GMAT Math : Calculating the area of a quadrilateral

Example Questions

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What is the area of a trapezoid with a height of 7, a base of 5, and another base of 13?

Explanation:

A circle can be circumscribed about each of the following figures except:

A regular hexagon

Each of the figures given in the other choices can have a circle circumscribed about it.

A triangle with sides 30, 40, 50.

A regular pentagon

triangle

Each of the figures given in the other choices can have a circle circumscribed about it.

Explanation:

A circle can be circumscribed about any triangle regardless of its sidelengths or angle measures, so we can eliminate the two triangle choices.

A circle can be circumscribed about any regular polygon, so we can eliminate those two choices as well.

The correct choice is that each figure can have a circle circumscribed about it.

What is the area of a quadrilateral on the coordinate plane with vertices  ?

Explanation:

As can be seen from this diagram, this is a parallelogram with base 8 and height 4:

The area of this parallelogram is the product of its base and its height:

What is the area of a quadrilateral on the coordinate plane with vertices  ?

Explanation:

As can be seen in this diagram, this is a trapezoid with bases 10 and 5 and height 8.

Setting  in the following formula, we can calculate the area of the trapezoid:

Note: Figure NOT drawn to scale

What is the area of Quadrilateral , above?

Explanation:

Quadrilateral  is a composite of two right triangles,  and , so we find the area of each and add the areas. First, we need to find  and , since the area of a right triangle is half the product of the lengths of its legs.

By the Pythagorean Theorem:

Also by the Pythagorean Theorem:

The area of  is .

The area of  is .

What is the area of the quadrilateral on the coordinate plane with vertices  ?

Explanation:

The quadrilateral formed is a trapezoid with two horizontal bases. One base connects (0,0) and (9,0) and therefore has length ; the other connects (4,7) and (7,7) and has length . The height is the vertical distance between the two bases, which is the difference of the  coorindates: . Therefore, the area of the trapezoid is

What is the area of the quadrilateral on the coordinate plane with vertices .

Explanation:

The quadrilateral is a trapezoid with horizontal bases; one connects  and  and has length , and the other connects  and  and has length . The height is the vertical distance between the bases, which is the difference of the -coordinates; this is . Substitute  in the formula for the area of a trapezoid:

Example Question #1 : Calculating The Area Of A Quadrilateral

What is the area of the quadrilateral on the coordinate plane with vertices  ?

Explanation:

The quadrilateral is a parallelogram with two vertical bases, each with length . Its height is the distance between the bases, which is the difference of the -coordinates: . The area of the parallelogram is the product of its base and its height:

Give the area of the above parallelogram if .

Explanation:

Multiply height  by base  to get the area.

By the 45-45-90 Theorem,

.

Since the product of the height and the base of a parallelogram is its area,

Example Question #1 : Calculating The Area Of A Quadrilateral

Give the area of the above parallelogram if .