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Intermediate Geometry : Parallelograms

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Example Question #18 : 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:

530

490

370

390

Correct answer:

490

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.

$\text{Area of Rectangle}=31\times 28$

$\text{Area of Rectangle}=868$

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.

$\text{height}=\frac{28}{2}$

$\text{height}=14$

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

$\text{Area of Parallelogram}=27\times 14$

$\text{Area of Parallelogram}=378$

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.

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

$\text{Area of Shaded Region}=868-378$

Solve.

$\text{Area of Shaded Region}=490$

Report an Error

Example Question #19 : How To Find The Area Of A Parallelogram

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

Possible Answers:

19.66

20.78

18.50

25.80

Correct answer:

20.78

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.

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

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

$\text{height}=\frac{12(\sqrt3)}{2}=6\sqrt3$

Now, find the area of the equilateral triangle.

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

$\text{Area of Equilateral Triangle}=\frac{1}{2}(12\times6\sqrt3)$

$\text{Area of Equilateral Triangle}=36\sqrt3$

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

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

$\text{height of parallelogram}=\frac{6\sqrt3}{2}$

$\text{height of parallelogram}=3\sqrt3$

Next, find the area of the parallelogram.

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

$\text{Area of Parallelogram}=8\times 3\sqrt3$

$\text{Area of Parallelogram}=24\sqrt3$

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.

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

$\text{Area of Shaded Region}=36\sqrt3-24\sqrt3$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=20.78$

Report an Error

Example Question #20 : How To Find The Area Of A Parallelogram

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

Possible Answers:

109.63

95.08

114.32

100.82

Correct answer:

114.32

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.

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

Report an Error

Example Question #21 : How To Find The Area Of A Parallelogram

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

Possible Answers:

151.55

174.89

159.87

162.53

Correct answer:

151.55

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.

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

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

$\text{height}=\frac{25(\sqrt3)}{2}$

Now, find the area of the equilateral triangle.

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

$\text{Area of Equilateral Triangle}=\frac{1}{2}(25\times\frac{25\sqrt3}{2})$

$\text{Area of Equilateral Triangle}=\frac{625\sqrt3}{4}$

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

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

$\text{height of parallelogram}=\frac{25\sqrt3}{2}\times\frac{1}{2}$

$\text{height of parallelogram}=\frac{25\sqrt3}{4}$

Next, find the area of the parallelogram.

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

$\text{Area of Parallelogram}=11\times \frac{25\sqrt3}{4}$

$\text{Area of Parallelogram}=\frac{275\sqrt3}{4}$

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.

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

$\text{Area of Shaded Region}=\frac{625\sqrt3}{4}-\frac{275\sqrt3}{4}$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=151.55$

Report an Error

Example Question #22 : How To Find The Area Of A Parallelogram

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

Possible Answers:

46.88

40.95

50.74

42.44

Correct answer:

42.44

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.

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

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

$\text{height}=\frac{14(\sqrt3)}{2}=7\sqrt3$

Now, find the area of the equilateral triangle.

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

$\text{Area of Equilateral Triangle}=\frac{1}{2}(14\times7\sqrt3)$

$\text{Area of Equilateral Triangle}=49\sqrt3$

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

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

$\text{height of parallelogram}=\frac{7\sqrt3}{2}$

Next, find the area of the parallelogram.

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

$\text{Area of Parallelogram}=7\times \frac{7\sqrt3}{2}$

$\text{Area of Parallelogram}=\frac{49\sqrt3}{2}$

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.

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

$\text{Area of Shaded Region}=49\sqrt3-\frac{49\sqrt3}{2}$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=42.44$

Report an Error

Example Question #401 : Intermediate Geometry

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

Possible Answers:

32.09

40.03

42.72

36.37

Correct answer:

36.37

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.

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

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

$\text{height}=\frac{14(\sqrt3)}{2}$

$\text{height}=7\sqrt3$

Now, find the area of the equilateral triangle.

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

$\text{Area of Equilateral Triangle}=\frac{1}{2}(14\times7\sqrt3)$

$\text{Area of Equilateral Triangle}=49\sqrt3$

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

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

$\text{height of parallelogram}=\frac{7\sqrt3}{2}$

Next, find the area of the parallelogram.

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

$\text{Area of Parallelogram}=8\times \frac{7\sqrt3}{2}$

$\text{Area of Parallelogram}=28\sqrt3$

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.

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

$\text{Area of Shaded Region}=49\sqrt3-28\sqrt3$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=36.37$

Report an Error

Example Question #24 : How To Find The Area Of A Parallelogram

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

Possible Answers:

35.09

34.64

31.94

32.28

Correct answer:

34.64

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.

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

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

$\text{height}=\frac{10(\sqrt3)}{2}=5\sqrt3$

Now, find the area of the equilateral triangle.

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

$\text{Area of Equilateral Triangle}=\frac{1}{2}(10\times5\sqrt3)$

$\text{Area of Equilateral Triangle}=25\sqrt3$

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

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

$\text{height of parallelogram}=\frac{5\sqrt3}{2}$

Next, find the area of the parallelogram.

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

$\text{Area of Parallelogram}=2\times \frac{5\sqrt3}{2}$

$\text{Area of Parallelogram}=5\sqrt3$

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.

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

$\text{Area of Shaded Region}=25\sqrt3-5\sqrt3$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=34.64$

Report an Error

Example Question #25 : How To Find The Area Of A Parallelogram

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

Possible Answers:

233.93

260.87

249.42

256.38

Correct answer:

249.42

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.

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

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

$\text{height}=\frac{36(\sqrt3)}{2}$

$\text{height}=18\sqrt3$

Now, find the area of the equilateral triangle.

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

$\text{Area of Equilateral Triangle}=\frac{1}{2}(36\times18\sqrt3)$

$\text{Area of Equilateral Triangle}=324\sqrt3$

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

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

$\text{height of parallelogram}=\frac{18\sqrt3}{2}$

$\text{height of parallelogram}=9\sqrt3$

Next, find the area of the parallelogram.

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

$\text{Area of Parallelogram}=20\times 9\sqrt3$

$\text{Area of Parallelogram}=180\sqrt3$

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.

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

$\text{Area of Shaded Region}=324\sqrt3-180\sqrt3$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=249.42$

Report an Error

Example Question #26 : How To Find The Area Of A Parallelogram

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

Possible Answers:

46.77

50.08

42.36

47.09

Correct answer:

46.77

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.

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

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

$\text{height}=\frac{18(\sqrt3)}{2}$

$\text{height}=9\sqrt3$

Now, find the area of the equilateral triangle.

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

$\text{Area of Equilateral Triangle}=\frac{1}{2}(18\times9\sqrt3)$

$\text{Area of Equilateral Triangle}=81\sqrt3$

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

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

$\text{height of parallelogram}=\frac{9\sqrt3}{2}$

Next, find the area of the parallelogram.

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

$\text{Area of Parallelogram}=12\times \frac{9\sqrt3}{2}$

$\text{Area of Parallelogram}=54\sqrt3$

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.

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

$\text{Area of Shaded Region}=81\sqrt3-54\sqrt3$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=46.77$

Report an Error

Example Question #21 : How To Find The Area Of A Parallelogram

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

Possible Answers:

6.09

6.50

7.11

12.00

Correct answer:

6.50

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.

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

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

$\text{height}=\frac{5(\sqrt3)}{2}$

Now, find the area of the equilateral triangle.

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

$\text{Area of Equilateral Triangle}=\frac{1}{2}\left(5\times\frac{5\sqrt3}{2}\right)$

$\text{Area of Equilateral Triangle}=\frac{25\sqrt3}{4}$

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

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

$\text{height of parallelogram}=\frac{5\sqrt3}{2}\times\frac{1}{2}$

$\text{height of parallelogram}=\frac{5\sqrt3}{4}$

Next, find the area of the parallelogram.

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

$\text{Area of Parallelogram}=2\times \frac{5\sqrt3}{4}$

$\text{Area of Parallelogram}=\frac{5\sqrt3}{2}$

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.

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

$\text{Area of Shaded Region}=\frac{25\sqrt3}{4}-\frac{5\sqrt3}{2}$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=6.50$

Report an Error

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Intermediate Geometry : Parallelograms

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #18 : How To Find The Area Of A Parallelogram

Example Question #19 : How To Find The Area Of A Parallelogram

Example Question #20 : How To Find The Area Of A Parallelogram

Example Question #21 : How To Find The Area Of A Parallelogram

Example Question #22 : How To Find The Area Of A Parallelogram

Example Question #401 : Intermediate Geometry

Example Question #24 : How To Find The Area Of A Parallelogram

Example Question #25 : How To Find The Area Of A Parallelogram

Example Question #26 : How To Find The Area Of A Parallelogram

Example Question #21 : How To Find The Area Of A Parallelogram

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