Areas and Perimeters of Polygons

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Triangle: Surface Area and Perimeter

Surface Area and Perimeter: Triangle
D. Russell

A triangle is any geometric object with three sides connecting to one another to form one cohesive shape. Triangles are commonly found in modern architecture, design, and carpentry, making the ability to determine the perimeter and area of a triangle centrally important.

Calculate the perimeter of a triangle by adding the distance around its three outer sides: a + b + c = Perimeter

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:
b (h+h) / 2 = A (*NOTE: Remember PEMDAS!)

To best understand why a triangle 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 with a pair of opposite parallel sides. The perimeter of a trapezoid is found simply by adding the sum of all four of its sides: a + b + c + d = P

Determining the surface area of a trapezoid is a bit more challenging. 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: (l/2) h = S

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 Theorem 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 consists of four interior 90-degree angles and parallel sides that are equal in length, though not necessarily equal to the lengths of the sides to which each is directly connected. 

Calculate the perimeter of a rectangle by adding 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, 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 "quadrilateral" with two pairs of opposite and parallel sides 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, l is the length, and w is the width.

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

The circle's circumference -- the measure of the total length around the shape -- is determined based on the fixed ratio of Pi. In degrees, a circle is equal to 360° and Pi (p) is the fixed ratio equal to 3.14.

The perimeter of a circle can be determined one of two ways:

  • C = pd
  • C = p2r

wherein C - circumference, d = diameter, r i= radius (which is half of the diameter), and p = 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.