Cogito
Geometry · Chapter 6 · Lesson 1
Angles, Arcs, and Sectors
Everything in a circle is a fraction of the whole.
10 problems · about 20 minutes · G-C.A.2, G-C.B.5
What this lesson teaches
The student relates central and inscribed angles to arcs and computes arc length and sector area.
- 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.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — Special Right Triangles: 30-60-90 with hypotenuse 18. Shortest side?
Answer 9
Why 9.
Review — Solving Right Triangles: Legs 7 and 24. Hypotenuse?
Answer 25
Why 25.
Warm Up
Straightforward practice. Get the method working first.
4 problemsCentral angle 140°. Inscribed angle on the same arc?
Answer 70
Why 70°.
How does an inscribed angle compare with the central angle on the same arc?
Answer It is half.
Why Half the central angle.
Central angle 100°. Inscribed angle on the same arc?
Answer 50
Why Half.
A 90° sector is what fraction of the circle? Give the bottom number.
Answer 4
Why 90 ÷ 360.
Build It Up
The same ideas with more to keep track of.
2 problemsInscribed angle 35°. The central angle on the same arc?
Answer 70
Why Double.
A 120° sector is what fraction? Give the bottom number.
Answer 3
Why 120 ÷ 360.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Inscribed Angle: A central angle is 80°. What is the inscribed angle on the same arc?
Answer 40 degrees
Why 40°.
The Semicircle: An angle inscribed in a semicircle is how many degrees?
Answer 90 degrees
Why 90°.