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Understanding Areas of Circles, Sectors, and Segments: Formulas and Examples

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kapila
18 diapositives · 323 Views · Oct 12, 2014

This chapter delves into the calculation of areas for various components of circles, including sectors and segments. You'll find essential definitions, such as sectors being regions bounded by two radii and an arc, and segments formed by a chord and arc. Key formulas are provided: for the area of a circle (A = πr²), sector (A = (M/360)πr²), and segment (Aseg = Asec - Atri). Examples illustrate practical applications, and there's homework to reinforce learning. Perfect for mastering circle geometry.

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  1. Jacob Epp Chapter 11.6 Sivam Bhatt Areas of Circles, Sectors, and Segments Justin Rosales Tim Huxtable

  2. Summary In this section we will learn how to find the area of various parts of circles.

  3. Definitions Sector - a region bounded by two radii and an arc of the circle Segment - a region bounded by a chord of the circle and its corresponding arc

  4. Circles There are several important formulae you will need to know for this chapter. The first is the formula for the area of a circle, which is ... A = πr2 r … radius

  5. Sectors The formula for area of a sector is … A = (M/360)πr2 r - radius m - measure of arc

  6. Segments The formula for area of a segment is … Aseg = Asec - Atri

  7. Segments - Step 1 1. Find the area of the sector that intercepts the segment’s arc.

  8. Segments - Step 2 2. Find the area of the triangle formed by the two radii and the chord.

  9. Segments - Step 3 3. Subtract the area of the triangle from the sector. minus =

  10. Example 1: Circles Find the area of the circle with radius 7 cm. 7 cm

  11. A = πr2 A = π72 A = 49πcm2 Solution Ex. 1

  12. Example 2: Sectors Find the area of the sector Measure of Angle = 90 Radius = 4 90° 4 cm.

  13. Solution Ex. 2 A=(M/360)πr2 A=(90/360)π42 A=(1/4)π42 A=(1/4)16π A=4π

  14. Example 3: Segments Find the area of the segment. 90° 8 cm.

  15. Solution Ex. 3 A = (M/360)πr2-Area of triangle A = (90/360)π82 -8(8)/2 A = (¼)64π-32 A = 16π-32

  16. Example 4 Find the area of the sector AOB. A 10o C O 12 B

  17. Solution to example 4 1) As you can see the radius of the circle is 12 you can find the area of the circle with πr2. Doing so you will get the area of the circle as 144π. 2) Since the inscribed angle is 10o arc AB would be 20o. 3) In order to solve for this you must use the formula of a sector which is (measure of arc/360)(πr2). Since we have both pieces of information we would get (1/18)(144π). Simplifying we would get 8π.

  18. Homework Page 539 problems 1-3, 5, 11, 13-15, 18, 20