Formula (Radians)
If the arc length = s and radius = r
👉 θ = s / r (in radians)
🟢 Full Circle Facts
A circle has:
🔸 360° (degrees)
🔸 2π radians
👉 Central angles add up to 360°.
🟡 Relationship With Inscribed Angles
A central angle is twice the inscribed angle subtending the same arc.
👉 ∠Central = 2 × ∠Inscribed
🟤 Sector Area and Central Angle
A sector is the slice of a circle formed by a central angle.
If θ is in degrees:
📘 Sector Area = (θ/360) × πr²
🔵 Real-Life Uses of Central Angles
🌗 Clock angles
📊 Pie charts
🎡 Ferris wheel rotations
⚙️ Engineering and construction
Learn with an example
What is ∠GFI ?
∠GFI = ______°
Look at the diagram:
Write an equation showing that the measures of ∠GFH,∠GFI and ∠HFI sum to 360°, since they span an entire circle. Then solve for ∠GFI.
∠GFH + ∠GFI + ∠HFI = 360°
150° + ∠GFI + 80° = 360° Plug in ∠GFH = 150° and ∠HFI = 80°
∠GFI + 230° = 360° Combine like terms
∠GFI = 130° Subtract 230° from both sides
So, ∠GFI = 130°
What is ∠RQS?
∠RQS = _______°
Look at the diagram:
Write an equation showing that the measures of ∠RQS , ∠RQT and ∠SQT sum to 360°, since they span an entire circle. Then solve for ∠RQS.
∠RQS + ∠RQT + ∠SQT = 360°
∠RQS+ 140° + 120° = 360° Plug in ∠RQT = 140° and ∠SQT = 120°
∠RQS + 260° = 360° Combine like terms
∠RQS = 100° Subtract 260° from both sides
So, ∠RQS = 100°
What is ∠IHJ?
∠IHJ = _______°
Look at the diagram:
Write an equation showing that the measures of ∠IHK , ∠JHK and ∠IHJ sum to 360°, since they span an entire circle. Then solve for ∠IHJ.
∠IHK + ∠JHK + ∠IHJ= 360°
140° + 80° +∠IHJ = 360° Plug in ∠IHK = 140° and ∠JHK = 80°
∠IHJ + 220° = 360° Combine like terms
∠IHJ = 140° Subtract 220° from both sides
So, ∠IHJ= 140°
let’s practice!
