How to find the area of a sector - ISEE Upper Level (grades 9-12) Quantitative Reasoning

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Question

Circle A has twice the radius of Circle B. Which is the greater quantity?

(a) The area of a $90^{\circ }$ sector of Circle A

(b) The area of Circle B

Answer

Let be the radius of Circle B. The radius of Circle A is therefore .

A $90^{\circ }$ sector of a circle comprises $\frac{90}{360} = \frac{1}{4}$ of the circle. The $90^{\circ }$ sector of circle A has area $\frac{1}{4} \pi (2r)^{2} = \frac{1}{4} \cdot 4 \pi r^{2} = \pi r^{2}$ , the area of Circle B. The two quantities are equal.

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Question

What is the area, rounded to the nearest hundredth, of the sector shaded in circle O in the diagram above?

Answer

To find the area of a sector, you need to find a percentage of the total area of the circle. You do this by dividing the sector angle by the total number of degrees in a full circle (i.e. ˚). Thus, for our circle, which has a sector with an angle of ˚, we have a percentage of:

$\frac{81}{360}$

Now, we will multiply this by the total area of the circle. Recall that we find such an area according to the equation:

$A = \pi * r^2$

For our problem,

Therefore, our equation is:

$\frac{81}{360} * \pi * 12.3^2 = \frac{12254.49\pi}{360}$

Using your calculator, you can determine that this is approximately .

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Question

What is the area, rounded to the nearest hundredth, of the sector shaded in circle O in the diagram above?

Answer

To find the area of a sector, you need to find a percentage of the total area of the circle. You do this by dividing the sector angle by the total number of degrees in a full circle (i.e. ˚). Thus, for our circle, which has a sector with an angle of ˚, we have a percentage of:

$\frac{122}{360}$

Now, we will multiply this by the total area of the circle. Recall that we find such an area according to the equation:

$A = \pi * r^2$

For our problem,

Therefore, our equation is:

$\frac{122}{360} * \pi * 9.12^2 = \frac{10147.2768\pi}{360}$

Using your calculator, you can determine that this is approximately .

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Question

Refer to the above figure, Which is the greater quantity?

(a) The area of the orange semicircle

(b) The area of $\bigtriangleup ABC$

Answer

$\bigtriangleup ABC$ has two angles of degree measure 60; its third angle must also have measure 60, making $\bigtriangleup ABC$ an equilateral triangle

For the sake of simplicity, let ; the reasoning is independent of the actual length. The area of equilateral $\bigtriangleup ABC$ can be found by substituting in the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

$= \frac{1^{2}\sqrt{3}}{4}$

$= \frac{1 \cdot \sqrt{3}}{4}$

$\approx \frac{1.7}{4}$

$\approx 0.425$

Also, if , then the orange semicircle has diameter 1 and radius $\frac{1}{2}$ . Its area can be found by substituting $r = \frac{1}{2}$ in the formula:

$A = \frac{1}{2} \cdot \pi r^{2}$

$= \frac{1}{2} \cdot \pi \left ( \frac{1}{2} \right ) ^{2}$

$= \frac{1}{2} \cdot \pi \cdot \left ( \frac{1}{4 } \right )$

$= \frac{1}{8} \cdot \pi$

$\approx 0. 125 \cdot 3.14$

$\approx 0. 3925$

$\bigtriangleup ABC$ has a greater area than the orange semicircle.

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Question

Refer to the above figure, Which is the greater quantity?

(a) The area of $\bigtriangleup ABC$

(b) The area of the orange semicircle

Answer

$\bigtriangleup ABC$ has two angles of degree measure 45; the third angle must measure 90 degrees, making $\bigtriangleup ABC$ a right triangle.

For the sake of simplicity, let ; the reasoning is independent of the actual length. The legs of a 45-45-90 triangle are congruent, so . The area of a right triangle is half the product of its legs, so

$A = \frac{1}{2} \cdot 1 \cdot 1 = \frac{1}{2} = 0.5$

Also, if , then the orange semicircle has diameter 1 and radius $\frac{1}{2}$ . Its area can be found by substituting $r = \frac{1}{2}$ in the formula:

$A = \frac{1}{2} \cdot \pi r^{2}$

$= \frac{1}{2} \cdot \pi \left ( \frac{1}{2} \right ) ^{2}$

$= \frac{1}{2} \cdot \pi \cdot \left ( \frac{1}{4 } \right )$

$= \frac{1}{8} \cdot \pi$

$\approx 0. 125 \cdot 3.14$

$\approx 0. 3925$

$\bigtriangleup ABC$ has a greater area than the orange semicircle.

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Question

The above circle, which is divided into sectors of equal size, has diameter 20. Give the area of the shaded region.

Answer

The radius of a circle is half its diameter; the radius of the circle in the diagram is half of 20, or 10.

To find the area of the circle, set in the area formula:

$A = \pi r ^{2} = \pi \cdot 10 ^{2} = 100 \pi$

The circle is divided into sixteen sectors of equal size, five of which are shaded; the shaded portion is

$\frac{5}{16} \cdot 100 \pi = \frac{500}{16} \pi = \frac{125 \pi }{4 }$ .

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