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Intermediate Geometry : Plane 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 #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

Example Question #251 : Triangles

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:

45.67

51.73

48.35

42.91

Correct answer:

48.35

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.

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

$\text{Area of Equilateral Triangle}=144\sqrt3$

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

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

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

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

$\text{Area of Circle}=64\pi$

Finally, find the area of the shaded region.

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

$\text{Area of Shaded Region}=144\sqrt3-64\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=48.35$

Report an Error

Example Question #30 : 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:

17.54

16.16

14.23

9.10

Correct answer:

16.16

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.

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

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

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.

$\text{Area of Circle}=\pi\times 9^2$

$\text{Area of Circle}=81\pi$

Finally, find the area of the shaded region.

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

$\text{Area of Shaded Region}=\frac{625\sqrt3}{4}-81\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=16.16$

Report an Error

Example Question #51 : Equilateral Triangles

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

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

Possible Answers:

5.98

6.73

5.44

7.12

Correct answer:

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:

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

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

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

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

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

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

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

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

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

Finally, find the area of the shaded region.

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

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

Solve and round to two decimal places.

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

Report an Error

Example Question #691 : Intermediate Geometry

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

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

Possible Answers:

0.12

0.51

0.08

1.02

Correct answer:

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:

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

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

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

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

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

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

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

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

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

Finally, find the area of the shaded region.

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

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

Solve and round to two decimal places.

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

Report an Error

Example Question #692 : Intermediate Geometry

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:

0.76

0.66

0.79

0.84

Correct answer:

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:

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

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

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

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

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

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

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

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

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

Finally, find the area of the shaded region.

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

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

Solve and round to two decimal places.

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

Report an Error

Example Question #54 : Equilateral Triangles

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:

72.30

69.20

71.00

75.60

Correct answer:

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:

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

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

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

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

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

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

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.

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

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

Solve and round to two decimal places.

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

Report an Error

Example Question #55 : Equilateral Triangles

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:

24.05

28.09

22.91

19.51

Correct answer:

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:

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

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

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

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

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

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

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

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

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

Finally, find the area of the shaded region.

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

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

Solve and round to two decimal places.

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

Report an Error

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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

Example Question #251 : Triangles

Example Question #30 : How To Find The Area Of An Equilateral Triangle

Example Question #51 : Equilateral Triangles

Example Question #691 : Intermediate Geometry

Example Question #692 : Intermediate Geometry

Example Question #54 : Equilateral Triangles

Example Question #55 : Equilateral Triangles

All Intermediate Geometry Resources

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