Areas and Perimeters of Polygons

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 A triangle is any geometric object with three sides that connect to one another to form one cohesive shape and can be found commonly in modern architecture, design, and carpentry, which is why it is important to be able to determine the perimeter and area of a triangle.

Triangle: Surface Area and Perimeter

Surface Area and Perimeter: Triangle. D. Russell

The perimeter of a triangle is calculated by adding up the distance around its three outer sides where if the side lengths are equal to A, B and C, the perimeter of a triangle is A + B + C.

The area of a triangle, on the other hand, is determined by multiplying the base length (the bottom) of the triangle by the height (sum of the two sides) of the triangle and dividing it by two—to best understand why it is divided by two, consider that a triangle forms one half of a rectangle!

Trapezoid: Surface Area and Perimeter

Surface Area and Perimeter: Trapezoid. D. Russell

A trapezoid is a flat shape with four straight sides that has a pair of opposite sides that are parallel, and you can find the perimeter of a trapezoid by simply adding the sum of all four of its sides.

Determining the surface area of a trapezoid is a little bit more difficult because of its strange shape, though. In order to do so, mathematicians must multiply the average width (the length of each base, or parallel line, divided by two) by the height of the trapezoid.

The area of a trapezoid can be expressed in the formula A = 1/2 (b1 + b2) h where A is the area, b1 is the length of the first parallel line and b2 is the length of the second, and h is the height of the trapezoid. 

If the height of the trapezoid is missing, one can use the Pythagorean Theory to determine the missing length of a right triangle formed by cutting the trapezoid along the edge to form a right triangle.

Rectangle: Surface Area and Perimeter

Surface Area and Perimeter: Rectangle. D. Russell

A rectangle has four interior angles that are 90 degrees and opposite sides that are parallel and equal in length, though not necessarily equal to the lengths of the sides connected directly to it.

To calculate the perimeter of a rectangle, one simply adds two times the width and two times the height of the rectangle, which is written as P = 2l + 2w where P is the perimeter, l is the length, and w is the width.

To find the surface area of a rectangle, simply multiply its length by its width, expressed as A = lw, where A is the area, l is the length, and w is the width.​​

Parallelogram: Area and Perimeter

Surface Area and Perimeter: Parallelogram. D. Russell

A parallelogram is a considered a "quadrilateral" that has two pairs of opposite sides that are parallel but whose internal angles are not 90 degrees, as are rectangles'. However, like a rectangle, one simply adds twice the length of each of the sides of a parallelogram, expressed as P = 2l + 2w where P is the perimeter, is the length, and w is the width.

Because the opposite sides of a parallelogram are equal to one another, the calculation for the surface area is very much like that of a rectangle but not like that of a trapezoid. Still, one might not know the height of the trapezoid, which is separate from its width (which slopes as at an angle as illustrated above).

Still, to find the surface area of a parallelogram, multiply the base of the parallelogram by the height.

Circle: Circumference and Surface Area

Surface Area and Perimeter: Circle. D. Russell

Unlike other polygons, the circle's perimeter is determined according to the fixed ratio of Pi and called the circumference instead of its perimeter but still is used to describe the measurement of the total length around the shape. In degrees, a circle is equal to 360° and Pi (p) is the fixed ratio that is equal to 3.14.

There are two formulas for finding the perimeter of a circle:

  • C = pd or C = p2r wherein C is the circumference, d is the diameter, r is the radius (which is half of the diameter), and p is Pi, which equals 3.1415926.
  • Use Pi to find the perimeter of a circle. Pi is the ratio of a circle's circumference to it's diameter. If the diameter is 1, the circumference is pi.

For the measurement of the area of a circle, simply multiply the radius squared by Pi, expressed as A = pr2.

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Russell, Deb. "Areas and Perimeters of Polygons." ThoughtCo, Oct. 26, 2017, thoughtco.com/area-and-perimeter-of-a-triangle-2312244. Russell, Deb. (2017, October 26). Areas and Perimeters of Polygons. Retrieved from https://www.thoughtco.com/area-and-perimeter-of-a-triangle-2312244 Russell, Deb. "Areas and Perimeters of Polygons." ThoughtCo. https://www.thoughtco.com/area-and-perimeter-of-a-triangle-2312244 (accessed November 25, 2017).