SM3.1 Areas related to circles
You can find arc lengths, sector and segment areas, and the areas of shaded regions built from circles and polygons together.
Topics shown in module order.
A sector is a slice of the circle and its area is the circle's area taken in proportion to the angle; an arc is the same idea applied to the circumference. The segment, the region between a chord and its arc, is a sector with a triangle removed. Almost every question in the chapter is a picture of overlapping shapes, and the whole method is naming the region as a sum and difference of shapes you already know.
Work through these
Compute arc lengths and sector areas for a range of angles, keeping the fraction of the full circle visible in each
Writing the angle over 360 as an explicit fraction, rather than substituting into a memorised formula, keeps the idea attached to the arithmetic. It also survives the day the formula is misremembered.
Find the area of a segment by writing it as sector minus triangle before touching any numbers
The subtraction written in words first is the plan; the numbers only carry it out. Segments are where this chapter's answers get long, and a stated plan is what keeps a long answer from wandering.
Solve three shaded-region problems, decomposing each region in words before calculating
A shaded region has no formula; it has a decomposition. Saying the decomposition aloud, this is the square minus the four quarter circles, is the entire skill, and the calculation that follows is arithmetic you already own.
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