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

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Example Questions

Example Question #231 : Triangles

Find the area of an equilateral triangle with a perimeter of 24cm. Leave answer in simplest radical form.

Possible Answers:

$32\ cm^{2}$

$32\sqrt3\ cm^{2}$

$64\ cm^{2}$

$16\sqrt3\ cm^{2}$

Correct answer:

$16\sqrt3\ cm^{2}$

Explanation:

To find the area of an equilateral triangle, one must find the base and the height.

$Area=\frac{1}{2}bh$

All the sides of an equilateral triangle are congruent, so if the perimeter of the equilaterail triangle is 24, then each side must equal one third of that total which is 8cm.

This will produce a triangle that includes the following information below:

1 ans2

Dropping an altitude down the center of the equilateral triangle will result in two 30-60-90 triangles with a hypotenuse of 8.

1 ans 3

In every 30-60-90 triangle the following formulas apply:

hypotenuse=2(shortleg)

$longleg=shortleg\sqrt3$

When we plug in the given information on the triangle we get:

8=2(shortleg)

Dividing both sides by 2 gives the below result.

shortleg=4

We can now plug this into the long leg formula to get the height of the triangle:

$longleg=4\sqrt3$

1 anspng

Now that we have all of the information needed to find the area we plug these values into the area formula.

$Area=\frac{1}{2}bh$

$Area=\frac{1}{2}(8)(4\sqrt3)$

$Area=16\sqrt3$

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Example Question #11 : 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 in the figure below.

Find the area of the shaded region.

Possible Answers:

1.99

3.02

3.68

2.01

Correct answer:

3.02

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.

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

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.

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

$\text{Area of Shaded Region}=9\sqrt3-4\pi=3.02$

Make sure to round to places after the decimal.

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Example Question #12 : 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 in the figure below.

Find the area of the shaded region.

Possible Answers:

5.87

6.80

7.25

6.33

Correct answer:

6.80

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.

$\text{Area of Circle}=\pi\times 3^2=9\pi$

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.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(9)^2=\frac{81\sqrt3}{4}$

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.

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

$\text{Area of Shaded Region}=\frac{81\sqrt3}{4}-9\pi=6.80$

Make sure to round to places after the decimal.

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Example Question #13 : 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 in the figure below.

Find the area of the shaded region.

Possible Answers:

13.08

10.27

12.09

16.33

Correct answer:

12.09

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.

$\text{Area of Circle}=\pi\times 4^2=16\pi$

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.

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

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.

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

$\text{Area of Shaded Region}=36\sqrt3-16\pi=12.09$

Make sure to round to places after the decimal.

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Example Question #14 : 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 in the figure below.

Find the area of the shaded region.

Possible Answers:

100.36

99.07

97.86

94.67

Correct answer:

94.67

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.

$\text{Area of Circle}=\pi\times 5^2=25\pi$

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.

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

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.

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

$\text{Area of Shaded Region}=100\sqrt3-25\pi=94.67$

Make sure to round to places after the decimal.

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Example Question #15 : 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 in the figure below.

Find the area of the shaded region.

Possible Answers:

132.02

120.14

136.32

128.95

Correct answer:

136.32

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.

$\text{Area of Circle}=\pi\times 6^2=36\pi$

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.

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

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.

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

$\text{Area of Shaded Region}=144\sqrt3-36\pi=136.32$

Make sure to round to places after the decimal.

Report an Error

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:

51.28

62.39

55.22

60.11

Correct answer:

60.11

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.

$\text{Area of Circle}=\pi\times 6^2=36\pi$

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.

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

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.

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

$\text{Area of Shaded Region}=100\sqrt3-36\pi=60.11$

Make sure to round to places after the decimal.

Report an Error

Example Question #241 : Triangles

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:

56.60

52.20

58.90

52.31

Correct answer:

56.60

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.

$\text{Area of Circle}=\pi\times 3^2=9\pi$

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.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(14)^2=49\sqrt3$

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.

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

$\text{Area of Shaded Region}=49\sqrt3-9\pi=56.60$

Make sure to round to places after the decimal.

Report an Error

Example Question #242 : Triangles

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:

240.43

256.31

199.68

225.31

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 #243 : Triangles

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:

10.25

12.68

11.03

10.04

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.

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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 #231 : Triangles

Example Question #11 : How To Find The Area Of An Equilateral Triangle

Example Question #12 : How To Find The Area Of An Equilateral Triangle

Example Question #13 : How To Find The Area Of An Equilateral Triangle

Example Question #14 : How To Find The Area Of An Equilateral Triangle

Example Question #15 : How To Find The Area Of An Equilateral Triangle

Example Question #681 : Intermediate Geometry

Example Question #241 : Triangles

Example Question #242 : Triangles

Example Question #243 : Triangles

All Intermediate Geometry Resources

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