How to find the area of an equilateral triangle

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Geometry › How to find the area of an equilateral triangle

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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.

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.

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?

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.

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

An equilateral triangle is placed on top of a regular hexagon as shown by the figure below.

Find the area of the entire figure.

Explanation

Recall that a regular hexagon can be divided into congruent equilateral triangles.

Since the extra equilateral triangle on top has the same side lengths as of the equilateral triangles made by dividing the hexagon, the entire figure has a total of congruent equilateral triangles.

Thus, to find the area of the entire figure, we must first find the area of the equilateral triangle.

Recall how to find the area of an equilateral triangle:

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

Plug in the length of a side of the equilateral triangle.

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

Now, multiply this area by to find the area of the entire figure.

$\text{Area of Figure}=7(\text{Area of Equilateral Triangle})$

$\text{Area of Figure}=7(\sqrt3)=12.12$

Make sure to round to places after the decimal.

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.

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.

An equilateral triangle is placed on top of a regular hexagon as shown by the figure below.

Find the area of the entire figure.

The area of the entire figure cannot be determined by the given information.

Explanation

Recall that a regular hexagon can be divided into congruent equilateral triangles.

Since the extra equilateral triangle on top has the same side lengths as of the equilateral triangles made by dividing the hexagon, the entire figure has a total of congruent equilateral triangles.

Thus, to find the area of the entire figure, we must first find the area of the equilateral triangle.

Recall how to find the area of an equilateral triangle:

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

Plug in the length of a side of the equilateral triangle.

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

Now, multiply this area by to find the area of the entire figure.

$\text{Area of Figure}=7(\text{Area of Equilateral Triangle})$

$\text{Area of Figure}=7(4\sqrt3)=48.50$

Make sure to round to places after the decimal.

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

$13.5 ; cm^{2}$

$27 ; cm^{2}$

$18 ; cm^{2}$

$9 ; cm^{2}$

$6.75 ; cm^{2}$

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.

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?

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.

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

An equilateral triangle is placed on top of a regular hexagon as shown by the figure below.

Find the area of the entire figure.

Explanation

Recall that a regular hexagon can be divided into congruent equilateral triangles.

Since the extra equilateral triangle on top has the same side lengths as of the equilateral triangles made by dividing the hexagon, the entire figure has a total of congruent equilateral triangles.

Thus, to find the area of the entire figure, we must first find the area of the equilateral triangle.

Recall how to find the area of an equilateral triangle:

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

Plug in the length of a side of the equilateral triangle.

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

Now, multiply this area by to find the area of the entire figure.

$\text{Area of Figure}=7(\text{Area of Equilateral Triangle})$

$\text{Area of Figure}=7(\frac{25\sqrt3}{4})=75.78$

Make sure to round to places after the decimal.

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.

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.

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.

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.

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

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