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Intermediate Geometry : Equilateral Triangles

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Example Question #681 : Intermediate Geometry

A circle with a radius of is inscribed in an equilateral triangle with side lengths of as shown by the figure below.

Find the area of the shaded region.

Possible Answers:

225.31

240.43

256.31

199.68

Correct answer:

240.43

Explanation:

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

Recall how to find the area of a circle:

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

Plug in the given radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 12^2=144\pi$

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

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

Plug in the length of the side of the triangle to find the area.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(40)^2=400\sqrt3$

In order to find the area of the shaded region, we will need to subtract the area of the circle from the area of the triangle.

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

$\text{Area of Shaded Region}=400\sqrt3-144\pi=240.43$

Make sure to round to places after the decimal.

Report an Error

Example Question #20 : How To Find The Area Of An Equilateral Triangle

A circle with a radius of $\frac{1}{2}$ is inscribed in an equilateral triangle with side lengths of as shown by the figure below.

Find the area of the shaded region.

Possible Answers:

12.68

10.04

10.25

11.03

Correct answer:

10.04

Explanation:

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

Recall how to find the area of a circle:

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

Plug in the given radius to find the area of the circle.

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

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

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

Plug in the length of the side of the triangle to find the area.

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

In order to find the area of the shaded region, we will need to subtract the area of the circle from the area of the triangle.

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

$\text{Area of Shaded Region}=\frac{25\sqrt3}{4}-\frac{1}{4}\pi=10.04$

Make sure to round to places after the decimal.

Report an Error

Example Question #21 : How To Find The Area Of An Equilateral Triangle

A circle with a radius of is inscribed in an equilateral triangle with side lengths of as shown by the figure below.

Find the area of the shaded region.

Possible Answers:

15.15

17.69

14.03

16.28

Correct answer:

15.15

Explanation:

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

Recall how to find the area of a circle:

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

Plug in the given radius to find the area of the circle.

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

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

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

Plug in the length of the side of the triangle to find the area.

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

In order to find the area of the shaded region, we will need to subtract the area of the circle from the area of the triangle.

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

$\text{Area of Shaded Region}=16\sqrt3-4\pi=15.15$

Make sure to round to places after the decimal.

Report an Error

Example Question #22 : How To Find The Area Of An Equilateral Triangle

A circle with a radius of is inscribed in an equilateral triangle with side lengths of as shown by the figure below.

Find the area of the shaded region.

Possible Answers:

77.08

62.38

70.38

69.57

Correct answer:

69.57

Explanation:

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

Recall how to find the area of a circle:

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

Plug in the given radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 8^2=64\pi$

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

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

Plug in the length of the side of the triangle to find the area.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(25)^2=\frac{625\sqrt3}{4}$

In order to find the area of the shaded region, we will need to subtract the area of the circle from the area of the triangle.

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

$\text{Area of Shaded Region}=\frac{625\sqrt3}{4}-64\pi=69.57$

Make sure to round to places after the decimal.

Report an Error

Example Question #23 : How To Find The Area Of An Equilateral Triangle

A circle with a radius of is inscribed in an equilateral triangle with side lengths of as shown by the figure below.

Find the area of the shaded region.

Possible Answers:

158.69

247.90

238.02

169.99

Correct answer:

169.99

Explanation:

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

Recall how to find the area of a circle:

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

Plug in the given radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 15^2=225\pi$

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

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

Plug in the length of the side of the triangle to find the area.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(45)^2=\frac{2025\sqrt3}{4}$

In order to find the area of the shaded region, we will need to subtract the area of the circle from the area of the triangle.

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

$\text{Area of Shaded Region}=\frac{2025\sqrt3}{4}-225\pi=169.99$

Make sure to round to places after the decimal.

Report an Error

Example Question #24 : 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 , what is the area of the shaded region?

Possible Answers:

3.90

4.15

5.67

4.60

Correct answer:

4.60

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:

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

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

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

$\text{Area of Equilateral Triangle}=48\sqrt3$

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

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

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

$\text{Area of Circle}=\pi\times 5^2$

$\text{Area of Circle}=25\pi$

Finally, find the area of the shaded region.

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

$\text{Area of Shaded Region}=48\sqrt3-25\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=4.60$

Report an Error

Example Question #25 : How To Find The Area Of An Equilateral Triangle

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

If the radius of the circle is , what is the area of the shaded region?

Possible Answers:

14.29

15.15

16.29

13.11

Correct answer:

15.15

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:

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

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

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

$\text{Area of Equilateral Triangle}=16\sqrt3$

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

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

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

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

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

Finally, find the area of the shaded region.

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

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

Solve and round to two decimal places.

$\text{Area of Shaded Region}=15.15$

Report an Error

Example Question #26 : 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 , find the area of the shaded region.

Possible Answers:

15.03

16.79

14.23

18.50

Correct answer:

15.03

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:

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

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

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(10)^2$

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

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

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

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

$\text{Area of Circle}=\pi\times 3^2$

$\text{Area of Circle}=9\pi$

Finally, find the area of the shaded region.

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

$\text{Area of Shaded Region}=25\sqrt3-9\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=15.03$

Report an Error

Example Question #27 : 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 , find the area of the shaded region.

Possible Answers:

30.58

29.91

32.31

35.12

Correct answer:

32.31

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:

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

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

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

$\text{Area of Equilateral Triangle}=64\sqrt3$

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

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

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

$\text{Area of Circle}=\pi\times 5^2$

$\text{Area of Circle}=25\pi$

Finally, find the area of the shaded region.

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

$\text{Area of Shaded Region}=64\sqrt3-25\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=32.31$

Report an Error

Example Question #28 : How To Find The Area Of An Equilateral Triangle

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

If the radius of the circle is , find the area of the shaded region.

Possible Answers:

29.95

27.20

30.91

25.46

Correct answer:

27.20

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:

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

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

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(18)^2$

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

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

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

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

$\text{Area of Circle}=\pi\times 6^2$

$\text{Area of Circle}=36\pi$

Finally, find the area of the shaded region.

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

$\text{Area of Shaded Region}=81\sqrt3-36\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=27.20$

Report an Error

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Intermediate Geometry : Equilateral Triangles

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #681 : Intermediate Geometry

Example Question #20 : How To Find The Area Of An Equilateral Triangle

Example Question #21 : How To Find The Area Of An Equilateral Triangle

Example Question #22 : How To Find The Area Of An Equilateral Triangle

Example Question #23 : How To Find The Area Of An Equilateral Triangle

Example Question #24 : How To Find The Area Of An Equilateral Triangle

Example Question #25 : How To Find The Area Of An Equilateral Triangle

Example Question #26 : How To Find The Area Of An Equilateral Triangle

Example Question #27 : How To Find The Area Of An Equilateral Triangle

Example Question #28 : How To Find The Area Of An Equilateral Triangle

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