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Intermediate Geometry : How to find the area of an equilateral triangle

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Example Questions

Example Question #696 : Intermediate Geometry

A circle is placed in an equilateral triangle as shown by the figure.

If the radius of the circle is $\frac{1}{4}$ $\displaystyle \frac{1}{4}$, find the area of the shaded region.

Possible Answers:

7.12 $\displaystyle 7.12$

5.44 $\displaystyle 5.44$

5.98 $\displaystyle 5.98$

6.73 $\displaystyle 6.73$

Correct answer:

6.73 $\displaystyle 6.73$

Explanation:

In order to find the area of the shaded region, you must first find the areas of the triangle and circle.

First, recall how to find the area of an equilateral triangle:

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

Substitute in the value of the side to find the area of the triangle.

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

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

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

Substitute in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times \left(\frac{1}{4}\right)^2$ $\displaystyle \text{Area of Circle}=\pi\times \left(\frac{1}{4}\right)^2$

$\text{Area of Circle}=\frac{1}{16}\pi$ $\displaystyle \text{Area of Circle}=\frac{1}{16}\pi$

Finally, find the area of the shaded region.

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

$\text{Area of Shaded Region}=4\sqrt3-\frac{1}{16}\pi$ $\displaystyle \text{Area of Shaded Region}=4\sqrt3-\frac{1}{16}\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=6.73$ $\displaystyle \text{Area of Shaded Region}=6.73$

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Example Question #697 : Intermediate Geometry

A circle is placed in an equilateral triangle as shown by the figure.

If the radius of the circle is $\frac{1}{3}$ $\displaystyle \frac{1}{3}$, find the area of the shaded region.

Possible Answers:

0.51 $\displaystyle 0.51$

0.12 $\displaystyle 0.12$

1.02 $\displaystyle 1.02$

0.08 $\displaystyle 0.08$

Correct answer:

0.08 $\displaystyle 0.08$

Explanation:

In order to find the area of the shaded region, you must first find the areas of the triangle and circle.

First, recall how to find the area of an equilateral triangle:

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

Substitute in the value of the side to find the area of the triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(1)^2$ $\displaystyle \text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(1)^2$

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

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

Substitute in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times \left(\frac{1}{3}\right)^2$ $\displaystyle \text{Area of Circle}=\pi\times \left(\frac{1}{3}\right)^2$

$\text{Area of Circle}=\frac{1}{9}\pi$ $\displaystyle \text{Area of Circle}=\frac{1}{9}\pi$

Finally, find the area of the shaded region.

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

$\text{Area of Shaded Region}=\frac{\sqrt3}{4}-\frac{1}{9}\pi$ $\displaystyle \text{Area of Shaded Region}=\frac{\sqrt3}{4}-\frac{1}{9}\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=0.08$ $\displaystyle \text{Area of Shaded Region}=0.08$

Report an Error

Example Question #698 : Intermediate Geometry

A circle is placed in an equilateral triangle as shown by the figure.

If the radius of the circle is $\displaystyle 1$, find the area of the shaded region.

Possible Answers:

0.76 $\displaystyle 0.76$

0.79 $\displaystyle 0.79$

0.66 $\displaystyle 0.66$

0.84 $\displaystyle 0.84$

Correct answer:

0.76 $\displaystyle 0.76$

Explanation:

In order to find the area of the shaded region, you must first find the areas of the triangle and circle.

First, recall how to find the area of an equilateral triangle:

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

Substitute in the value of the side to find the area of the triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(3)^2$ $\displaystyle \text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(3)^2$

$\text{Area of Equilateral Triangle}=\frac{9\sqrt3}{4}$ $\displaystyle \text{Area of Equilateral Triangle}=\frac{9\sqrt3}{4}$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

Substitute in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 1^2$ $\displaystyle \text{Area of Circle}=\pi\times 1^2$

$\text{Area of Circle}=\pi$ $\displaystyle \text{Area of Circle}=\pi$

Finally, find the area of the shaded region.

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

$\text{Area of Shaded Region}=\frac{9\sqrt3}{4}-\pi$ $\displaystyle \text{Area of Shaded Region}=\frac{9\sqrt3}{4}-\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=0.76$ $\displaystyle \text{Area of Shaded Region}=0.76$

Report an Error

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

A circle is placed in an equilateral triangle as shown by the figure.

If the radius of the circle is $\displaystyle 2$, find the area of the shaded region.

Possible Answers:

71.00 $\displaystyle 71.00$

75.60 $\displaystyle 75.60$

69.20 $\displaystyle 69.20$

72.30 $\displaystyle 72.30$

Correct answer:

72.30 $\displaystyle 72.30$

Explanation:

In order to find the area of the shaded region, you must first find the areas of the triangle and circle.

First, recall how to find the area of an equilateral triangle:

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

Substitute in the value of the side to find the area of the triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(14)^2$ $\displaystyle \text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(14)^2$

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

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

Substitute in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 2^2$ $\displaystyle \text{Area of Circle}=\pi\times 2^2$

$\text{Area of Circle}=4\pi$ $\displaystyle \text{Area of Circle}=4\pi$

Finally, find the area of the shaded region.

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

$\text{Area of Shaded Region}=49\sqrt3-4\pi$ $\displaystyle \text{Area of Shaded Region}=49\sqrt3-4\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=72.30$ $\displaystyle \text{Area of Shaded Region}=72.30$

Report an Error

Example Question #700 : Intermediate Geometry

A circle is placed in an equilateral triangle as shown by the figure.

If the radius of the circle is $\displaystyle 4$, find the area of the shaded region.

Possible Answers:

28.09 $\displaystyle 28.09$

19.51 $\displaystyle 19.51$

24.05 $\displaystyle 24.05$

22.91 $\displaystyle 22.91$

Correct answer:

22.91 $\displaystyle 22.91$

Explanation:

In order to find the area of the shaded region, you must first find the areas of the triangle and circle.

First, recall how to find the area of an equilateral triangle:

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

Substitute in the value of the side to find the area of the triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(13)^2$ $\displaystyle \text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(13)^2$

$\text{Area of Equilateral Triangle}=\frac{169\sqrt3}{4}$ $\displaystyle \text{Area of Equilateral Triangle}=\frac{169\sqrt3}{4}$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

Substitute in the value of the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 4^2$ $\displaystyle \text{Area of Circle}=\pi\times 4^2$

$\text{Area of Circle}=16\pi$ $\displaystyle \text{Area of Circle}=16\pi$

Finally, find the area of the shaded region.

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

$\text{Area of Shaded Region}=\frac{169\sqrt3}{4}-16\pi$ $\displaystyle \text{Area of Shaded Region}=\frac{169\sqrt3}{4}-16\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=22.91$ $\displaystyle \text{Area of Shaded Region}=22.91$

Report an Error

Example Question #51 : Equilateral Triangles

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:

10.51 $\displaystyle 10.51$

12.12 $\displaystyle 12.12$

8.82 $\displaystyle 8.82$

14.21 $\displaystyle 14.21$

Correct answer:

12.12 $\displaystyle 12.12$

Explanation:

Recall that a regular hexagon can be divided into $\displaystyle 6$ 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 $\displaystyle 7$ 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:

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

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

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

Now, multiply this area by $\displaystyle 7$ to find the area of the entire figure.

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

$\text{Area of Figure}=7(\sqrt3)=12.12$ $\displaystyle \text{Area of Figure}=7(\sqrt3)=12.12$

Make sure to round to $\displaystyle 2$ places after the decimal.

Report an Error

Example Question #52 : Equilateral Triangles

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:

48.50 $\displaystyle 48.50$

The area of the entire figure cannot be determined by the given information.

55.22 $\displaystyle 55.22$

42.19 $\displaystyle 42.19$

Correct answer:

48.50 $\displaystyle 48.50$

Explanation:

Recall that a regular hexagon can be divided into $\displaystyle 6$ 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:

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

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

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

Now, multiply this area by $\displaystyle 7$ to find the area of the entire figure.

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

$\text{Area of Figure}=7(4\sqrt3)=48.50$ $\displaystyle \text{Area of Figure}=7(4\sqrt3)=48.50$

Make sure to round to $\displaystyle 2$ places after the decimal.

Report an Error

Example Question #53 : Equilateral Triangles

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:

82.30 $\displaystyle 82.30$

72.30 $\displaystyle 72.30$

79.77 $\displaystyle 79.77$

75.78 $\displaystyle 75.78$

Correct answer:

75.78 $\displaystyle 75.78$

Explanation:

Recall that a regular hexagon can be divided into $\displaystyle 6$ 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:

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

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

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(\text{5})^2}{4}=\frac{25\sqrt3}{4}$ $\displaystyle \text{Area of Equilateral Triangle}=\frac{\sqrt3(\text{5})^2}{4}=\frac{25\sqrt3}{4}$

Now, multiply this area by $\displaystyle 7$ to find the area of the entire figure.

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

$\text{Area of Figure}=7(\frac{25\sqrt3}{4})=75.78$ $\displaystyle \text{Area of Figure}=7(\frac{25\sqrt3}{4})=75.78$

Make sure to round to $\displaystyle 2$ places after the decimal.

Report an Error

Example Question #61 : Equilateral Triangles

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

Possible Answers:

120.41 $\displaystyle 120.41$

104.97 $\displaystyle 104.97$

109.12 $\displaystyle 109.12$

112.59 $\displaystyle 112.59$

Correct answer:

109.12 $\displaystyle 109.12$

Explanation:

Recall that a regular hexagon can be divided into $\displaystyle 6$ 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:

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

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

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

Now, multiply this area by $\displaystyle 7$ to find the area of the entire figure.

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

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

Make sure to round to $\displaystyle 2$ places after the decimal.

Report an Error

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.

Find the area of the entire figure.

Possible Answers:

176.58 $\displaystyle 176.58$

198.41 $\displaystyle 198.41$

193.99 $\displaystyle 193.99$

185.23 $\displaystyle 185.23$

Correct answer:

193.99 $\displaystyle 193.99$

Explanation:

Recall that a regular hexagon can be divided into $\displaystyle 6$ 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:

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

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

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

Now, multiply this area by $\displaystyle 7$ to find the area of the entire figure.

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

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

Make sure to round to $\displaystyle 2$ places after the decimal.

Report an Error

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Intermediate Geometry : How to find the area of an equilateral triangle

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #696 : Intermediate Geometry

Example Question #697 : Intermediate Geometry

Example Question #698 : Intermediate Geometry

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

Example Question #700 : Intermediate Geometry

Example Question #51 : Equilateral Triangles

Example Question #52 : Equilateral Triangles

Example Question #53 : Equilateral Triangles

Example Question #61 : Equilateral Triangles

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

All Intermediate Geometry Resources

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