Geometry - 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.

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:

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

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

$\text{Area of Rectangle}=8 \times 14$

$\text{Area of Rectangle}=112$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

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

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

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

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

$\text{height}=\frac{8}{2}$

$\text{height}=4$

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

$\text{Area of Parallelogram}=10 \times 4$

$\text{Area of Parallelogram}=40$

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.

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

$\text{Area of Shaded Region}=112-40$

Find the area of the parallelogram.

Explanation

Recall how to find the area of a parallelogram:

$\text{Area}=\text{base}\times\text{height}$

From the given parallelogram, we will need to use the Pythagorean Theorem to find the length of the height of the parallelogram.

$\text{Hypotenuse}^2=\text{Triangle base}^2+\text{height}^2$

$\text{height}^2=\text{Hypotenuse}^2-\text{Triangle base}^2$

$\text{height}=\sqrt{\text{Hypotenuse}^2-\text{Triangle base}^2}$

Plug in the given values to find the length of the height.

$\text{height}=\sqrt{5^2-2^2}=\sqrt{25-4}=\sqrt{21}$

Now, use the height to find the area of the parallelogram.

$\text{Area}=12\times\sqrt{21}=54.99$

Remember to round to places after the decimal.

In the figure, the area of the parallelogram is . Find the length of the base.

Cannot be determined

Explanation

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=(\text{base})(\text{height})$

Now, substitute in the area, base, and height values that are given by the question.

Expand this equation.

Now factor this equation.

Solve for .

Since the values of lengths of shapes can only be positive, we know that the base of the parallelogram must be .

In the figure, the height of the parallelogram is half the height of the equilateral triangle. Find the area of the shaded region.

Explanation

First, we will need to find the height of the equilateral triangle.

Recall that the height of an equilateral triangle will cut the equilateral triangle into two congruent $30^\circ-60^\circ-90^\circ$ triangles that have side lengths in the the following ratio:

$1:\sqrt3:2$

Use the given side length of the equilateral triangle in order to find the length of the height.

$\frac{\text{side length}}{2}=\frac{\text{height}}{\sqrt3}$

$\text{height}=(\sqrt3)\times \frac{\text{side length}}{2}$

$\text{height}=\frac{22(\sqrt3)}{2}$

$\text{height}=11\sqrt3$

Now, find the area of the equilateral triangle.

$\text{Area of Equilateral Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Equilateral Triangle}=\frac{1}{2}(22\times11\sqrt3)$

$\text{Area of Equilateral Triangle}=121\sqrt3$

Now, use the height of the equilateral triangle to find the height of the parallelogram.

$\text{height of parallelogram}=\frac{\text{height of triangle}}{2}$

$\text{height of parallelogram}=\frac{11\sqrt3}{2}$

Next, find the area of the parallelogram.

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

$\text{Area of Parallelogram}=10\times \frac{11\sqrt3}{2}$

$\text{Area of Parallelogram}=55\sqrt3$

Now, from the figure, we can see that we will need to subtract the area of the parallelogram from the area of the triangle in order to find the area of the shaded region.

$\text{Area of Shaded Region}=\text{Area of Equilateral Triangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=121\sqrt3-55\sqrt3$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=114.32$

Find the area of the parallelogram.

Explanation

Recall how to find the area of a parallelogram:

$\text{Area}=\text{base}\times\text{height}$

From the given parallelogram, we will need to use the Pythagorean Theorem to find the length of the height of the parallelogram.

$\text{Hypotenuse}^2=\text{Triangle base}^2+\text{height}^2$

$\text{height}^2=\text{Hypotenuse}^2-\text{Triangle base}^2$

$\text{height}=\sqrt{\text{Hypotenuse}^2-\text{Triangle base}^2}$

Plug in the given values to find the length of the height.

$\text{height}=\sqrt{9^2-4^2}=\sqrt{81-16}=\sqrt{65}$

Now, use the height to find the area of the parallelogram.

$\text{Area}=12\times\sqrt{65}=96.75$

Remember to round to places after the decimal.

Find the area of the parallelogram.

Explanation

Recall how to find the area of a parallelogram:

$\text{Area}=\text{base}\times\text{height}$

From the given parallelogram, we will need to use the Pythagorean Theorem to find the length of the height of the parallelogram.

$\text{Hypotenuse}^2=\text{Triangle base}^2+\text{height}^2$

$\text{height}^2=\text{Hypotenuse}^2-\text{Triangle base}^2$

$\text{height}=\sqrt{\text{Hypotenuse}^2-\text{Triangle base}^2}$

Plug in the given values to find the length of the height.

$\text{height}=\sqrt{15^2-9^2}=\sqrt{225-81}=12$

Now, use the height to find the area of the parallelogram.

$\text{Area}=12\times20=240$

Remember to round to places after the decimal.

In the figure, the area of the parallelogram is . Find the length of the base.

Explanation

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=(\text{base})(\text{height})$

Now, substitute in the area, base, and height values that are given by the question.

Expand this equation.

Now factor this equation.

Solve for .

Since the values of lengths of shapes can only be positive, we know that the base of the parallelogram must be .

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.

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:

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

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

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

$\text{Area of Rectangle}=12$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

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

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

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

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

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

$\text{height}=1$

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

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

$\text{Area of Parallelogram}=3$

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.

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

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

Solve.

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

The perimeter of a square is . If the sides of the square are reduced by a factor of two, what is the area of the new square?

$9; in^{2}$

$36; in^{2}$

$27; in^{2}$

$18; in^{2}$

$45; in^{2}$

Explanation

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

Thus

so .

The original side is reduced by a factor of which results in a new side of . The area of the new square is geven by

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

In the figure, the area of the parallelogram is . Find the length of the base.

Explanation

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=(\text{base})(\text{height})$

Now, substitute in the area, base, and height values that are given by the question.

Expand this equation.

Now factor this equation.

Solve for .

Since the values of lengths of shapes can only be positive, we know that the base of the parallelogram must be .

How to find the area of a parallelogram

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