# ISEE Upper Level Quantitative : How to find the area of a parallelogram

## Example Questions

### Example Question #1 : Parallelograms

In the above parallelogram,  is acute. Which is the greater quantity?

(A) The area of the parallelogram

(B) 120 square inches

It is impossible to determine which is greater from the information given

(B) is greater

(A) and (B) are equal

(A) is greater

(B) is greater

Explanation:

Since  is acute, a right triangle can be constructed with an altitude as one leg and a side as the hypotenuse, as is shown here. The height of the triangle must be less than its sidelength of 8 inches.

The height of the parallelogram must be less than its sidelength of 8 inches.

The area of the parallelogram is the product of the base and the height - which is

Therefore,

(B) is greater.

### Example Question #1 : Parallelograms

Parallelogram A is below:

Parallelogram B is below:

Note: These figures are NOT drawn to scale.

Refer to the parallelograms above. Which is the greater quantity?

(A) The area of parallelogram A

(B) The area of parallelogram B

(A) is greater

It is impossible to determine which is greater from the information given

(A) and (B) are equal

(B) is greater

(A) and (B) are equal

Explanation:

The area of a parallelogram is the product of its height and its base; its slant length is irrelevant. Both parallelograms have the same height (8 inches) and the same base (1 foot, or 12 inches), so they have the same areas.

### Example Question #3 : Parallelograms

Figure NOT drawn to scale

The above figure shows Rhombus  and  are midpoints of their respective sides. Rectangle  has area 150.

Give the area of Rhombus .

Explanation:

A rhombus, by definition, has four sides of equal length. Therefore, . Also, since  and  are the midpoints of their respective sides,

We will assign  to the common length of the four half-sides of the rhombus.

Also, both  and  are altitudes of the rhombus; the are congruent, and we will call their common length  (height).

The figure, with the lengths, is below.

Rectangle  has dimensions  and ; its area, 150, is the product of these dimensions, so

The area of the entire Rhombus  is the product of its height  and the length of a base , so

.