Fix the mistake: Sectors
Sectors ① Terms for Circles and Sectors
A figure that looks like part of a circle cut out is a sector. It looks like a slice of pizza. First, let us organize the basic terms for circles and the terms related to sectors.
Basic Terms for Circles
- Center O: The point at the center of a circle
- Radius: A segment from the center to the circumference. All radii have the same length.
- Diameter: A segment connecting two points on the circumference that passes through the center. Diameter ＝ radius×2
- Circumference: The boundary of a circle
- Chord: A segment connecting two points on the circumference
- Arc: A part of the circumference. Symbol ⌒AB
Terms for Sectors
Sector: A figure enclosed by two radii and the arc they cut off.
- Central angle: The angle formed by the two radii of a sector (the angle at center O)
- Arc: The curved part on the outside of a sector
- Radius: The same radius as that of the original circle
Within one circle, the size of a sector is determined by the size of its central angle.
Example ①
For a circle with radius 6 cm, answer the following.
(1) Diameter (2) Organize the elements of a sector with central angle 90° (central angle, radius, and arc).
Solution:
(1) 6×2 ＝ 12 cm.
(2) Central angle 90°, radius 6 cm, and the arc is 90/360 ＝ 1/4 of the circumference.
For a circle with radius 6 cm, answer the following.
(1) Diameter (2) Organize the elements of a sector with central angle 90° (central angle, radius, and arc).
Solution:
(1) 6×2 ＝ 12 cm.
(2) Central angle 90°, radius 6 cm, and the arc is 90/360 ＝ 1/4 of the circumference.
Proportional Relationships Among Central Angle, Arc, and Area
Within the same circle, a sector’s central angle, arc length, and area are proportional to one another.
・Central angle becomes 2 times larger → arc length and area also become 2 times larger
・When the central angle is a degrees, the sector is a/360 of the entire circle.
Example ②
Sector A of a circle has a central angle of 60° and an arc length of 3 cm. Sector B of the same circle has a central angle of 120°. Find the arc length of sector B.
Solution:
The central angle becomes 2 times larger, so the arc length also becomes 2 times larger. 3 × 2 ＝ 6 cm.
Sector A of a circle has a central angle of 60° and an arc length of 3 cm. Sector B of the same circle has a central angle of 120°. Find the arc length of sector B.
Solution:
The central angle becomes 2 times larger, so the arc length also becomes 2 times larger. 3 × 2 ＝ 6 cm.
Key Points
- A sector is enclosed by two radii and an arc.
- Central angle = the angle formed by two radii
- In one circle, the central angle, arc, and area are proportional.
- A sector with central angle a degrees is a/360 of the entire circle.
Practice Problems
- Find the diameter of a circle with radius 10 cm.
- In a sector of a circle, the central angle is 45°. What fraction of the entire circle is it?
- In the same circle, a sector with central angle 30° has an arc length of 2 cm. Find the arc length of a sector with central angle 90°.
Answers and Explanations
- Answer: 20 cm
Explanation: Diameter = radius×2 = 10×2 = 20. - Answer: 1/8
Explanation: 45/360 = 1/8. - Answer: 6 cm
Explanation: The central angle becomes 3 times larger (30→90), so the arc also becomes 3 times larger: 2×3 = 6 cm.
Previous and next lessons in this course
- Previous lesson: Fix the mistake: Sectors
- Next lesson: Teach: Solid Figures ① Types and Names of Solids