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Intermediate Geometry : Equilateral Triangles

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #1 : How To Find The Perimeter Of An Equilateral Triangle

An equilateral triangle has a side length of cm.

What is the perimeter of the triangle?

Possible Answers:

125cm

10cm

15cm

20cm

Correct answer:

15cm

Explanation:

The perimeter of a triangle is the sum of all three sides.

Because an equilateral triangle has all three sides of equal length, we have

P = 5+5+5=15cm

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Example Question #5 : How To Find The Perimeter Of An Equilateral Triangle

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Possible Answers:

10.28

12.37

14.68

13.22

Correct answer:

12.37

Explanation:

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent 30-60-90 triangles.

Recall that the side lengths in a 30-60-90 triangle are in a $1:\sqrt3:2$ ratio. Thus, the radius of the circle, which is also the base of the triangle, the height of the triangle, and the side length of the triangle are in the same ratio.

We can then set up the following to determine the length of the side of the equilateral triangle:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

Rearrange the equation to solve for the length of the side.

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the length of the height to find the length of the side.

$\text{side}=\frac{2\sqrt3(3)}{3}=2\sqrt3$

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

$\text{side}=\text{diameter}$

We have two sides of the equilateral triangle and the circumference of a semi-circle.

$\text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}$

Plug in the length of the side to find the perimeter.

$\text{Perimeter}=2(2\sqrt3)+\frac{\pi(2\sqrt3)}{2}=4\sqrt3+\pi\sqrt3=12.37$

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Example Question #6 : How To Find The Perimeter Of An Equilateral Triangle

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Possible Answers:

25.09

22.38

24.69

24.74

Correct answer:

24.74

Explanation:

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent 30-60-90 triangles.

We can then set up the following to determine the length of the side of the equilateral triangle:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

Rearrange the equation to solve for the length of the side.

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the length of the height to find the length of the side.

$\text{side}=\frac{2\sqrt3(6)}{3}=4\sqrt3$

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

$\text{side}=\text{diameter}$

We have two sides of the equilateral triangle and the circumference of a semi-circle.

$\text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}$

Plug in the length of the side to find the perimeter.

$\text{Perimeter}=2(4\sqrt3)+\frac{\pi(4\sqrt3)}{2}=8\sqrt3+2\pi\sqrt3=24.74$

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Example Question #7 : How To Find The Perimeter Of An Equilateral Triangle

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Possible Answers:

28.86

31.59

24.09

30.60

Correct answer:

28.86

Explanation:

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent 30-60-90 triangles.

We can then set up the following to determine the length of the side of the equilateral triangle:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

Rearrange the equation to solve for the length of the side.

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the length of the height to find the length of the side.

$\text{side}=\frac{2\sqrt3(7)}{3}=\frac{14\sqrt3}{3}$

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

$\text{side}=\text{diameter}$

We have two sides of the equilateral triangle and the circumference of a semi-circle.

$\text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}$

Plug in the length of the side to find the perimeter.

$\text{Perimeter}=2(\frac{14\sqrt3}{3})+\frac{\pi(\frac{14\sqrt3}{3})}{2}=\frac{28\sqrt3}{3}+\frac{7\pi\sqrt3}{3}=28.86$

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Example Question #8 : How To Find The Perimeter Of An Equilateral Triangle

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Possible Answers:

40.09

41.28

47.09

45.36

Correct answer:

45.36

Explanation:

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent 30-60-90 triangles.

We can then set up the following to determine the length of the side of the equilateral triangle:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

Rearrange the equation to solve for the length of the side.

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the length of the height to find the length of the side.

$\text{side}=\frac{2\sqrt3(11)}{3}=\frac{22\sqrt3}{3}$

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

$\text{side}=\text{diameter}$

We have two sides of the equilateral triangle and the circumference of a semi-circle.

$\text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}$

Plug in the length of the side to find the perimeter.

$\text{Perimeter}=2(\frac{22\sqrt3}{3})+\frac{\pi(\frac{22\sqrt3}{3})}{2}=\frac{44\sqrt3}{3}+\frac{11\pi\sqrt3}{3}=45.36$

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Example Question #9 : How To Find The Perimeter Of An Equilateral Triangle

An equilateral triangle is placed together with a semicircle as shown in the figure below.

Find the perimeter of the figure.

Possible Answers:

51.24

55.82

53.60

56.17

Correct answer:

53.60

Explanation:

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent 30-60-90 triangles.

We can then set up the following to determine the length of the side of the equilateral triangle:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

Rearrange the equation to solve for the length of the side.

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the length of the height to find the length of the side.

$\text{side}=\frac{2\sqrt3(13)}{3}=\frac{26\sqrt3}{3}$

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

$\text{side}=\text{diameter}$

We have two sides of the equilateral triangle and the circumference of a semi-circle.

$\text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}$

Plug in the length of the side to find the perimeter.

$\text{Perimeter}=2(\frac{26\sqrt3}{3})+\frac{\pi(\frac{26\sqrt3}{3})}{2}=\frac{52\sqrt3}{3}+\frac{13\pi\sqrt3}{3}=53.60$

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Example Question #10 : How To Find The Perimeter Of An Equilateral Triangle

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Possible Answers:

62.81

70.09

65.34

71.91

Correct answer:

70.09

Explanation:

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent 30-60-90 triangles.

We can then set up the following to determine the length of the side of the equilateral triangle:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

Rearrange the equation to solve for the length of the side.

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the length of the height to find the length of the side.

$\text{side}=\frac{2\sqrt3(17)}{3}=\frac{34\sqrt3}{3}$

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

$\text{side}=\text{diameter}$

We have two sides of the equilateral triangle and the circumference of a semi-circle.

$\text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}$

Plug in the length of the side to find the perimeter.

$\text{Perimeter}=2(\frac{34\sqrt3}{3})+\frac{\pi(\frac{34\sqrt3}{3})}{2}=\frac{68\sqrt3}{3}+\frac{17\pi\sqrt3}{3}=70.09$

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Example Question #731 : Plane Geometry

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Possible Answers:

71.08

78.34

74.28

76.31

Correct answer:

78.34

Explanation:

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent 30-60-90 triangles.

We can then set up the following to determine the length of the side of the equilateral triangle:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

Rearrange the equation to solve for the length of the side.

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the length of the height to find the length of the side.

$\text{side}=\frac{2\sqrt3(19)}{3}=\frac{38\sqrt3}{3}$

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

$\text{side}=\text{diameter}$

We have two sides of the equilateral triangle and the circumference of a semi-circle.

$\text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}$

Plug in the length of the side to find the perimeter.

$\text{Perimeter}=2(\frac{38\sqrt3}{3})+\frac{\pi(\frac{38\sqrt3}{3})}{2}=\frac{76\sqrt3}{3}+\frac{19\pi\sqrt3}{3}=78.34$

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Example Question #732 : Plane Geometry

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Possible Answers:

90.61

95.27

91.11

94.83

Correct answer:

94.83

Explanation:

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent 30-60-90 triangles.

We can then set up the following to determine the length of the side of the equilateral triangle:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

Rearrange the equation to solve for the length of the side.

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the length of the height to find the length of the side.

$\text{side}=\frac{2\sqrt3(23)}{3}=\frac{46\sqrt3}{3}$

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

$\text{side}=\text{diameter}$

We have two sides of the equilateral triangle and the circumference of a semi-circle.

$\text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}$

Plug in the length of the side to find the perimeter.

$\text{Perimeter}=2(\frac{46\sqrt3}{3})+\frac{\pi(\frac{46\sqrt3}{3})}{2}=\frac{92\sqrt3}{3}+\frac{23\pi\sqrt3}{3}=94.83$

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Example Question #733 : Plane Geometry

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Possible Answers:

80.09

91.37

82.33

86.59

Correct answer:

86.59

Explanation:

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent 30-60-90 triangles.

We can then set up the following to determine the length of the side of the equilateral triangle:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

Rearrange the equation to solve for the length of the side.

$\text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}$

Plug in the length of the height to find the length of the side.

$\text{side}=\frac{2\sqrt3(32)}{3}=\frac{42\sqrt3}{3}$

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

$\text{side}=\text{diameter}$

We have two sides of the equilateral triangle and the circumference of a semi-circle.

$\text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}$

Plug in the length of the side to find the perimeter.

$\text{Perimeter}=2(\frac{42\sqrt3}{3})+\frac{\pi(\frac{42\sqrt3}{3})}{2}=\frac{84\sqrt3}{3}+\frac{21\pi\sqrt3}{3}=86.59$

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Intermediate Geometry : Equilateral Triangles

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1 : How To Find The Perimeter Of An Equilateral Triangle

Example Question #5 : How To Find The Perimeter Of An Equilateral Triangle

Example Question #6 : How To Find The Perimeter Of An Equilateral Triangle

Example Question #7 : How To Find The Perimeter Of An Equilateral Triangle

Example Question #8 : How To Find The Perimeter Of An Equilateral Triangle

Example Question #9 : How To Find The Perimeter Of An Equilateral Triangle

Example Question #10 : How To Find The Perimeter Of An Equilateral Triangle

Example Question #731 : Plane Geometry

Example Question #732 : Plane Geometry

Example Question #733 : Plane Geometry

All Intermediate Geometry Resources

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