Area of a Parallelogram - Pre-Algebra

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Question

A parallelogram has the base length of and the altitude of . Give the area of the parallelogram.

Answer

The area of a parallelogram is given by:

Where is the base length and is the corresponding altitude. So we can write:

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Question

A parallelogram has a base length of which is 3 times longer than its corresponding altitude. The area of the parallelogram is 12 square inches. Give the .

Answer

Base length is so the corresponding altitude is $\frac{t}{3}$ .

The area of a parallelogram is given by:

Where:

is the length of any base
is the corresponding altitude

So we can write:

$12=t\times \frac{t}{3}\Rightarrow t\times t=12\times 3\Rightarrow t^2=36\Rightarrow t=6$

$t\times t=12\times 3$

$t^{2}=36$

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Question

What is the area of the parallelogram?

Answer

The area of a parallelogram is determined using the equation:

In this problem:

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Question

Find the area:

Answer

The area of a parallelogram can be determined using the following equation:

$\small A=bh$

Therefore,

$\small A=8\times3=24$

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Question

The length of the shorter diagonal of a rhombus is 40% that of the longer diagonal. The area of the rhombus is . Give the length of the longer diagonal in terms of .

Answer

Let be the length of the longer diagonal. Then the shorter diagonal has length 40% of this. Since 40% is equal to $\frac{40}{100 } = \frac{40 \div 20 }{100 \div 20 } = \frac{2}{5}$ , 40% of is equal to $\frac{2}{5}D$ .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up, and solve for , in the equation:

$\frac{1}{2} \cdot \frac{2}{5}D \cdot D = K$

$\frac{1}{5}D^{2}= K$

$5\cdot \frac{1}{5}D^{2}=5\cdot K$

$D^{2}=5K$

$D =\sqrt{5K}$

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Question

The length of the shorter diagonal of a rhombus is two-thirds that of the longer diagonal. The area of the rhombus is square yards. Give the length of the longer diagonal, in inches, in terms of .

Answer

Let be the length of the longer diagonal in yards. Then the shorter diagonal has length two-thirds of this, or $\frac{2}{3}D$ .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up the following equation and solve for :

$\frac{1}{2} \cdot \frac{2}{3}D \cdot D = Q$

$\frac{1}{3}D ^{2} = Q$

$3\cdot \frac{1}{3}D ^{2} = 3 \cdot Q$

$D ^{2} = 3Q$

$D = \sqrt{3Q}$

To convert yards to inches, multiply by 36:

$\sqrt{ 3Q } \times 36 = 36\sqrt{ 3Q }$

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Question

The longer diagonal of a rhombus is 20% longer than the shorter diagonal; the rhombus has area . Give the length of the shorter diagonal in terms of .

Answer

Let be the length of the shorter diagonal. If the longer diagonal is 20% longer, then it measures 120% of the length of the shorter diagonal; this is

$\frac{120}{100} = \frac{120 \div 20}{100 \div 20} = \frac{6}{5}$

of , or $\frac{6}{5}D$ .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up an equation and solve for :

$\frac{1}{2} \cdot \frac{6}{5}D \cdot D = N$

$\frac{3}{5}D^{2} = N$

$\frac{5}{3} \cdot \frac{3}{5}D^{2} = \frac{5}{3} \cdot N$

$D^{2} = \frac{5N}{3}$

$D =\sqrt{ \frac{5N}{3}}$

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Question

Refer to the above figure, which shows Parallelogram . You are given that $\overline{AX} \perp \overline{BC}$ and $\overline{AY} \perp \overline{CD}$ .

If you know the length of $\overline{AD}$ , then, of the following segments, choose the one whose length, if known, will allow us to calculate the area of Parallelogram .

Answer

The area of a parallelogram is the product of the length of any one side, or its base, and the perpendicular distance to the opposite side, or its height. If we know $\overline{AD}$ , then we also know $\overline{BC}$ , which is of the same length. We can take $\overline{BC}$ to be the base, and the segement perpendicular to it, $\overline{AX}$ , as the altitude. Therefore, $\overline{AX}$ is the segment whose length we need to know.

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Question

Note: Figure NOT drawn to scale.

Refer to the above figure, which shows Parallelogram . You are given that $\overline{AX} \perp \overline{BC}$ and $\overline{AY} \perp \overline{CD}$ .

, . .

Evaluate .

Answer

The area of a parallelogram is the product of the length of any one side, or its base, and the length of a segment perpendicular to that side, or its height.

One way to find the area is to multiply the length of side $\overline{CD}$ by its corresponding altitude, $\overline{AY}$ . Since and ,

$A = (CD)(AY) = 120 \cdot 80 = 9,600$ .

Another way to find the area is to multiply the length of side $\overline{BC}$ by its corresponding altitude, $\overline{AX}$ . Since and the area is 9,600, we set up this equation and solve for :

$100 \cdot AX =9,600$

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Question

Find the area of a parallelogram with a base of length 8 and a height of length 6.

Answer

The area of a parallelogram is

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Question

What is the area of a parallelogram with a base of and a height of ?

Answer

Write the formula for the area of a parallelogram.

Substitute the dimensions.

$A=7\times 8 = 56$

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Question

Find the area of a parallelogram if the base length is and the height is .

Answer

Write the formula for thearea of a parallelogram.

Substitute the dimensions.

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Question

Find the area of a parallelogram with a base of 4 and a height of 40.

Answer

Write the formula to find the area of a parallelogram.

Substitute the dimensions.

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Question

Find the area of a parallelogram if the base and height are and , respectively.

Answer

Write the formula for the area of a parallelogram.

Substitute the dimensions. Use the distributive property. The distibutive property means to multiply the monomial term, , to each term within the parentheses. Remember when like bases are multiplied, their exponents are added together. Thus resulting in the following area.

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Question

Find the area of a parallelogram if both the base and height have a length of 10.

Answer

Write the area of the parallelogram.

Substitute the dimensions.

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Question

Find the area of a parallelogram in $\textup{inches}$ if the base is $\textup{3 feet}$ and the height is $\textup{3 inches}$ .

Answer

Convert the base from feet to inches. There are $\textup{12 inches}$ in a foot. Multiply the base by .

$b=3 \textup{ feet }\times \frac{12\textup{ inches }}{1 \textup{ foot}} = 36 \textup{ inches}$

Write the area for a parallelogram.

Substitute the new base and the height.

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Question

Find the area of a parallelogram with a base of and a height of .

Answer

Write the formula to find the area of a parallelogram.

Substitute the dimensions. Use the distributive property.

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Question

Find the area of a parallelogram if the base is 3 inches and the height is 8 inches.

Answer

Write the formula for the area of a parallelogram.

Substitute the dimensions.

$A= (3 \textup{ inches})(8\textup{ inches}) = 24 \textup{ inches} ^2$

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Question

Find the area of a parallelogram with the base of $\frac{8}{3}$ and a height of $\frac{2}{7}$ .

Answer

Write the formula for the area of a parallelogram.

Substitute the dimensions.

$A=\frac{8}{3} \times \frac{2}{7} = \frac{16}{21}$

Remember, when multiplying fractions, multiply the numerators together and multiply the denominators together.

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Question

Determine the area of a parallelogram with a base and height of $3\sqrt{5}$ .

Answer

Write the formula to find the area of a parallelogram. Substitute the dimensions.

$A=bh = (3\sqrt5)(3\sqrt5) = 3\times 3\times 5 = 45$

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