How to find the area of a sector - Math

Card 0 of 232

Question

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

(a) The area of a 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 sector of a circle comprises $\frac{90}{360}$ = $\frac{1}{4}$ of the circle. The 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.

Compare your answer with the correct one above

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 .

Compare your answer with the correct one above

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 .

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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 = $\frac{125 \pi }{4 }$ .

Compare your answer with the correct one above

Question

Find the area of a sector if it has an arc length of $3\pi$ and a radius of .

Answer

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}$=\frac{\text{Arc Length}$}{2\pi r}\pi $r^2$

The equation can be simplified to the following:

$$\text{Area}$=\frac{\text{Arc Length}$(r)}{2}$

Plug in the given arc length and radius to find the area of the sector.

$$\text{Area}$=\frac{3\pi(5.7)}{2}$=26.86$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

Find the area of a sector if it has an arc length of $10\pi$ and a radius of .

Answer

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}$=\frac{\text{Arc Length}$}{2\pi r}\pi $r^2$

The equation can be simplified to the following:

$$\text{Area}$=\frac{\text{Arc Length}$(r)}{2}$

Plug in the given arc length and radius to find the area of the sector.

$$\text{Area}$=\frac{12\pi(10)}{2}$=188.50$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

The radius of the circle above is and $\angle $A=45^{\circ}$$ . What is the area of the shaded section of the circle?

Answer

Area of Circle = πr2 = π42 = 16π

Total degrees in a circle = 360

Therefore 45 degree slice = 45/360 fraction of circle = 1/8

Shaded Area = 1/8 * Total Area = 1/8 * 16π = 2π

Compare your answer with the correct one above

Question

Find the approximate area of the shaded portion of the figure.

Answer

The answer is approximately $6200\ $cm^{2}$$ .

First, you would need to find the diameter of the circle. Use the Pythagorean Theorem to get

or

Since the diameter is 130, we divide by 2 to get 65 cm for our radius. Then the area of the circle is

$\pi $r^{2}$ = \pi \cdot $65^{2}$ \approx 13,273 \ $cm^{2}$$

Next we would find the area of each triangle:

$\frac{1}{2}$ (50)(120) = $3000cm^{2}$

and

$\frac{1}{2}$ (78)(104) = $4056cm^{2}$

Then we would subtract these from our answer above to get:

$13273-3000 - 4056 \approx 6200\ $cm^{2}$$ .

Compare your answer with the correct one above

Question

Give the area of a sector of a circle if its central angle is and the length of its arc is 10 inches. Write the answer in terms of $\pi$ , if applicable.

Answer

The arc of the sector measures 10 inches, so the circumference of the entire circle is

$C=10 \cdot$\frac{360}{20}$=180$

The radius is therefore

$r=\frac{C}{2\pi }$=\frac{180}{2\pi }$=\frac{90}{\pi }$$

The area of a sector of this circle measures:

$A = $\frac{20}{360}$\cdot\pi $r^{2}$= $\frac{20}{360}$\cdot\pi \cdot \left($\frac{90}{\pi }$ \right $)^{2}$$

$=\frac{20\cdot 90\cdot 90\cdot \pi }{360\cdot \pi \cdot \pi }$=\frac{450}{\pi }$$

Compare your answer with the correct one above

Question

A circle has a diameter of meters. A certain sector of the circle has a central angle of $\small 20^\circ$ . Find the area of the sector.

Answer

The formula for the area of a sector is.

$\small S=\frac{m}{360}$\pi $r^2$$ where $\small r$ is the radius and $\small m$ is the measure of the central angle of the sector.

We are given that the diameter of the circle is 60. Therefore its radius is simply half as long, or 30.

Substituting into our equation gives

$\small \small $S=\frac{20}{360}$\pi(30^2$)=\frac{1}{18}$\pi(900)=50\pi$

Therefore our area is $\small $50\pi,m^2$$

Compare your answer with the correct one above

Question

If the sector in the provided illustration has an angle of and the circle has a radius of , what is the area of the sector? Round to the nearest tenth.

Answer

To solve for the area of the sector, it helps to solve for the area of the complete circle and multiple that value by the sector angle over the full of a complete circle:

Compare your answer with the correct one above

Question

Find the area of a sector with a central angle of degrees and a radius of .

Answer

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}$=\frac{\text{Central Angle}$}{360}\times\pi\times $r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}$=\frac{124}{360}$\times\pi\times $12^2$

Solve and round to two decimal places.

$$\text{Area of Sector}$=155.82$

Compare your answer with the correct one above

Question

Find the area of a sector that has a central angle of degrees and a radius of .

Answer

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}$=\frac{\text{Central Angle}$}{360}\times\pi\times $r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}$=\frac{129}{360}$\times\pi\times $13^2$

Solve and round to two decimal places.

$$\text{Area of Sector}$=190.25$

Compare your answer with the correct one above

Question

Find the area of a sector that has a central angle of degrees and a radius of .

Answer

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}$=\frac{\text{Central Angle}$}{360}\times\pi\times $r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}$=\frac{130}{360}$\times\pi\times $19^2$

Solve and round to two decimal places.

$$\text{Area of Sector}$=409.54$

Compare your answer with the correct one above

Question

Find the area of a sector that has a central angle of degrees and a radius of .

Answer

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}$=\frac{\text{Central Angle}$}{360}\times\pi\times $r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}$=\frac{135}{360}$\times\pi\times $21^2$

Solve and round to two decimal places.

$$\text{Area of Sector}$=519.54$

Compare your answer with the correct one above

Question

Find the area of a sector that has a central angle of degrees and a radius of .

Answer

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}$=\frac{\text{Central Angle}$}{360}\times\pi\times $r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}$=\frac{115}{360}$\times\pi\times $8^2$

Solve and round to two decimal places.

$$\text{Area of Sector}$=64.23$

Compare your answer with the correct one above

Question

Find the area of a sector that has a central angle of degrees and a radius of .

Answer

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}$=\frac{\text{Central Angle}$}{360}\times\pi\times $r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}$=\frac{118}{360}$\times\pi\times $10^2$

Solve and round to two decimal places.

$$\text{Area of Sector}$=102.97$

Compare your answer with the correct one above

Question

Find the area of a sector that has a central angle of degrees and a radius of .

Answer

The circle in question could be depicted as shown in the figure.

Recall the formula for finding the area of a sector of a circle:

$\text{Area of Sector}$=\frac{\text{Central Angle}$}{360}\times\pi\times $r^2$

Since the central angle and the radius are given in the question, plug them in to find the area of the sector.

$\text{Area of Sector}$=\frac{34}{360}$\times\pi\times $5^2$

Solve and round to two decimal places.

$$\text{Area of Sector}$=7.42$

Compare your answer with the correct one above

Tap the card to reveal the answer