# Area - GMAT Math Study Guide

## Area

The area of a shape refers to the space contained inside of the lines creating the shape.

### Triangle

The area, A, of a triangle can be found using the equation A = .5*B*H where B stands for the base of the triangle and H stands for the height. Any side can be chosen to be the base, but the height is the line that is perpendicular to the base and goes through the opposing vertex.

Area = 1/2(Base)(Height)
Area = 1/2(B)(H)

### Circle

In a circle, each point is equidistant from a central point (i.e., the center). The area, A, of a circle can be found using the equation A = π*R2 where R is the radius of the circle and π is pi = 3.1415....

Area = π*(radius)2 Area = π*R2

### Square

In a square: (1) every side is equal (2) the opposing sides are parallel (3) all the angles are 90°. The area, A, of a square can be found using the equation A = a2, where a is the length of a side of the square. Notice that the equation for the area of a square is a specific version of the rectangle equation below.

Area = (side)2
Area = a2

### Rectangle

In a rectangle: (1) opposing sides of a rectangle are equal and parallel (2) every side is not necessarily equal in length (3) all of the angles are 90°. The area, A, of a rectangle can be found using the equation A = l*w. In this case, l stands for the length and w stands for the width. It does not matter which sides are labeled the length or the width.

Area = (length)(width)
Area = lw

### Parallelogram

In a parallelogram, opposing sides are parallel and equal in length. Opposing angles are equal but not necessarily 90°. The area, A, of a parallelogram can be found using the equation A = b*h. In this case, b stands for the base of the parallelogram and h stands for the height. Any side can be chosen to be the base, but the height is the line that is perpendicular to the base and opposing line.

Area = (base)(height)
Area = bh

### Trapezoid

In a trapezoid, one set of opposing sides is parallel, but not necessarily equal. The area, A, of a trapezoid can be found using the equation A = .5*(a+c)*h. In this case, h stands for the line that is perpendicular to parallel sides a and c.

Area = .5*(Base1+Base2)*h
Area = .5*(a+c)*h

### Rhombus

In a rhombus: (1) all sides are equal in length (2) opposite sides are parallel (3) diagonals bisect each other (4) the intersection of diagonals forms a 90° angle.

Area = .5*(Diagonal1*Diagonal2)

## Types of GMAT Problems

1. Calculating the Area of Objects with Objects.

If there are multiple shapes within shapes, it is best to determine what areas are solvable with the given information and what pieces are needed in order to solve for the area requested in the problem. It is often possible to determine the area of a large figure by finding the sum of smaller sub-sections.

A circle with a radius of 6 has a square inscribed inside of it. What is the area of the black region?
 A) 23π - 31 B) 36π - 72 C) 63π - 42 D) 24π - 16 E) 30π - 62
Correct Answer: B
1. The general way to approach this problem is to find the area of the circle and subtract the area of the square.
2. The area of the circle is A = πr2 = π62 = 36π
3. To find the area of the square, the length of the sides must first be determined.
Use the triangle in the bottom of the diagram and the pythagorean theorem.
62+62 = a2 where a is the base of the triangle (and the side of the square).
a2 = 36 + 36 = 72
a = the square root of 72
area of square = (side)2 = (the square root of 72)2 = 72
4. Subtract the area of the square from the area of the circle to find the area of the shaded region.
36π - 72