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High School Math : How to find the area of a sector

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Example Questions

Example Question #22 : Plane Geometry

A circular, 8-slice pizza is placed in a square box that has dimensions four inches larger than the diameter of the pizza. If the box covers a surface area of 256 in², what is the surface area of one piece of pizza?

Possible Answers:

9π in²

4.5π in²

36π in²

18π in²

144π in²

Correct answer:

4.5π in²

Explanation:

The first thing to do is calculate the dimensions of the pizza box. Based on our data, we know 256 = s². Solving for s (by taking the square root of both sides), we get 16 = s (or s = 16).

Now, we know that the diameter of the pizza is four inches less than 16 inches. That is, it is 12 inches. Be careful! The area of the circle is given in terms of radius, which is half the diameter, or 6 inches. Therefore, the area of the pizza is π * 6² = 36π in². If the pizza is 8-slices, one slice is equal to 1/8 of the total pizza or (36π)/8 = 4.5π in².

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Example Question #1 : Sectors

If B is a circle with line AC = 12 and line BC = 16, then what is the area formed by DBE?

Possible Answers:

$144$ $\displaystyle 144$

$5\pi$ $\displaystyle 5\pi$

$200$ $\displaystyle 200$

$256\pi$ $\displaystyle 256\pi$

$100\pi$ $\displaystyle 100\pi$

Correct answer:

$100\pi$ $\displaystyle 100\pi$

Explanation:

Line AB is a radius of Circle B, which can be found using the Pythagorean Theorem:

$AB^2=AC^2+BC^2\rightarrow AB=\sqrt{AC^2+BC^2}=\sqrt{16^2+12^2}=\sqrt{400}=20$ $\displaystyle AB^2=AC^2+BC^2\rightarrow AB=\sqrt{AC^2+BC^2}=\sqrt{16^2+12^2}=\sqrt{400}=20$

Since AB is a radius of B, we can find the area of circle B via:

$Area=\pi R^2=\pi(20^2)=400\pi$ $\displaystyle Area=\pi R^2=\pi(20^2)=400\pi$

Angle DBE is a right angle, and therefore $\frac{1}{4}$ $\displaystyle \frac{1}{4}$ of the circle so it follows:

$Area(DBE)=\frac{400}{4}\pi=100\pi$ $\displaystyle Area(DBE)=\frac{400}{4}\pi=100\pi$

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Example Question #1 : How To Find The Area Of A Sector

To the nearest tenth, give the area of a $60^{\circ }$ $\displaystyle 60^{\circ }$ sector of a circle with diameter 18 centimeters.

Possible Answers:

$42.4 \textrm{ cm}^{2}$ $\displaystyle 42.4 \textrm{ cm}^{2}$

$254.5 \textrm{ cm}^{2}$ $\displaystyle 254.5 \textrm{ cm}^{2}$

$169.6 \textrm{ cm}^{2}$ $\displaystyle 169.6 \textrm{ cm}^{2}$

$21.2 \textrm{ cm}^{2}$ $\displaystyle 21.2 \textrm{ cm}^{2}$

$84.8 \textrm{ cm}^{2}$ $\displaystyle 84.8 \textrm{ cm}^{2}$

Correct answer:

$42.4 \textrm{ cm}^{2}$ $\displaystyle 42.4 \textrm{ cm}^{2}$

Explanation:

The radius of a circle with diameter 18 centimeters is half that, or 9 centimeters. The area of a $60^{\circ }$ $\displaystyle 60^{\circ }$ sector of the circle is

$A = \frac{60}{360 }\cdot \pi \cdot 9^{2} =\frac{1}{6} \cdot \pi \cdot 81 \approx 42.4 \textrm{ cm}^{2}$ $\displaystyle A = \frac{60}{360 }\cdot \pi \cdot 9^{2} =\frac{1}{6} \cdot \pi \cdot 81 \approx 42.4 \textrm{ cm}^{2}$

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Example Question #1 : Sectors

Find the area of a sector that has an angle of 120 degrees and radius of 3.

Possible Answers:

$9\Pi$ $\displaystyle 9\Pi$

$27\Pi$ $\displaystyle 27\Pi$

$12\Pi$ $\displaystyle 12\Pi$

$\frac{1}{3}\Pi$ $\displaystyle \frac{1}{3}\Pi$

$3\Pi$ $\displaystyle 3\Pi$

Correct answer:

$3\Pi$ $\displaystyle 3\Pi$

Explanation:

The equation to find the area of a sector is $(\frac{x}{360})\Pi r^2$ $\displaystyle (\frac{x}{360})\Pi r^2$.

Substitute the given radius in for $\displaystyle r$ and the given angle in for $\displaystyle x$ to get:

$(\frac{120}{360})\Pi (3^2)$ $\displaystyle (\frac{120}{360})\Pi (3^2)$

Simplify the equation to get the area:

$(\frac{1}{3})\Pi9=3\Pi$ $\displaystyle (\frac{1}{3})\Pi9=3\Pi$

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Example Question #5 : Circles

What is the area of the following sector of a full circle?

Arch

Note: Figure is not drawn to scale.

Possible Answers:

$21.5\pi m^2$ $\displaystyle 21.5\pi m^2$

$27\pi m^2$ $\displaystyle 27\pi m^2$

$23.5\pi m^2$ $\displaystyle 23.5\pi m^2$

$10\pi m^2$ $\displaystyle 10\pi m^2$

$13.5\pi m^2$ $\displaystyle 13.5\pi m^2$

Correct answer:

$13.5\pi m^2$ $\displaystyle 13.5\pi m^2$

Explanation:

In order to find the fraction of a sector from an angle, you need to know that a full circle is $360^{\circ}$ $\displaystyle 360^{\circ}$.

Therefore, we can find the fraction by dividing the angle of the sector by $360^{\circ}$ $\displaystyle 360^{\circ}$:

$Percentage = \frac{\measuredangle}{360^{\circ}}$ $\displaystyle Percentage = \frac{\measuredangle}{360^{\circ}}$

$Percentage = \frac{60^{\circ}}{360^{\circ}}=\frac{1}{6}$ $\displaystyle Percentage = \frac{60^{\circ}}{360^{\circ}}=\frac{1}{6}$

The formula to find the area of a sector is:

$A=\pi (r^2)(fraction\ of\ whole\ circle)$ $\displaystyle A=\pi (r^2)(fraction\ of\ whole\ circle)$

where $\displaystyle r$ is the radius of the circle.

Plugging in our values, we get:

$A=\pi (r^2)(fraction)$ $\displaystyle A=\pi (r^2)(fraction)$

$A=\pi (9m^2)\left(\frac{1}{6}\right)=13.5\pi m^2$ $\displaystyle A=\pi (9m^2)\left(\frac{1}{6}\right)=13.5\pi m^2$

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Example Question #1 : How To Find The Area Of A Sector

Find the area of the shaded region:

Screen_shot_2014-02-27_at_6.35.30_pm

Possible Answers:

$108\pi-81\sqrt{3}m^2$ $\displaystyle 108\pi-81\sqrt{3}m^2$

$81\pi-108\sqrt{3}m^2$ $\displaystyle 81\pi-108\sqrt{3}m^2$

$108\pi-81\sqrt{2}m^2$ $\displaystyle 108\pi-81\sqrt{2}m^2$

$81\pi-81\sqrt{3}m^2$ $\displaystyle 81\pi-81\sqrt{3}m^2$

$108\pi-108\sqrt{3}m^2$ $\displaystyle 108\pi-108\sqrt{3}m^2$

Correct answer:

$108\pi-81\sqrt{3}m^2$ $\displaystyle 108\pi-81\sqrt{3}m^2$

Explanation:

To find the area of the shaded region, you must subtract the area of the triangle from the area of the sector.

The formula for the shaded area is:

$A=A_{sector}-A_{triangle}$ $\displaystyle A=A_{sector}-A_{triangle}$

$A = \pi(r^2)(part)-\frac{1}{2}(b)(h)$ $\displaystyle A = \pi(r^2)(part)-\frac{1}{2}(b)(h)$,

where $\displaystyle r$ is the radius of the circle, part $\displaystyle part$ is the fraction of the sector, $\displaystyle b$ is the base of the triangle, and $\displaystyle h$ is the height of the triangle.

In order to the find the base and height of the triangle, use the formula for a $\displaystyle 30-60-90$ triangle:

$a-a\sqrt{3}-2a$ $\displaystyle a-a\sqrt{3}-2a$, where $\displaystyle a$ is the side opposite the $\measuredangle30$ $\displaystyle \measuredangle30$.

Plugging in our final values, we get:

$A = \pi(18m^2)(\frac{120}{360})-\frac{1}{2}(18\sqrt{3}m)(9m)$ $\displaystyle A = \pi(18m^2)(\frac{120}{360})-\frac{1}{2}(18\sqrt{3}m)(9m)$

$A = 108\pi-81\sqrt{3}m^2$ $\displaystyle A = 108\pi-81\sqrt{3}m^2$

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Example Question #161 : High School Math

Find the area of the following sector:

Possible Answers:

$9 \pi\ m^2$ $\displaystyle 9 \pi\ m^2$

$3 \pi\ m^2$ $\displaystyle 3 \pi\ m^2$

$6 \pi\ m^2$ $\displaystyle 6 \pi\ m^2$

$12 \pi\ m^2$ $\displaystyle 12 \pi\ m^2$

$18 \pi\ m^2$ $\displaystyle 18 \pi\ m^2$

Correct answer:

$9 \pi\ m^2$ $\displaystyle 9 \pi\ m^2$

Explanation:

The formula for the area of a sector is

$A = \pi r^2 (part)$ $\displaystyle A = \pi r^2 (part)$,

where $\displaystyle r$ is the radius of the circle and part $\displaystyle part$ is the fraction of the sector.

Plugging in our values, we get:

$A = \pi (6\ m)^2 (\frac{90}{360})$ $\displaystyle A = \pi (6\ m)^2 (\frac{90}{360})$

$A=9 \pi\ m^2$ $\displaystyle A=9 \pi\ m^2$

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Example Question #511 : Geometry

Circle

The radius of the circle above is $\displaystyle 4$ and $\angle A=45^{\circ}$ $\displaystyle \angle A=45^{\circ}$. What is the area of the shaded section of the circle?

Possible Answers:

$4\pi$ $\displaystyle 4\pi$

$2\pi$ $\displaystyle 2\pi$

$8\pi$ $\displaystyle 8\pi$

$\pi$ $\displaystyle \pi$

$16\pi$ $\displaystyle 16\pi$

Correct answer:

$2\pi$ $\displaystyle 2\pi$

Explanation:

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

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Example Question #1 : How To Find The Area Of A Sector

Find the area of the shaded segment of the circle. The right angle rests at the center of the circle.

Possible Answers:

$9\pi$ $\displaystyle 9\pi$

$9\pi-18$ $\displaystyle 9\pi-18$

$36\pi-18$ $\displaystyle 36\pi-18$

$9\pi-36$ $\displaystyle 9\pi-36$

$9\pi+36$ $\displaystyle 9\pi+36$

Correct answer:

$9\pi-18$ $\displaystyle 9\pi-18$

Explanation:

We know that the right angle rests at the center of the circle; thus, the sides of the triangle represent the radius of the circle.

r=6 $\displaystyle r=6$

Because the sector of the circle is defined by a right triangle, the region corresponds to one-fourth of the circle.

$\frac{degrees\ in\ angle}{degrees\ in\ circle}=\frac{90^o}{360^o}=\frac{1}{4}$ $\displaystyle \frac{degrees\ in\ angle}{degrees\ in\ circle}=\frac{90^o}{360^o}=\frac{1}{4}$

First, find the total area of the circle and divide it by four to find the area of the depicted sector.

$\small A(circle)=\pi(r)^2$ $\displaystyle \small A(circle)=\pi(r)^2$

$\small A(sector)=\frac{\pi(r)^2}{4}$ $\displaystyle \small A(sector)=\frac{\pi(r)^2}{4}$

$\small A(sector)=\frac{\pi(6)^2}{4}=\frac{\pi(36)}{4}=9\pi$ $\displaystyle \small A(sector)=\frac{\pi(6)^2}{4}=\frac{\pi(36)}{4}=9\pi$

Next, calculate the area of the triangle.

$\small A(tri)=\frac{1}{2}bh=\frac{1}{2}(6)(6)=18$ $\displaystyle \small A(tri)=\frac{1}{2}bh=\frac{1}{2}(6)(6)=18$

Finally, subtract the area of the triangle from the area of the sector.

$\small A(shaded)=9\pi-18$ $\displaystyle \small A(shaded)=9\pi-18$

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Example Question #1 : Plane Geometry

Square-missing

SRAD $\displaystyle SRAD$ is a square.

$\overline{AB}=\overline{BC}=\overline{CD}$ $\displaystyle \overline{AB}=\overline{BC}=\overline{CD}$

The arc from $\displaystyle B$ to $\displaystyle C$ is a semicircle with a center at the midpoint of $\overline{BC}$ $\displaystyle \overline{BC}$.

All units are in feet.

The diagram shows a plot of land.

The cost of summer upkeep is $2.50 per square foot.

In dollars, what is the total upkeep cost for the summer?

Possible Answers:

$\$900$ $\displaystyle \$900$

$\$(900-25\pi )$ $\displaystyle \$(900-25\pi )$

$\$(2250-31.25\pi )$ $\displaystyle \$(2250-31.25\pi )$

$\$(2250-250\pi )$ $\displaystyle \$(2250-250\pi )$

$\$(900-12.5\pi )$ $\displaystyle \$(900-12.5\pi )$

Correct answer:

$\$(2250-31.25\pi )$ $\displaystyle \$(2250-31.25\pi )$

Explanation:

To solve this, we must begin by finding the area of the diagram, which is the area of the square less the area of the semicircle.

The area of the square is straightforward:

30 * 30 = 900 square feet

Because each side is 30 feet long, AB + BC + CD = 30.

We can substitute BC for AB and CD since all three lengths are the same:

BC + BC + BC = 30

3BC = 30

BC = 10

Therefore the diameter of the semicircle is 10 feet, so the radius is 5 feet.

The area of the semi-circle is half the area of a circle with radius 5. The area of the full circle is 5²π = 25π, so the area of the semi-circle is half of that, or 12.5π.

The total area of the plot is the square less the semicircle: 900 - 12.5π square feet

The cost of upkeep is therefore 2.5 * (900 – 12.5π) = $(2250 – 31.25π).

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High School Math : How to find the area of a sector

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #22 : Plane Geometry

Example Question #1 : Sectors

Example Question #1 : How To Find The Area Of A Sector

Example Question #1 : Sectors

Example Question #5 : Circles

Example Question #1 : How To Find The Area Of A Sector

Example Question #161 : High School Math

Example Question #511 : Geometry

Example Question #1 : How To Find The Area Of A Sector

Example Question #1 : Plane Geometry

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