Calculate segment areas by decomposing the region into a sector minus a central triangle, utilizing trigonometry for non-right angles.
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Visual of sector decomposing into a triangle and a segment.
Clean scientific diagram of a circle with center O, showing a sector OAPB. A chord AB is drawn, creating a central trian…
Introduces the composite method for finding segment area.
When a chord is drawn across a circle, it divides the circular region into two parts called segments. The smaller region is the minor segment, and the larger one is the major segment. Unless specified otherwise, "segment" usually refers to the minor segment.
Step-by-step segment area calculation involving trigonometry.
Problem. Find the area of the segment AYB if the radius of the circle is and . (Use )
Partially solved calculation for a 90-degree segment.
Let's find the area of the minor segment when a chord of a circle with radius 10 cm subtends a right angle at the centre. First, calculate the area of the sector using the angle . The sector area is . Next, find the area of the central triangle . Because it's a right triangle, we can directly use the formula . Here, both the base and height are the radius, so the area is . Finally, we calculate the area of the minor segment by the triangle area from the sector area. The final area is .
Closure question linking minor segment calculation to major segment calculation.
A chord of a circle of radius subtends an angle of at the centre. If the calculated minor segment area is approximately , how do you correctly calculate the area of the major segment?
Guided problem for segment area utilizing trig.
A chord of a circle of radius subtends an angle of at the centre. Find the area of the corresponding segment of the circle. (Use and ).
List the radius and the central angle provided in the problem.
State the relationship between the segment, sector, and triangle.
Substitute the values to find the area of the entire sector.
Show the steps to find the triangle's base and height using trig, then calculate its area.
Subtract the triangle area from the sector area.
Confirm your units are correct and the final number makes logical sense.