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Intermediate Geometry : Intermediate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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8 Diagnostic Tests 250 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #131 : Hexagons

A rectangle is attached to a regular hexagon as shown by the figure.

If the diagonal of the hexagon is , find the area of the entire figure.

Possible Answers:

1624.90

1490.63

1560.98

1519.23

Correct answer:

1519.23

Explanation:

When all of the diagonals of a regular hexagon are drawn in, you should notice that the hexagon is divided into six congruent equilateral triangles. The length of the diagonal is twice the length of a side of one of the equilateral triangles.

Thus, the diagonal of a hexagon is also twice the length of a side of the regular hexagon.

$\text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$

Substitute in the given diagonal to find the side length of the hexagon.

$\text{Side Length of Hexagon}=\frac{40}{2}=20$

Now, recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3(\text{side length})^2}{2}$

Substitute in the value of the side length to find the area of the hexagon.

$\text{Area of Hexagon}=\frac{3\sqrt3(20)^2}{2}=600\sqrt3$

Next, notice that the hexagon shares a side with the length of the rectangle.

Recall how to find the area of a rectangle.

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

Substitute in the length and the width of the rectangle to find the area.

$\text{Area of Rectangle}=20\times 24=480$ .

In order to find the area of the entire figure, add the areas of the hexagon and rectangle together.

$\text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$

$\text{Area of Entire Figure}=600\sqrt3+480$

Solve and round to two decimal places.

$\text{Area of Entire Figure}=1519.23$

Report an Error

Example Question #132 : Hexagons

A rectangle is attached to a regular hexagon as shown by the figure.

If the diagonal of the hexagon is , find the area of the entire figure.

Possible Answers:

379.01

373.44

351.20

369.81

Correct answer:

369.81

Explanation:

Thus, the diagonal of a hexagon is also twice the length of a side of the regular hexagon.

$\text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$

Substitute in the given diagonal to find the side length of the hexagon.

$\text{Side Length of Hexagon}=\frac{20}{2}=10$

Now, recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3(\text{side length})^2}{2}$

Substitute in the value of the side length to find the area of the hexagon.

$\text{Area of Hexagon}=\frac{3\sqrt3(10)^2}{2}=150\sqrt3$

Next, notice that the hexagon shares a side with the length of the rectangle.

Recall how to find the area of a rectangle.

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

Substitute in the length and the width of the rectangle to find the area.

$\text{Area of Rectangle}=10\times 11=110$ .

In order to find the area of the entire figure, add the areas of the hexagon and rectangle together.

$\text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$

$\text{Area of Entire Figure}=150\sqrt3+110$

Solve and round to two decimal places.

$\text{Area of Entire Figure}=369.81$

Report an Error

Example Question #133 : Hexagons

A rectangle is attached to a regular hexagon as shown by the figure.

If the diagonal of the hexagon is , find the area of the entire figure.

Possible Answers:

491.24

590.67

512.20

566.12

Correct answer:

566.12

Explanation:

Thus, the diagonal of a hexagon is also twice the length of a side of the regular hexagon.

$\text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$

Substitute in the given diagonal to find the side length of the hexagon.

$\text{Side Length of Hexagon}=\frac{24}{2}=12$

Now, recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3(\text{side length})^2}{2}$

Substitute in the value of the side length to find the area of the hexagon.

$\text{Area of Hexagon}=\frac{3\sqrt3(12)^2}{2}=216\sqrt3$

Next, notice that the hexagon shares a side with the length of the rectangle.

Recall how to find the area of a rectangle.

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

Substitute in the length and the width of the rectangle to find the area.

$\text{Area of Rectangle}=12\times 16=192$ .

In order to find the area of the entire figure, add the areas of the hexagon and rectangle together.

$\text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$

$\text{Area of Entire Figure}=216\sqrt3+192$

Solve and round to two decimal places.

$\text{Area of Entire Figure}=566.12$

Report an Error

Example Question #134 : Hexagons

A rectangle is attached to a regular hexagon as shown by the figure.

If the diagonal of the hexagon is , find the area of the entire figure.

Possible Answers:

990.08

984.21

1002.11

985.11

Correct answer:

985.11

Explanation:

Thus, the diagonal of a hexagon is also twice the length of a side of the regular hexagon.

$\text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$

Substitute in the given diagonal to find the side length of the hexagon.

$\text{Side Length of Hexagon}=\frac{32}{2}=16$

Now, recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3(\text{side length})^2}{2}$

Substitute in the value of the side length to find the area of the hexagon.

$\text{Area of Hexagon}=\frac{3\sqrt3(16)^2}{2}=384\sqrt3$

Next, notice that the hexagon shares a side with the length of the rectangle.

Recall how to find the area of a rectangle.

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

Substitute in the length and the width of the rectangle to find the area.

$\text{Area of Rectangle}=16\times 20=320$ .

In order to find the area of the entire figure, add the areas of the hexagon and rectangle together.

$\text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$

$\text{Area of Entire Figure}=384\sqrt3+320$

Solve and round to two decimal places.

$\text{Area of Entire Figure}=985.11$

Report an Error

Example Question #135 : Hexagons

A rectangle is attached to a regular hexagon as shown by the figure.

If the diagonal of the hexagon is , find the area of the entire figure.

Possible Answers:

1805.68

1577.63

1790.24

1639.23

Correct answer:

1639.23

Explanation:

Thus, the diagonal of a hexagon is also twice the length of a side of the regular hexagon.

$\text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$

Substitute in the given diagonal to find the side length of the hexagon.

$\text{Side Length of Hexagon}=\frac{40}{2}=20$

Now, recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3(\text{side length})^2}{2}$

Substitute in the value of the side length to find the area of the hexagon.

$\text{Area of Hexagon}=\frac{3\sqrt3(20)^2}{2}=600\sqrt3$

Next, notice that the hexagon shares a side with the length of the rectangle.

Recall how to find the area of a rectangle.

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

Substitute in the length and the width of the rectangle to find the area.

$\text{Area of Rectangle}=20\times 30=600$ .

In order to find the area of the entire figure, add the areas of the hexagon and rectangle together.

$\text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$

$\text{Area of Entire Figure}=699\sqrt3+600$

Solve and round to two decimal places.

$\text{Area of Entire Figure}=1639.23$

Report an Error

Example Question #136 : Hexagons

A rectangle is attached to a regular hexagon as shown by the figure.

If the diagonal of the hexagon is , find the area of the entire figure.

Possible Answers:

624.29

639.60

644.41

621.07

Correct answer:

621.07

Explanation:

Thus, the diagonal of a hexagon is also twice the length of a side of the regular hexagon.

$\text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$

Substitute in the given diagonal to find the side length of the hexagon.

$\text{Side Length of Hexagon}=\frac{26}{2}=13$

Now, recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3(\text{side length})^2}{2}$

Substitute in the value of the side length to find the area of the hexagon.

$\text{Area of Hexagon}=\frac{3\sqrt3(13)^2}{2}=\frac{507\sqrt3}{2}$

Next, notice that the hexagon shares a side with the length of the rectangle.

Recall how to find the area of a rectangle.

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

Substitute in the length and the width of the rectangle to find the area.

$\text{Area of Rectangle}=13\times 14=182$ .

In order to find the area of the entire figure, add the areas of the hexagon and rectangle together.

$\text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$

$\text{Area of Entire Figure}=\frac{507\sqrt3}{2}+182$

Solve and round to two decimal places.

$\text{Area of Entire Figure}=621.07$

Report an Error

Example Question #137 : Hexagons

A rectangle is attached to a regular hexagon as shown by the figure.

If the diagonal of the hexagon is , find the area of the entire figure.

Possible Answers:

327.44

328.95

314.15

290.57

Correct answer:

327.44

Explanation:

Thus, the diagonal of a hexagon is also twice the length of a side of the regular hexagon.

$\text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$

Substitute in the given diagonal to find the side length of the hexagon.

$\text{Side Length of Hexagon}=\frac{18}{2}=9$

Now, recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3(\text{side length})^2}{2}$

Substitute in the value of the side length to find the area of the hexagon.

$\text{Area of Hexagon}=\frac{3\sqrt3(9)^2}{2}=\frac{243\sqrt3}{2}$

Next, notice that the hexagon shares a side with the length of the rectangle.

Recall how to find the area of a rectangle.

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

Substitute in the length and the width of the rectangle to find the area.

$\text{Area of Rectangle}=9\times 13=117$ .

In order to find the area of the entire figure, add the areas of the hexagon and rectangle together.

$\text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$

$\text{Area of Entire Figure}=\frac{243\sqrt3}{2}+117$

Solve and round to two decimal places.

$\text{Area of Entire Figure}=327.44$

Report an Error

Example Question #41 : How To Find The Area Of A Hexagon

A rectangle is attached to a regular hexagon as shown by the figure.

If the diagonal of the hexagon is , find the area of the entire figure.

Possible Answers:

3110.92

3190.59

3328.27

3224.44

Correct answer:

3328.27

Explanation:

Thus, the diagonal of a hexagon is also twice the length of a side of the regular hexagon.

$\text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$

Substitute in the given diagonal to find the side length of the hexagon.

$\text{Side Length of Hexagon}=\frac{60}{2}=30$

Now, recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3(\text{side length})^2}{2}$

Substitute in the value of the side length to find the area of the hexagon.

$\text{Area of Hexagon}=\frac{3\sqrt3(30)^2}{2}=1350\sqrt3$

Next, notice that the hexagon shares a side with the length of the rectangle.

Recall how to find the area of a rectangle.

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

Substitute in the length and the width of the rectangle to find the area.

$\text{Area of Rectangle}=30\times 33=990$ .

In order to find the area of the entire figure, add the areas of the hexagon and rectangle together.

$\text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$

$\text{Area of Entire Figure}=1350\sqrt3+990$

Solve and round to two decimal places.

$\text{Area of Entire Figure}=3328.27$

Report an Error

Example Question #42 : How To Find The Area Of A Hexagon

If a regular hexagon has a side length of $8.7\;cm$ , what would be the area? Assume all sides are of equal length and round to the nearest tenth.

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Possible Answers:

$75.7\;cm^2$

$90.4\;cm^2$

$454.1\;cm^2$

$121.7\;cm^2$

$196.7\;cm^2$

Correct answer:

$196.7\;cm^2$

Explanation:

The following equation can be used to solve for area of a regular polygon, given the side length (s) and number of sides (N) :

$\\Area=\frac{s^2N}{4tan(\frac{180^{\circ}}{N})}\\ \\ \\Area=\frac{(8.7\;cm)^2(6)}{4tan(\frac{180^{\circ}}{6})}$

$Area=\frac{454.14\;cm^2}{2.3094}\approx196.7\;cm^2$

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Example Question #941 : Intermediate Geometry

A regular hexagon has a side length of $7.6\;cm$ . Assuming all sides are of equal length, what is the area of the hexagon rounded to the nearest tenth?

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Possible Answers:

$150.1\;cm^2$

$178.4\;cm^2$

$180.3\;cm^2$

$79\;cm^2$

$223.2\;cm^2$

Correct answer:

$150.1\;cm^2$

Explanation:

We can solve for area using the given formula that works for all regular polygons, with being the side length and representing the number of sides:

$\\Area=\frac{s^2N}{4tan(\frac{180^{\circ}}{N})}\\ \\ \\Area=\frac{(7.6\;cm)^2(6)}{4tan(\frac{180^{\circ}}{6})}$

$Area=\frac{346.56\;cm^2}{2.3094}\approx150.1\;cm^2$

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Intermediate Geometry : Intermediate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #131 : Hexagons

Example Question #132 : Hexagons

Example Question #133 : Hexagons

Example Question #134 : Hexagons

Example Question #135 : Hexagons

Example Question #136 : Hexagons

Example Question #137 : Hexagons

Example Question #41 : How To Find The Area Of A Hexagon

Example Question #42 : How To Find The Area Of A Hexagon

Example Question #941 : Intermediate Geometry

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