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Intermediate Geometry : Equilateral Triangles

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

Asd

If $a=5\:cm$ , what is the area of this equilateral triangle?

Possible Answers:

$\frac{25\sqrt{3}}{4}\:cm^2$

$15\sqrt{3}\:cm^2$

$5\sqrt{3}\:cm^2$

$20\sqrt{3}\:cm^2$

$\sqrt{3}\:cm^2$

Correct answer:

$\frac{25\sqrt{3}}{4}\:cm^2$

Explanation:

Given that the sides of our equilateral triangle are each $5\:cm$ long, we can just plug the value in to the formula for the area of an equilateral triangle and solve for the area of the triangle:

$Area=\frac{\sqrt{3}}{4}a^2$ if is a side of the triangle.

$Area=\frac{\sqrt{3}}{4}a^2=\frac{\sqrt{3}}{4}5^2=\frac{\sqrt{3}}{4}25=\frac{25\sqrt{3}}{4}\:cm^2$

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Example Question #31 : Equilateral Triangles

An equilateral triangle has sides of length 6cm. If the height of the triangle is 4.5cm what is the area of the triangle?

Possible Answers:

$18 \; cm^{2}$

$9 \; cm^{2}$

$6.75 \; cm^{2}$

$27 \; cm^{2}$

$13.5 \; cm^{2}$

Correct answer:

$13.5 \; cm^{2}$

Explanation:

To find the area of any triangle we can use the formula 1/2 (base x height) , that is the base times the height divided by two. It is important to remember any of the sides of an equaliateral triangle can be used as the base when the hieght is given. The area can be found by (6 x 4.5) divided by 2; which gives 13.5 square centimeters.

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

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

Possible Answers:

$32\ cm^{2}$

$64\ cm^{2}$

$32\sqrt3\ cm^{2}$

$16\sqrt3\ cm^{2}$

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 #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:

3.68

1.99

3.02

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 #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:

7.25

6.33

6.80

5.87

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 #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:

12.09

10.27

16.33

13.08

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 #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:

94.67

99.07

97.86

100.36

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 #16 : 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:

120.14

136.32

128.95

132.02

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.

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Example Question #17 : 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:

62.39

51.28

60.11

55.22

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 #18 : 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:

58.90

52.20

52.31

56.60

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.

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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 Area Of An Equilateral Triangle

Example Question #31 : Equilateral 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 #16 : How To Find The Area Of An Equilateral Triangle

Example Question #17 : How To Find The Area Of An Equilateral Triangle

Example Question #18 : How To Find The Area Of An Equilateral Triangle

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