Understanding Circles: Key Concepts and Formulas for Geometry
This resource delves into the fundamental aspects of circles in geometry, including key terms such as the center, diameter, radius, chords, arcs, and area calculations. A circle is defined as the collection of all points equidistant from a single center point. We explore related concepts like semicircles, minor arcs, major arcs, and real-world examples such as wheels and plates. Furthermore, we provide essential formulas for calculating circumference and area, alongside worked examples for clarity and practice.
Understanding Circles: Key Concepts and Formulas for Geometry
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Presentation Transcript
UNIT 5 Circles
Key Term (only write what’s in RED) • Circle: the set of a all points that are a given distance from a given point called the center KEEP IN MIND: A circle is a shape with all points the same distance from its center. A circle is named by its center. Thus, the circle below is called circle A since its center is at point A. Some real world examples of a circle are a wheel, a dinner plate and (the surface of) a coin. written as: A (circle A)
Key Terms Do you remember…? • Arc: part of the circumference of a circle • Semicircle: half of a circle (180°) • Minor Arc: smaller than a semicircle (2 letters) • Major Arc: greater than a semicircle (3 letters)
Identify: • Semicircles ________________________________ • Minor Arcs ________________________________ • Major Arcs ________________________________ B C E BONUS What’s the name of the circle?? A D
*the measure of an arc is EQUAL to the measure of its CENTRAL ANGLE
EX. 1a) Find the measure of each arc in Q. mCD: 40° mAD: 180° - 40° = 140° mCAD: 360° - 40° = 320° or 140° + 180° = 320° mDCA: 360° - 140° = 220° or 180° + 40° = 220° D 40° A C Q
EX. 1 (YOU TRY)B) Find the measure of each arc in Q. B mDC: ___________ mEB: ___________ mDAC: ___________ mACD: ___________ Q C A 35° 75° E D
Parts of a Circle KEEP IN MIND: The distance across a circle through the center is called the diameter. A real-world example of diameter is a 9-inch plate. The radius of a circle is the distance from the center of a circle to any point on the circle. If you place two radii end-to-end in a circle, you would have the same length as one diameter. Thus, the diameter of a circle is twice as long as the radius. A chord (pronounce CORD) is a line segment that joins two points on a curve. In geometry, a chord is often used to describe a line segment joining two endpoints that lie on a circle. The circle to the top right contains chord AB.
CIRCLE FORMULAS CIRCUMFERENCE AREA • C = πd or • C = 2πr Why are these equations the same?? • A = πr×r or • A = πr2 Why are these equations the same??
Ex. 2Find the circumference and area of each .(Fill in the blanks) 2.3 cm 15m 3 in. 5 in. C = πd C = π(15) C = 47.1 m A = πr2 A = π(7.5)2 A = 176. 7 m2 *hint: use PT C = πd C = π( ____ ) C = _____cm A = πr2 A = π(2.3)2 A = 16.6 cm2 C = πd C = π( ____ ) C = _____ in. A = πr2 A = π( ____ )2 A = _______in2
Ex. 3) The diameter of a bicycle wheel is 17 inches. If the wheel makes 10,500 revolutions, how far did the bike travel? d = 17; C = πd C = π(17) C ≈ 53.4 For 1 revolution, the bike traveled about 53.4 in. To find the distance traveled for 10,500 revolutions, multiply: C ≈ 53.4 × 10,500 C ≈ 560,774.3 inches
