Perimeter - GMAT Math Study Guide

The Concept of Perimeter

The perimeter of an object refers to the total length of the lines forming the outside of the shape. In order to calculate the perimeter of an object, add up the length of each side of the object.

Circle

The perimeter or circumference, C, of a circle is found using the equation C = 2πr, where r is the radius.

Consider another example:

Triangle

The perimeter of a triangle is the sum of its three sides. It should be noted that the sum of any two sides of a triangle must be larger than the length of the third side.

What is the perimeter of the following triangle?

Since the triangle is right, one can solve for the length of the unknown side.
x² + 5² = 13²
x² = 13² - 5²
x² = 169 - 25 = 144
x = 12

Quadrilateral

The perimeter, P, of a quadrilateral is the sum of the four sides. There are shortcuts to finding the perimeter for some quadrilaterals. For squares, P = 4*e where e is the length of a side. For rectangles, P = 2w + 2l. For parallelograms, P = 2x + 2y where x and y are the lengths of the parallel sides.

Square

Consider the following example:

If the figure below is a square and e is 5cm, what is the perimeter of the square below?


Perimeter_Square = 4(side) = 4(5cm) = 20cm

Parallelogram

Consider the following example:

If the length of side gh is 10cm and the length of side kg is 6cm, what is the perimeter of the parallelogram below?

Perimeter_{Parallelogram} = 2(10) + 2(6) = 32cm
Note: In a parallelogram, opposite sides have the same length.

Other Shapes

In some cases, the figure does not fit into any specific geometric shape. In these cases, one can find the perimeter by splitting the shape into recognized geometric figures.

What is the perimeter of figure ABCDE, shown below, if AC = 4, BC = 3, ED = 2.5, AE = 3, DC = 3

The outer perimeter of the trapezoid: ED + AE + DC
2.5 + 3 + 3 = 8.5

The outer perimeter of the triangle:
In order to find this, one must first find the length of AB, which is 5 by the Pythagorean theorem.
Outside perimeter of a triangle = AB + BC (do not count AC since AC is on the interior).
AB + BC = 5 + 3 = 8

Total Perimeter = outer perimeter of the trapezoid + outer perimeter of the triangle
Total Perimeter = 8.5 + 8 = 16.5

Types of GMAT Problems

1. Combining Multiple Perimeters

   GMAT questions often require the use of several perimeters. If stuck on a difficult problem, it is often best to write out all the information possible as well as the specific information needed to solve the problem. By attacking the problem one piece at a time, it is often possible to arrive at the final answer through discovering important intermediate information.

   If square ABCD is inscribed inside the circle and DB and AC are both diameters of the circle, what is the circumference of the circle if the perimeter of the square is 4(200)^(1/2)?
   

   A)	25π
   B)	20π
   C)	5π
   D)	10π
   E)	12π

   Correct Answer: B

   1. Since the perimeter of the square is 4*(200)^(1/2), each side must be 200^(1/2) since there are four equal sides to a square.
   2. By use of the Pythagorean theorem, the radius of the circle can be determined. Chose one of the right angled triangles formed by two radii and one side of the square. If the radii are labeled r, then r² + r² = (200^(1/2))²
      2r² = 200
      r² = 100
      r = 10 (r = -10 can be discarded as negative lengths have no meaning).
   3. C = circumference = 2πr
      C = 2π10 = 20π