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

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

Example Question #1 : Parallelograms

Parallelogram

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

(A) The area of the parallelogram

(B) 120 square inches

Possible Answers:

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

(B) is greater

(A) is greater

(A) and (B) are equal

Correct answer:

(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.

Parallelogram

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 #2 : How To Find The Area Of A Parallelogram

Parallelogram A is below:

Rhombus_1

Parallelogram B is below:

Parallelogram

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

Possible Answers:

(A) and (B) are equal

(B) is greater

(A) is greater

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

Correct answer:

(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 : How To Find The Area Of A Parallelogram

Parallelogram

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 .

Possible Answers:

Correct answer:

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.

Rhombus

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

.

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