☰ Contents · Geometry

The circle. The central angle

Lessons 34 · 1 lessons · A. A. Rahimqoriyev, M. A. To‘xtaxo‘jayeva. Geometry, Grade 8 (textbook for general secondary schools), revised 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2014
34

The circle. The central angle

Textbook: pp. 107–108
GoalKnow the circle, radius, chord, diameter, central angle and the angular measure of an arc.
New words
circle · aylanachord · vatarcentral angle · markaziy burchakangular measure of an arc · yoyning burchak kattaligi
Explanation

A circle is the line made of all points of a plane lying at the same distance from one point; that point is the centre of the circle. A segment from the centre to a point of the circle is a radius, a segment joining two points of the circle is a chord, and a chord through the centre is a diameter; a diameter is twice a radius. An angle with its vertex at the centre is a central angle. Two points of the circle split it into two arcs. If an arc is not larger than a semicircle, its angular measure equals the central angle; if it is larger, it equals 360° minus the central angle. The measures of two arcs with the same endpoints add up to 360°.

Worked examples
The central angle ∠AOB = 72° gives the smaller arc 72°, and the other arc is 360° − 72° = 288°.
Two points split a circle into arcs in the ratio 2 : 3: 360° : 5 = 72°, so the arcs are 144° and 216°.
Class activity

“Circle with angles”: draw a circle with a compass (its point is sharp, so work with an adult) and use a protractor to divide it into 6 equal arcs.

Practice
1
One arc measures 125°. How many degrees is the other arc with the same endpoints?
2
A circle is divided into 6 equal arcs. How many degrees is each arc?
3
The radius of a circle is 9 cm. How long is its diameter (cm)?
4
Why do the measures of two arcs with the same endpoints add up to 360°?