Area formulas are very useful in geometry, math, and CAD. Many applications require the area of a figure, the amount of surface space it occupies.
Rectangles are one of the most common figures where the area is easily found. The width times the length is the total square inches or square meters it contains.
Circles are a little more complicated to find the area of, but still not too hard. The average person knows that the formula for finding the area of a circle is πr2, where r is the radius and π is pi or 3.14.
Surface Area Formulas are very useful for finding out how much surface space an object contains. They are not as straightforward as finding an object’s volume, but they help provide an answer to how much space an object has on the outside.
Calculate the area of an ellipse
An ellipse is a closed curve with a radial line extending from the center to either focus. The distance between the two foci is called the focal length.
The area of an ellipse is calculated by finding the sum of the areas of each rectangle formed by the axis and the curve. Then, you multiply this sum by one-half of the total length of the curve.
You can also find the area of an ellipse using its radius and circumference. Given these parameters, you can use Pi (∏) to find its area, as well as how many radii length cover one square inch of surface.
There are many uses for finding the area of an ellipse, including analyzing its shape, calculating how much material it takes to make it, and determining how much material it takes to cover it with something else.
Apply these equations to find the area enclosed by the curve
Finding the area enclosed by a curve is a bit more complex than finding the area of a rectangle or triangle. There are specific equations you can use, however.
Area of circle = πr2 where r is the radius of the circle and π is pi, or approximately 3.14.
where r is the radius of the circle and π is pi, or approximately 3.14. Area enclosed by a curve = length x width
Where length and width are dependent on what type of curve you have. These equations can be swapped around depending on which one makes more sense for your situation.
Another way to find the area enclosed by a curve is to first find the inner loop of the curve, then multiply that by two times the sin of angle θ .
Example using algebraic equations
A more complicated example using algebraic equations is finding the area of a region enclosed by an arbitrary curve. Consider the region enclosed by the curve y = x2, shown in the figure below.
Given any point on the curve, you can find its x-coordinate by looking at its y-coordinate and solving for x. For instance, if we consider a point on the curve where its y-coordinate is 2, then its x-coordinate is 2.
We can use this information to write an equation for all of the interior angles of the triangle formed by this point on the curve and the X-axis. We’ll call these angles θ 1 and θ 2 . The following equations solve for these angles: cos(θ 1 ) = x / y , cos(θ 2 ) = x / y .
Example using a graph
A practical example of finding an area enclosed by a curve is finding the area enclosed by a circle. A circle is defined as the set of all points in a two-dimensional plane that are a given distance from a common point, the center.
How do we find the area of a circle? By calculating the radius multiplied by itself times pi (3.14).
So, if we had a circle with a radius of 2 and calculated the area using this formula, we would get 8pi square units. How did we get that?
We first drew out our diagram to make it clear which components were which: Next, we calculated the length of each component: Then, we used our calculator to multiply 2 x 3.14 and got 8pi! We could also just divide 8pi by 2 to get our answer of 4pi square units, which is exactly how much area one side of the circle has.
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