In geometry, the term angle refers to the measure of turn between two lines or vectors. There are several types of angles, each one defined by its measure.
Angles are typically measured in degrees (°), minutes (′), and seconds (‴) or radians. A radian is the standard unit of angle measurement, where there is a complete circle.
When looking at an angle in the plane, there is a unique relationship between the length of the side opposite the angle, called the hypotenuse, and the size of the angle itself. This relationship is called trigonometry, and it is very useful in many fields.
In this article, we will discuss some situations where you need to find the area of an arc-shaped region that lies inside a circle with a given diameter.
Calculate the length of one complete circle
The first step to finding the area of the sector is to calculate the length of one complete circle in the rectangle. You can do this by finding the central angle measurement in radians.
The central angle Xyz measures 1.25 radians, so Xyz=1.25.
Now, you can use the formula for circumference of a circle, which is 2πr, where r is the radius of the circle. So, you need to find the radius!
You can find the radius by dividing 1 by the central angle measurement in radians. So, 1/1.25=0.8, so the radius of one circle in this shape is 0.8.
Find the area of the shaded sector
The final step of this problem is to find the area of the shaded sector. You can do this by multiplying the length of the side of the sector by the radius and then multiplying that number by pi.
Area of sector = (side length) × (radius) × π
Once you have found the radius, you can use that in this formula to find the area of the entire circle. Then you can divide that number by two to find how much area is in each half-circle.
For this problem, we will assume that the circle has a diameter of 4 units. To find out how much area is in each half-circle, we will divide 8 unit2 by 2, which gives us 4 unit2 as the area of each half-circle.
Divide the central angle into 360 degrees
The first step in finding the area of the shaded sector is to divide the central angle into 360 degrees. In this problem, the central angle is 1.25 radians, which makes the degree measure of one radian 1.
Radians are a way of measuring angles that describe how far around a circle you go while moving in the same direction. A full rotation around a circle is 2pi (2π) radians, or 360 degrees.
To make things easier, there are 2π equal to 6 facts . This means that there are 6 times as many degrees in a radian as there are in a circle.
You can use this fact to find out how many degrees there are in any other angle: just divide the number of degrees in an angle by 6 and you’ll get the number of radians in that angle.
Convert degrees to radians
In order to calculate the area of the shaded sector, you need to know how many radians the central angle covers. To find this, you need to know how many degrees the central angle covers and then convert that into radians.
Radians are a different way to measure an angle. One radian is equal to approximately 7.5 degrees. So one radian is a pretty small measurement of an angle!
You can use a calculator to convert degrees to radians if you do not know how to do it on your own. Most calculators have a button that says “Rad” next to the degrees button. Press this and then select what type of unit you want converted and it will do the job for you!
Remember: Radians are a little bit bigger than degrees, so when you calculate the area of a circle using radians, you will get a little bit more than what you would get using degrees.
Multiply angles by radius and divide by 2pi
When you know the length of the radius and the measure of the central angle, you can find the area of the sector formed by the radius and the diagonal line.
You just need to remember that you need to multiply angles by radius and then divide by 2π (2π is approximately 6.28).
This is because you have traversed a full circle, or moved through a full radian, which is equal to 2π.
When you divide by 2π, you are leaving out some decimal places, so you are taking away some area. You are cutting off some of the sector created by the angle and the line.
Use a calculator with trig functions
Once you have the central angle measurement, you can use a calculator to find the area of the shaded sector. Most calculators have a trig function that works for angles measured in radians.
Simply enter the radius of the circle into the calculator, then enter the central angle measurement in radians as the function. Your calculator should then give you the area of the shaded sector.
For example, if the circle has a radius of 2 and the central angle is 1.25 radians, then enter 2 for r, 1.25 for α, and press “sin” on your calculator. You will get an area of 0.8π, which is correct because one square pyramid covering one quarter of the circle’s surface area takes up half of the entire surface area in theory.
Graph y = x and use software to find area under curve
Area can be calculated by finding the area under the curve of y = x. Most software programs have an option to find the area under a curve, which can be extremely helpful in finding the true area of a sector.
Unfortunately, this method does not give you the A r , or absolute area of the sector, only R2 where R is the radius of the circle. The reason for this is because when calculating area under a curve, the length measurements are based on coordinate points, not lines as with finite areas.
This makes it difficult to determine exactly how long each line is, and therefore how large the A r is. While it is possible to find out how many square units there are, this may not match up with what you think the absolute area should be.
Simplify using algebraic expressions for central angle and radius
After you calculate the area of a circle, you can use that information to calculate the area of the shaded sector. You can do this by simplifying the expression for the central angle and radius.
First, simplify the expression for the central angle. Then, substitute the expressions for radius and circle area into this new expression for central angle. Finally, combine all of these expressions into one equation that can be solved with algebra.
For example, suppose you have an isosceles triangle with a side length of 1 and an angle measuring 1/2 radian. You want to find the area of the shaded sector formed by rotating the triangle 45° around its base. First, simplify =1/2. Then, =1+1/2. Simplify again to get =1+1. Then substitute r, or 1/2radian , and >area of circle=pr2>, so that >area of sector = (1+>1)pr^2>. Solve for pr to get 0.5.
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You can now return to your original question about finding the area of a sector with an unknown central angle.
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