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Intermediate Geometry : How to find the area of a parallelogram

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Example Questions

Example Question #1 : How To Find The Area Of A Parallelogram

The perimeter of a square is $24\; in$ $\displaystyle 24\; in$. If the sides of the square are reduced by a factor of two, what is the area of the new square?

Possible Answers:

$18\; in^{2}$ $\displaystyle 18\; in^{2}$

$9\; in^{2}$ $\displaystyle 9\; in^{2}$

$27\; in^{2}$ $\displaystyle 27\; in^{2}$

$45\; in^{2}$ $\displaystyle 45\; in^{2}$

$36\; in^{2}$ $\displaystyle 36\; in^{2}$

Correct answer:

$9\; in^{2}$ $\displaystyle 9\; in^{2}$

Explanation:

The perimeter of a square is geven by P=4s $\displaystyle P=4s$ and the area of a square is given by $A=s^{2}$ $\displaystyle A=s^{2}$.

Thus $P=4s=24\: in$ $\displaystyle P=4s=24\: in$

so $s=6\; in$ $\displaystyle s=6\; in$.

The original side is reduced by a factor of $\displaystyle 2$ which results in a new side of $3\; in$ $\displaystyle 3\; in$. The area of the new square is geven by

$A=s^{2}=3^{2}=9\; in^{2}$ $\displaystyle A=s^{2}=3^{2}=9\; in^{2}$

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Example Question #2 : How To Find The Area Of A Parallelogram

Find the area of the box in square inches:

Geo_box

Possible Answers:

84xy $\displaystyle 84xy$

12x +7y $\displaystyle 12x +7y$

$\displaystyle x + y + 19$

$\displaystyle 2x + 2y + 38$

$\displaystyle xy + 7x + 12y +84$

Correct answer:

$\displaystyle xy + 7x + 12y +84$

Explanation:

The answer is $\displaystyle xy + 7x + 12y +84$. You can find the area of this box by multiplying the length by its width:

Foil $\displaystyle (12 + x) (7 + y ) = xy + 7x + 12y +84$

If you chose $\displaystyle 2x + 2y + 38$, you added the sides to get the perimeter.

Just remember, the width is 12 added to $\displaystyle x$. Not 12 times the side of $\displaystyle x$.

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Example Question #2 : How To Find The Area Of A Parallelogram

If all angles are right angles in the figure below, what is the total area of the figure?

Box1

Possible Answers:

$256\ in^{2}$ $\displaystyle 256\ in^{2}$

$308\ in^{2}$ $\displaystyle 308\ in^{2}$

$152\ in^{2}$ $\displaystyle 152\ in^{2}$

$164\ in^{2}$ $\displaystyle 164\ in^{2}$

$184\ in^{2}$ $\displaystyle 184\ in^{2}$

Correct answer:

$164\ in^{2}$ $\displaystyle 164\ in^{2}$

Explanation:

The answer is $164\ in^{2}$ $\displaystyle 164\ in^{2}$.

To get the area, you would need to find the missing sides. Since the horizontal top is 8+8=16 $\displaystyle 8+8=16$ inches total, the bottom must be $\displaystyle 16$ inches total. Thus you would subtract 16-6 $\displaystyle 16-6$ to get $\displaystyle 10$ for the missing bottom horizontal side.

The vertical right side is $\displaystyle 4+12 = 16$ inches total. Then you would subtract 16-10 $\displaystyle 16-10$ to get $\displaystyle 6$ for the missing vertical side.

Then you would have to split the figure into 3 different boxes and find the area of each one. Then you would add the area of each box

$(32+60+72)=164\ in ^{2}$ $\displaystyle (32+60+72)=164\ in ^{2}$

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Example Question #1 : How To Find The Area Of A Parallelogram

What is the area of a parallelogram if the base is $1.5\:cm$ $\displaystyle 1.5\:cm$, and the height is $1.0\:cm$ $\displaystyle 1.0\:cm$?

Possible Answers:

$0.25\:cm$ $\displaystyle 0.25\:cm$

$0.75\:cm$ $\displaystyle 0.75\:cm$

$3\:cm$ $\displaystyle 3\:cm$

$1.5\sqrt2\:cm$ $\displaystyle 1.5\sqrt2\:cm$

$1.5\:cm$ $\displaystyle 1.5\:cm$

Correct answer:

$1.5\:cm$ $\displaystyle 1.5\:cm$

Explanation:

The formula for the area of a parallelogram is:

$A=base\cdot height$ $\displaystyle A=base\cdot height$

So, we are given all of the information that we need to solve this problem. Substitute the provided values into the equation and simplify:

$A= (1.5\:cm)(1\:cm) =1.5\:cm^2$ $\displaystyle A= (1.5\:cm)(1\:cm) =1.5\:cm^2$

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Example Question #5 : How To Find The Area Of A Parallelogram

What is the area of a parallelogram if the base is $2x\:cm$ $\displaystyle 2x\:cm$ and the height is $x^2\:cm$ $\displaystyle x^2\:cm$?

Possible Answers:

$2x^3\:cm^2$ $\displaystyle 2x^3\:cm^2$

$x^3\:cm^2$ $\displaystyle x^3\:cm^2$

$2x+x^2\:cm^2$ $\displaystyle 2x+x^2\:cm^2$

$2\sqrt2x^3\:cm^2$ $\displaystyle 2\sqrt2x^3\:cm^2$

$2x^2\:cm^2$ $\displaystyle 2x^2\:cm^2$

Correct answer:

$2x^3\:cm^2$ $\displaystyle 2x^3\:cm^2$

Explanation:

The formula for the area of a parallelogram is:

$A=base\cdot height$ $\displaystyle A=base\cdot height$

Substitute in the provided values and simplify to calculate the correct answer:

$A= (2x\:cm)(x^2\:cm) =2x^3\:cm^2$ $\displaystyle A= (2x\:cm)(x^2\:cm) =2x^3\:cm^2$

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Example Question #6 : How To Find The Area Of A Parallelogram

Find the area of a parallelogram if the base is (x-2) $\displaystyle (x-2)$ and the height is (x+4) $\displaystyle (x+4)$.

Possible Answers:

$\displaystyle x^2+2x-8$

4x+4 $\displaystyle 4x+4$

2x+2 $\displaystyle 2x+2$

$\displaystyle x^2-2x-8$

x^2-8 $\displaystyle x^2-8$

Correct answer:

$\displaystyle x^2+2x-8$

Explanation:

The formula for the area of a parallelogram is:

$A=base\cdot height$ $\displaystyle A=base\cdot height$

Substitute the values provided in the question:

$\displaystyle A= (x-2)(x+4)$

Use the FOIL method to simplify:

$\displaystyle A=(x)(x)+(x)(4) +(-2)(x)+(-2)(4)$

$\displaystyle A= x^2+4x-2x-8$

$\displaystyle A=x^2+2x-8$

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Example Question #3 : How To Find The Area Of A Parallelogram

What is the area of a parallelogram if the base is $\sqrt2\:cm$ $\displaystyle \sqrt2\:cm$ and the height is $\sqrt3\:cm$ $\displaystyle \sqrt3\:cm$?

Possible Answers:

$\frac{\sqrt6}{2}\:cm^2$ $\displaystyle \frac{\sqrt6}{2}\:cm^2$

$\frac{\sqrt5}{2}\:cm^2$ $\displaystyle \frac{\sqrt5}{2}\:cm^2$

$\sqrt2 + \sqrt3\:cm^2$ $\displaystyle \sqrt2 + \sqrt3\:cm^2$

$\sqrt6\:cm^2$ $\displaystyle \sqrt6\:cm^2$

$\sqrt5\:cm^2$ $\displaystyle \sqrt5\:cm^2$

Correct answer:

$\sqrt6\:cm^2$ $\displaystyle \sqrt6\:cm^2$

Explanation:

The formula for the area of a parallelogram is:

$A=base\cdot height$ $\displaystyle A=base\cdot height$

Substitute the values provided in the problem and solve to calculate the correct answer:

$A= (\sqrt2\:cm)(\sqrt3\:cm) =\sqrt6\:cm^2$ $\displaystyle A= (\sqrt2\:cm)(\sqrt3\:cm) =\sqrt6\:cm^2$

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Example Question #1 : How To Find The Area Of A Parallelogram

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Possible Answers:

$\displaystyle 6$

$\displaystyle 9$

$\displaystyle 8$

$\displaystyle 12$

Correct answer:

$\displaystyle 9$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=2 \times 6$ $\displaystyle \text{Area of Rectangle}=2 \times 6$

$\text{Area of Rectangle}=12$ $\displaystyle \text{Area of Rectangle}=12$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$ $\displaystyle \text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$ $\displaystyle \text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{2}{2}$ $\displaystyle \text{height}=\frac{2}{2}$

$\text{height}=1$ $\displaystyle \text{height}=1$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=3 \times 1$ $\displaystyle \text{Area of Parallelogram}=3 \times 1$

$\text{Area of Parallelogram}=3$ $\displaystyle \text{Area of Parallelogram}=3$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=12-3$ $\displaystyle \text{Area of Shaded Region}=12-3$

Solve.

$\text{Area of Shaded Region}=9$ $\displaystyle \text{Area of Shaded Region}=9$

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Example Question #1 : How To Find The Area Of A Parallelogram

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Possible Answers:

$\displaystyle 32$

$\displaystyle 24$

$\displaystyle 16$

$\displaystyle 40$

Correct answer:

$\displaystyle 32$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=4 \times 12$ $\displaystyle \text{Area of Rectangle}=4 \times 12$

$\text{Area of Rectangle}=48$ $\displaystyle \text{Area of Rectangle}=48$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$ $\displaystyle \text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$ $\displaystyle \text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{4}{2}$ $\displaystyle \text{height}=\frac{4}{2}$

$\text{height}=2$ $\displaystyle \text{height}=2$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=8 \times 2$ $\displaystyle \text{Area of Parallelogram}=8 \times 2$

$\text{Area of Parallelogram}=16$ $\displaystyle \text{Area of Parallelogram}=16$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=48-16$ $\displaystyle \text{Area of Shaded Region}=48-16$

Solve.

$\text{Area of Shaded Region}=32$ $\displaystyle \text{Area of Shaded Region}=32$

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Example Question #10 : How To Find The Area Of A Parallelogram

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Possible Answers:

180 $\displaystyle 180$

120 $\displaystyle 120$

$\displaystyle 90$

150 $\displaystyle 150$

Correct answer:

150 $\displaystyle 150$

Explanation:

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=12 \times 20$ $\displaystyle \text{Area of Rectangle}=12 \times 20$

$\text{Area of Rectangle}=240$ $\displaystyle \text{Area of Rectangle}=240$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$ $\displaystyle \text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$ $\displaystyle \text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{12}{2}$ $\displaystyle \text{height}=\frac{12}{2}$

$\text{height}=6$ $\displaystyle \text{height}=6$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=15 \times 6$ $\displaystyle \text{Area of Parallelogram}=15 \times 6$

$\text{Area of Parallelogram}=\90$ $\displaystyle \text{Area of Parallelogram}=\90$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=240-90$ $\displaystyle \text{Area of Shaded Region}=240-90$

Solve.

$\text{Area of Shaded Region}=150$ $\displaystyle \text{Area of Shaded Region}=150$

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Intermediate Geometry : How to find the area of a parallelogram

Study concepts, example questions & explanations for Intermediate Geometry

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Example Question #1 : How To Find The Area Of A Parallelogram

Example Question #2 : How To Find The Area Of A Parallelogram

Example Question #2 : How To Find The Area Of A Parallelogram

Example Question #1 : How To Find The Area Of A Parallelogram

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Example Question #6 : How To Find The Area Of A Parallelogram

Example Question #3 : How To Find The Area Of A Parallelogram

Example Question #1 : How To Find The Area Of A Parallelogram

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