Circle a has been dissected into 16 congruent sectors. the base of one sector is 1.17 units, and its height is 2.94 units. what is the approximate area of circle a?
27.14 units
27.52 units
48.92 units
76.44 units

  27.52 units

Step-by-step explanation:

The area of the 16-gon is 16 times the area of one of the sector triangles, so is ...

  16-gon area = 16×(1/2)bh = 8bh = 8(1.17 units)(2.94 units) = 27.5184 units²

  ≈ 27.52 units²

The circle area will be larger than the inscribed polygon and smaller than the circumscribing polygon. We have assumed the problem is describing an inscribed 16-gon, so the actual circle area can be estimated to be slightly larger than 27.52 units².

Comments on this answer and on the geometry

This is (presumably) the area of an inscribed 16-gon, so its area will be smaller than the area of the circumscribing circle.

We can figure the radius of the circle using the Pythagorean theorem:

  r² = h² +(b/2)² = (2.94 units)² +(0.585 units)² = 8.985825 units²

The area of the circumscribing circle is π times this, ...

  A = πr² ≈ 28.23 units²

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If we take the "height" of the sector to be the radius of the circle we are estimating, then the circle area is ...

  A = πr² = π(2.94 units)² ≈ 27.14 units² . . . . . . using 3.14 for π

Interestingly, this matches the smallest answer choice exactly. That is, the circle inscribed within the 16-gon has this area.

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Since a sector has a curved edge, we're not quite sure what the "base of one sector" is measuring. If it is 1/16 of the circumference of the circle, then the circle has an area of about 27.89 units². Then, we have no idea what the 2.94 is a measure of, since the circle's radius is 2.98 units. A 16-gon inscribed in this circle will have an apothem of 2.92 units.

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b. 2 and 4

step-by-step explanation: