How to find the area of a sector - Math

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).

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.

First, we figure out what fraction of the circle is contained in sector OPQ: $$\frac{30^{circ}$$$}{360^{circ}$}= $\frac{1}{12}$, so the total area of the circle is 12times 3pi=36 .

Using the formula for the area of a circle, ${pi}r^{2}$, we can see that r=6.

We can use this to solve for the circumference of the circle, 2{pi}r, or 12{pi}.

Now, OP and OQ are both equal to r, and PQ is equal to $\frac{1}{12}$ of the circumference of the circle, or {pi}.

To get the perimeter, we add OP + OQ + PQ, which give us 12+{pi}.

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

$AB^2$$=AC^2$$+BC^2rightarrow$ $AB=$\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$)=400pi

Angle DBE is a right angle, and therefore of the circle so it follows:

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

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.

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

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

Next, calculate the area of the triangle.

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

The radius of a circle with diameter 18 centimeters is half that, or 9 centimeters. The area of a sector of the circle is

The equation to find the area of a sector is .

Substitute the given radius in for and the given angle in for to get:

Simplify the equation to get the area:

In order to find the fraction of a sector from an angle, you need to know that a full circle is .

Therefore, we can find the fraction by dividing the angle of the sector by :

The formula to find the area of a sector is:

where is the radius of the circle.

Plugging in our values, we get:

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:

,

where is the radius of the circle, is the fraction of the sector, is the base of the triangle, and is the height of the triangle.

In order to the find the base and height of the triangle, use the formula for a triangle:

, where is the side opposite the .

Plugging in our final values, we get:

The formula for the area of a sector is

,

where is the radius of the circle and is the fraction of the sector.

Plugging in our values, we get:

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

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).

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.

First, we figure out what fraction of the circle is contained in sector OPQ: $$\frac{30^{circ}$$$}{360^{circ}$}= $\frac{1}{12}$, so the total area of the circle is 12times 3pi=36 .

Using the formula for the area of a circle, ${pi}r^{2}$, we can see that r=6.

We can use this to solve for the circumference of the circle, 2{pi}r, or 12{pi}.

Now, OP and OQ are both equal to r, and PQ is equal to $\frac{1}{12}$ of the circumference of the circle, or {pi}.

To get the perimeter, we add OP + OQ + PQ, which give us 12+{pi}.

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

$AB^2$$=AC^2$$+BC^2rightarrow$ $AB=$\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$)=400pi

Angle DBE is a right angle, and therefore of the circle so it follows:

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

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.

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

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

Next, calculate the area of the triangle.

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

The radius of a circle with diameter 18 centimeters is half that, or 9 centimeters. The area of a sector of the circle is

The equation to find the area of a sector is .

Substitute the given radius in for and the given angle in for to get:

Simplify the equation to get the area:

In order to find the fraction of a sector from an angle, you need to know that a full circle is .

Therefore, we can find the fraction by dividing the angle of the sector by :

The formula to find the area of a sector is:

where is the radius of the circle.

Plugging in our values, we get: