It is very easy to compute the angle of a sector if we know what it is as a percentage of the total circle.  To do this, you merely need to multiply $\displaystyle 0.37$ by $\displaystyle 360$˚.  This yields $\displaystyle 133.2$˚. 

It is very easy to compute the angle of a sector if we know what it is as a percentage of the total circle.  To do this, you merely need to multiply $\displaystyle 0.87$ by $\displaystyle 360$˚.  This yields $\displaystyle 313.2$˚. 

It is very easy to compute the angle of a sector if we know what it is as a percentage of the total circle.  To do this, you merely need to multiply $\displaystyle 0.37$ by $\displaystyle 360$˚.  This yields $\displaystyle 133.2$˚

Let $\displaystyle r$ be the radius of Circle B. The radius of Circle A is therefore $\displaystyle 2r$.

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

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. $\displaystyle 360$˚).  Thus, for our circle, which has a sector with an angle of $\displaystyle 81$˚, we have a percentage of:

$\displaystyle \frac{81}{360}$

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

$\displaystyle A = \pi * r^2$

For our problem, $\displaystyle r=12.3$

Therefore, our equation is:

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

Using your calculator, you can determine that this is approximately $\displaystyle 106.94$.

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. $\displaystyle 360$˚).  Thus, for our circle, which has a sector with an angle of $\displaystyle 122$˚, we have a percentage of:

$\displaystyle \frac{122}{360}$

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

$\displaystyle A = \pi * r^2$

For our problem, $\displaystyle r=9.12$

Therefore, our equation is:

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

Using your calculator, you can determine that this is approximately $\displaystyle 88.55$.

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

For the sake of simplicity, let $\displaystyle BC = 1$; the reasoning is independent of the actual length. The legs of a 45-45-90 triangle are congruent, so $\displaystyle AB = 1$. The area of a right triangle is half the product of its legs, so 

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

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

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

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

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

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

$\displaystyle \approx 0. 125 \cdot 3.14$

$\displaystyle \approx 0. 3925$

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

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

For the sake of simplicity, let $\displaystyle BC = 1$; the reasoning is independent of the actual length. The area of equilateral $\displaystyle \bigtriangleup ABC$ can be found by substituting $\displaystyle s = 1$ in the formula

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

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

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

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

$\displaystyle \approx 0.425$

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

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

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

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

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

$\displaystyle \approx 0. 125 \cdot 3.14$

$\displaystyle \approx 0. 3925$

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

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 $\displaystyle r = 10$ in the area formula:

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

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

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

This question is asking for the length of an arc corresponding to $\displaystyle \frac{58}{60}$ of a circle with radius eight feet. The question can be answered by evaluating for $\displaystyle r=8$:

$\displaystyle \frac{58}{60} \cdot 2 \pi r=\frac{58}{60} \cdot 2 \pi \cdot 8 \approx 48.6$