Area , Perimeter and Circumference

A sizeable number of GMAT math test questions belong to the Geometry section.  Some of these questions test  a candidate’s ability to understand 2-Dimensional Geometry by asking the candidate to calculate the area, perimeter or circumference of a geometrical shape.

The following geometrical shapes are most common – Triangles, Quadrilaterals, Rectangles, Rhombuses, Squares, Circles and Trapeziums.

Triangles – A triangle represents an enclosed shape made by joining three straight lines. The area of a triangle can be calculated as follows:

Area = ½*Base Side*Height of the triangle

In this formula, the Base Side can be any side of the triangle. However, depending on the base side chosen, height of the triangle needs to be ascertained. Height of the triangle is the shortest perpendicular distance from the Base side to the height of the Apex of that triangle.  Note that the height of a triangle may need to be calculated outside the triangle, depending on the base side chosen.  


Triangles can be of different shapes depending upon the length of each of the three straight lines that create the triangle. For example, if two lines are of equal length then it results in an isosceles triangle. If all three sides are equal, it results in an equilateral triangle. A scalene triangle has three sides that are of unequal lengths.   

Quadrilateral – A quadrilateral represents an enclosed shape made by joining four lines. The GMAT questions will ask the test taker to calculate areas for the following types of quadrilaterals, each of which has a specific formula for Area:

Square – A square is a quadrilateral that has four equal sides, and each side is perpendicular to two other sides of the square. The area of the Square is given by:
Area = (length of side)^2.
 
Rectangle – Rectangles have special properties:  each side connects to two other sides perpendicularly, and the parallel facing sides are equal. The area of a rectangle is given by:
Area = product of two adjacent sides

Trapezium –  The GMAT often tests a student’s ability to calculate the area of a trapezium, a shape that is slightly more difficult than the standard shapes I mentioned earlier, as it sometimes requires calculating the area of two such standard shapes. A trapezium is represented as follows:



The two sides of a trapezium are parallel as shown. However the other two sides can take any shape and length. The general formula for calculating Area of a trapezium is as follows:

Area = ½ x (Sum of two parallel side lengths) x Height of Trapezium.

The trick to remember this formula is that trapezium is really made up of two triangles.



And as discussed earlier, the area of any triangle is ½ x Base x Height. Therefore, the above trapezium formula is sum of the areas of the two triangles.

Circle – A circle represents an enclosed shape where each point on the circumference of the circle is equidistant from the center of the circle. The distance from the center to any point on the circumference is called Radius of the circle.

Area of a circle is given as a function of the radius of the circle.
 
Where ? is a constant with value = 3.14

Perimeter

Another set of problems in the Geometry section consists of calculating the perimeter of a shape. The perimeter of a geometrical shape is the length of the boundary that surrounds this shape. You may see questions asking about calculating the perimeter of standard shapes such as triangles, quadrilaterals, rectangles, rhombuses, squares, circles and trapeziums.

Triangles – A triangle represents an enclosed shape made by joining three straight lines. The perimeter of a triangle can be calculated as follows:
 
Perimeter = Length(a + b+ c) where a,b and c are three sides of the triangle

Triangles can be of different shapes depending upon the length of each of three straight lines that create the triangle. If all three sides are equal, it results in an equilateral triangle. For an equilateral triangle, the perimeter simply becomes three times the length of one side of that triangle.

Quadrilateral – A quadrilateral represents an enclosed shape made by joining four lines. GMAT questions will ask the test taker to calculate perimeter for the following types of quadrilaterals, each of which has a specific formula for its perimeter:

Square – A square is quadrilateral that has four equal sides and each side is perpendicular to two other sides of the square. The perimeter of the square is given by:

Perimeter = 4 x Length of a side.
 
Rectangle – Rectangles have special properties:  each side connects to two other sides perpendicularly, and the parallel facing sides are equal. The perimeter of a rectangle is given by:

Perimeter = 2 (sum of length of two adjacent sides)

Circle – A circle represents an enclosed shape where each point on the circumference of the circle is equidistant from the center of the circle. The distance from the center to any point on the circumference is called radius of the circle.

The perimeter of a circle is given as a function of the radius of the circle.

A = 2Pir

Where Pi is a constant with value = 3.14