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Intermediate Geometry : Equilateral Triangles

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Example Question #31 : How To Find The Area Of An Equilateral Triangle

An equilateral triangle is placed on top of a regular hexagon as shown by the figure below.

Possible Answers:

112.59

109.12

104.97

120.41

Correct answer:

109.12

Explanation:

Recall that a regular hexagon can be divided into congruent equilateral triangles.

Since the extra equilateral triangle on top has the same side lengths as of the equilateral triangles made by dividing the hexagon, the entire figure has a total of congruent equilateral triangles.

Thus, to find the area of the entire figure, we must first find the area of the equilateral triangle.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(\text{side})^2}{4}$

Plug in the length of a side of the equilateral triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(6)^2}{4}=\frac{36\sqrt3}{4}=9\sqrt3$

Now, multiply this area by to find the area of the entire figure.

$\text{Area of Figure}=7(\text{Area of Equilateral Triangle})$

$\text{Area of Figure}=7(9\sqrt3)=109.12$

Make sure to round to places after the decimal.

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Example Question #32 : How To Find The Area Of An Equilateral Triangle

An equilateral triangle is placed on top of a regular hexagon as shown by the figure below.

Find the area of the entire figure.

Possible Answers:

193.99

198.41

176.58

185.23

Correct answer:

193.99

Explanation:

Recall that a regular hexagon can be divided into congruent equilateral triangles.

Since the extra equilateral triangle on top has the same side lengths as of the equilateral triangles made by dividing the hexagon, the entire figure has a total of congruent equilateral triangles.

Thus, to find the area of the entire figure, we must first find the area of the equilateral triangle.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(\text{side})^2}{4}$

Plug in the length of a side of the equilateral triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(8)^2}{4}=\frac{64\sqrt3}{4}=16\sqrt3$

Now, multiply this area by to find the area of the entire figure.

$\text{Area of Figure}=7(\text{Area of Equilateral Triangle})$

$\text{Area of Figure}=7(16\sqrt3)=193.99$

Make sure to round to places after the decimal.

Report an Error

Example Question #41 : How To Find The Area Of An Equilateral Triangle

An equilateral triangle is placed on top of a regular hexagon as shown by the figure below.

Find the area of the entire figure.

Possible Answers:

501.28

491.50

436.48

421.82

Correct answer:

436.48

Explanation:

Recall that a regular hexagon can be divided into congruent equilateral triangles.

Since the extra equilateral triangle on top has the same side lengths as of the equilateral triangles made by dividing the hexagon, the entire figure has a total of congruent equilateral triangles.

Thus, to find the area of the entire figure, we must first find the area of the equilateral triangle.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(\text{side})^2}{4}$

Plug in the length of a side of the equilateral triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(12)^2}{4}=\frac{144\sqrt3}{4}=36\sqrt3$

Now, multiply this area by to find the area of the entire figure.

$\text{Area of Figure}=7(\text{Area of Equilateral Triangle})$

$\text{Area of Figure}=7(36\sqrt3)=436.48$

Make sure to round to places after the decimal.

Report an Error

Example Question #42 : How To Find The Area Of An Equilateral Triangle

An equilateral triangle is placed on top of a regular hexagon as shown by the figure below.

Find the area of the entire figure.

Possible Answers:

772.11

770.29

741.22

775.96

Correct answer:

775.96

Explanation:

Recall that a regular hexagon can be divided into congruent equilateral triangles.

Since the extra equilateral triangle on top has the same side lengths as of the equilateral triangles made by dividing the hexagon, the entire figure has a total of congruent equilateral triangles.

Thus, to find the area of the entire figure, we must first find the area of the equilateral triangle.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(\text{side})^2}{4}$

Plug in the length of a side of the equilateral triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(16)^2}{4}=\frac{256\sqrt3}{4}=64\sqrt3$

Now, multiply this area by to find the area of the entire figure.

$\text{Area of Figure}=7(\text{Area of Equilateral Triangle})$

$\text{Area of Figure}=7(64\sqrt3)=775.96$

Make sure to round to places after the decimal.

Report an Error

Example Question #43 : How To Find The Area Of An Equilateral Triangle

An equilateral triangle is placed on top of a regular hexagon as shown by the figure below.

Find the area of the entire figure.

Possible Answers:

1145.29

1122.01

1298.41

1212.44

Correct answer:

1212.44

Explanation:

Recall that a regular hexagon can be divided into congruent equilateral triangles.

Since the extra equilateral triangle on top has the same side lengths as of the equilateral triangles made by dividing the hexagon, the entire figure has a total of congruent equilateral triangles.

Thus, to find the area of the entire figure, we must first find the area of the equilateral triangle.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(\text{side})^2}{4}$

Plug in the length of a side of the equilateral triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(20)^2}{4}=\frac{400\sqrt3}{4}=100\sqrt3$

Now, multiply this area by to find the area of the entire figure.

$\text{Area of Figure}=7(\text{Area of Equilateral Triangle})$

$\text{Area of Figure}=7(100\sqrt3)=1212.44$

Make sure to round to places after the decimal.

Report an Error

Example Question #44 : How To Find The Area Of An Equilateral Triangle

An equilateral triangle is placed on top of a regular hexagon as shown by the figure below.

Find the area of the entire figure.

Possible Answers:

1721.40

1500.67

1745.91

1682.52

Correct answer:

1745.91

Explanation:

Recall that a regular hexagon can be divided into congruent equilateral triangles.

Since the extra equilateral triangle on top has the same side lengths as of the equilateral triangles made by dividing the hexagon, the entire figure has a total of congruent equilateral triangles.

Thus, to find the area of the entire figure, we must first find the area of the equilateral triangle.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(\text{side})^2}{4}$

Plug in the length of a side of the equilateral triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(24)^2}{4}=\frac{576\sqrt3}{4}=144\sqrt3$

Now, multiply this area by to find the area of the entire figure.

$\text{Area of Figure}=7(\text{Area of Equilateral Triangle})$

$\text{Area of Figure}=7(144\sqrt3)=1745.91$

Make sure to round to places after the decimal.

Report an Error

Example Question #45 : How To Find The Area Of An Equilateral Triangle

An equilateral triangle is placed on top of a regular hexagon as shown by the figure below.

Find the area of the entire figure.

Possible Answers:

2205.29

2424.54

2376.37

2149.20

Correct answer:

2376.37

Explanation:

Recall that a regular hexagon can be divided into congruent equilateral triangles.

Since the extra equilateral triangle on top has the same side lengths as of the equilateral triangles made by dividing the hexagon, the entire figure has a total of congruent equilateral triangles.

Thus, to find the area of the entire figure, we must first find the area of the equilateral triangle.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(\text{side})^2}{4}$

Plug in the length of a side of the equilateral triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(28)^2}{4}=\frac{784\sqrt3}{4}=196\sqrt3$

Now, multiply this area by to find the area of the entire figure.

$\text{Area of Figure}=7(\text{Area of Equilateral Triangle})$

$\text{Area of Figure}=7(196\sqrt3)=2376.37$

Make sure to round to places after the decimal.

Report an Error

Example Question #46 : How To Find The Area Of An Equilateral Triangle

An equilateral triangle is placed on top of a regular hexagon as shown by the figure below.

Find the area of the entire figure.

Possible Answers:

3098.59

2790.18

3199.26

3103.84

Correct answer:

3103.84

Explanation:

Recall that a regular hexagon can be divided into congruent equilateral triangles.

Since the extra equilateral triangle on top has the same side lengths as of the equilateral triangles made by dividing the hexagon, the entire figure has a total of congruent equilateral triangles.

Thus, to find the area of the entire figure, we must first find the area of the equilateral triangle.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(\text{side})^2}{4}$

Plug in the length of a side of the equilateral triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(32)^2}{4}=\frac{1024\sqrt3}{4}=256\sqrt3$

Now, multiply this area by to find the area of the entire figure.

$\text{Area of Figure}=7(\text{Area of Equilateral Triangle})$

$\text{Area of Figure}=7(256\sqrt3)=3103.84$

Make sure to round to places after the decimal.

Report an Error

Example Question #47 : How To Find The Area Of An Equilateral Triangle

An equilateral triangle is placed on top of a regular hexagon as shown by the figure below.

Find the area of the entire figure.

Possible Answers:

4510.20

3612.06

3938.29

3782.09

Correct answer:

3938.29

Explanation:

Recall that a regular hexagon can be divided into congruent equilateral triangles.

Since the extra equilateral triangle on top has the same side lengths as of the equilateral triangles made by dividing the hexagon, the entire figure has a total of congruent equilateral triangles.

Thus, to find the area of the entire figure, we must first find the area of the equilateral triangle.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(\text{side})^2}{4}$

Plug in the length of a side of the equilateral triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(36)^2}{4}=\frac{1296\sqrt3}{4}=324\sqrt3$

Now, multiply this area by to find the area of the entire figure.

$\text{Area of Figure}=7(\text{Area of Equilateral Triangle})$

$\text{Area of Figure}=7(324\sqrt3)=3928.29$

Make sure to round to places after the decimal.

Report an Error

Example Question #48 : How To Find The Area Of An Equilateral Triangle

Pyramid

The cube in the above diagram has edges of length 1. Give the area of $\bigtriangleup ABC$ .

Possible Answers:

$\frac{1}{2}$

$\frac{ 3\sqrt{3}}{4}$

$\frac{ \sqrt{3}}{3}$

$\frac{ \sqrt{3}}{2}$

$\frac{2}{3}$

Correct answer:

$\frac{ \sqrt{3}}{2}$

Explanation:

Each of the three triangles $\bigtriangleup ADB$ , $\bigtriangleup ADC$ , and $\bigtriangleup BDC$ has two legs of length 1; therefore, each is an isosceles right triangle, and each is a 45-45-90 triangle. By the 45-45-90 Triangle Theorem, each hypotenuse - $\overline{AB} , \overline{AC} , \overline{BC}$ - has length $\sqrt{2}$ times that of the legs, or simply $\sqrt{2}$ . This makes $\bigtriangleup ABC$ an equilateral triangle with common sidelength $s = \sqrt{2}$ . Using the area formula

$A = \frac{s^{2}\sqrt{3}}{4}$

and substituting, the area can be calculated as

$A = \frac{ (\sqrt{2})^{2}\sqrt{3}}{4} = \frac{2 \sqrt{3}}{4}= \frac{ \sqrt{3}}{2}$ .

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Intermediate Geometry : Equilateral Triangles

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #31 : How To Find The Area Of An Equilateral Triangle

Example Question #32 : How To Find The Area Of An Equilateral Triangle

Example Question #41 : How To Find The Area Of An Equilateral Triangle

Example Question #42 : How To Find The Area Of An Equilateral Triangle

Example Question #43 : How To Find The Area Of An Equilateral Triangle

Example Question #44 : How To Find The Area Of An Equilateral Triangle

Example Question #45 : How To Find The Area Of An Equilateral Triangle

Example Question #46 : How To Find The Area Of An Equilateral Triangle

Example Question #47 : How To Find The Area Of An Equilateral Triangle

Example Question #48 : How To Find The Area Of An Equilateral Triangle

All Intermediate Geometry Resources

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