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How to Find Area - skills needed and how it’s done

Area is a measurement of the amount of space inside a two-dimensional object. How to find area? This simply can be done by multiplying two numbers together, but sometimes it can take just more than multiplying the numbers. Area is 2-dimensional as it has a length and a width. Area is measured in square units such as square inches, square meters or square feet. Different shapes and objects have different methods used in finding their area. Some of those shapes are listed below

  • Rectangles
  • Squares
  • Parallelograms
  • Trapezoid
  • Triangles
  • Regular polygons
  • Circles
  • Pyramids
  • Cylinder

Just as a dissertation introduction requires some prerequisites and a level of precision, finding the surface area also requires some certain set of skills and they include the following

  • Multiplication
  • Addition
  • Subtraction
  • Division

Most people wonder how to find area of an irregular shape, such as a room. Just as most students worry about how to start a thesis till they actually do start it, finding the area of an object can be said to be nothing to worry much about as well. Either irregular shapes or the usual regular shapes, both objects have surface area. For instance, let’s try to find the area of the object below.

First, we notice it’s an irregular object and this means we can use one of two methods on how to find area of the object.

Description: 3- 2 -4 -6 irregular shape

 

    • Method 1 : Divide the object into 2 rectangles (as shown below) and find all missing lengths

Description: http://www.shmoop.com/images/prealgebra/unit4/pa.4.3101.png

 

    • The larger rectangle has an area of 4cm * 7cm = 28cm2
    • The smaller rectangle has an area of 4cm * 2cm = 8cm2
    • Combining these two rectangles we will find the total area: 28cm2 + 8cm2 = 36 cm2
    • Method 2 : Draw two lines to make the figure into one large rectangle as shown below

Description: http://www.shmoop.com/images/prealgebra/unit4/pa.4.3105.png

 

  • The area of the large rectangle is 7cm * 6cm = 42cm2
  • However, a 2 * 3cm rectangle is not included in the initial figure, so we need to take out the area of the white rectangle (2cm * 3cm = 6cm2)()
  • 42cm2 - 6cm2 = 36 cm2

How to find area of a rectangle is very easy. We already know that any rectangular shape has 2 pairs of sides that are equal to each other. First, we might want to label one of the sides as the base or horizontal while the other side is labelled the height. After doing that, the rest is pretty easy. Just use the formula for finding the area of any rectangle and all is done. The formula is.

  • Area=W x L

Where W is the width or the height of the rectangle and L is the length or the base of the rectangle. Therefore, the dimensions in a rectangle are its length and width. Given the length and width of a rectangle, we can find the area and given the area and one dimension of a rectangle, we will be able to find the other dimension.

Squares area can be calculated using the same formula as a rectangle. The only difference between this shape and that of the rectangle is that all the sides of a square are equal. This means that if we measure one side of the square, the same answer will be for the other side. Area is measured in square units. The area of an object is the number of squares needed to cover the entire object just like tiles used in designing the floor of a house. Area of a square = a side multiplied by the other side. Since each side of a square are equal, it can simply be the length of one side squared. If a square has one side of 4 inches, the area would be 4 inches multiplied by 4 inches or 16 square inches. Square inches can also be written in2.

Parallelograms: choose one of the sides of the base of the parallelogram. Find the length of this base. Then draw a perpendicular line to this base, and find the length of this line drawn between where it meets the base and the side opposite to the base. This length becomes the height. If the side opposite to the base is not long enough such that the perpendicular line crosses it, extend the side along the line until it traverses the perpendicular line.

Trapezoids: use to find the lengths of the two parallel sides. Assign these values to variables a and b. Find the height then draw a parallel line that spans both parallel sides and the length of the line segment on this line joining the two sides is the height of the parallelogram (h). Now that’s not all on how to find area, you still need to substitute the given values into the formula: A=0.5(a+b) x h.

Triangles: Find the base and height of the triangle. This consists of the length of one side of the triangle and the length of the line segment perpendicular to the end joining the base to the opposite vertex of the triangle. Now, substitute the given values into the equation: A=0.5b*h

  • Regular Polygons: Find the length of a side and the length of the line segment perpendicular to a side that connects the middle of the polygon to the center. The length of the apothem will be given the variable a. Multiply the length of one side of the polygon by the number of sides to get the perimeter of the polygon (p). Substitute the given values into the equation: A=0.5a*p and this is how to find area of a regular polygon.

     

    Circles: Find the radius of the circle (r). This is a line segment that joins the center to a point on the circle. This value is the same no matter what point you pick on the circle. Substitute the given values into the equation A=πr^2.

    Surface Area of a Pyramid: Find the area of the base of the rectangle using the formula shown above for finding the area of a rectangle: k=b*h. Find the area of each side: A=0.5b*h. Add up all the areas: the base and all the sides.

    Surface Area of a Cylinder: Find the radius of one of the base circles. Find the height of the cylinder. Find the area of the base using the formula of the area of a circle: A=πr^2. Find the area of the side by multiplying the height by the perimeter of the base. Therefore, the perimeter of a circle is P=2πr while the area of the side of the circle is A=2πhr. Adding up all the areas: this consists of two identical circular bases and the side of the circle. So, the surface area should be SA=2πr^2+2πhr.

    The Area Under a Function: is used if you want to find the area under a curve and above the x-axis denoted by function f(x) in the domain interval x within [a, b]. Although, this method requires the knowledge of integral calculus. Define f(x) in terms of x. Take the integral of: f(x) within [a, b]. Plug in a and b values into the integral equation. The area under f(x) between x [a, b] will then be defined as ∫abf(x). So, A=F(b))—F(a).

    How to find area using other approaches and its history

    Just as it is with writing a cover letter, there are other methods used if we are to find area of any object, shape or thing. Those methods include

    • Area by Counting Squares
    • Approximate area by counting squares
    • Areas of difficult shapes
    • Areas by adding up triangles
    • Area by coordinates

    Area by Counting Squares : using this method to find the area involves putting the shape on a grid for example, and counting the number of small squares we can find within the shape, say it was a rectangle. This is illustrated below.

    Description: Area Count

     

    In this case, the rectangle has an area of 15. Sometimes, the squares don't match the exact shape we put on the grid, but we can get an "approximate" answer.

    Approximate area by counting squares : one way to go on how to find area using this method is

    • If it is more than half a square count as 1
    • But if it is less than half a square count as 0
    Description: Area Count

     

    This pentagon has an area of approximately 17

    Hippocrates in the 5th century, was the first to prove that the area of a disk is proportional to the square of disks diameter. Subsequently, Book I of Euclidean geometry dealt with the equality of areas between 2-dimensional objects. This book was further worked on by a mathematician named Archimedes who used it to show the area inside a circle is equal to that of a right triangle with whose base has the length of the circle’s radius. The results of Archimedes’ thesis topics produced formulas which he used to calculate the areas of shapes that are of course regular, using a revolutionary method of abducting new shapes, by using shapes he already understood. For instance, to estimate the area of a circle, he built a bigger polygon outside the circle with a smaller one inside. He then enclosed the circle in a triangle, followed by in a square, as well as in a pentagon and also in a hexagon and each time approximating the area of the circle.

    Interestingly, Archimedes realized that all that could be established was a range and that the real value might never be known. The method he used for π estimation was taken to the extreme by Ludoph van Ceulen in the 16th Century. He used a polygon with an exceptional 4,611,686,018,427,387,904 sides to arrive at a conclusive value of π correct to 35 numerals. π is now known a number, so irrational to the point that its value can never be known with complete accuracy.

    Similarly, he calculated the approximate volume of a solid, such as spheres by cutting it up into a series of cylinders, and adding up volumes of the constituent cylinders. He noticed that the thinner he cuts the cylinders, his approximation became more exact, such that, in the limit, his approximation became the exact calculation.

    Heron of Alexandria found what is called Heron’s formula for resolving the problem of how to find area of a triangle in terms of its sides. In 499 Aryabhata one of the greatest mathematician-astronomer from the classical age of Indian mathematics and its astronomy expressed the area of a triangle as one-half the base multiplied by the height of the triangle.

    After going through the above-mentioned details for resolving the area of irregular and regular shapes, it is definitely clear that not every establishment offering dissertation writing services will be helpful in such cases. So one should make an effort to analyze and be sure that the one he/she chooses will be capable for the job.

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