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Intermediate Geometry : Plane Geometry

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Example Questions

Example Question #261 : Triangles

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

Find the area of the entire figure.

Possible Answers:

3098.59

2790.18

3199.26

3103.84

Correct answer:

3103.84

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.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(32)^2}{4}=\frac{1024\sqrt3}{4}=256\sqrt3$

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

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

$\text{Area of Figure}=7(256\sqrt3)=3103.84$

Make sure to round to places after the decimal.

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Example Question #62 : Equilateral Triangles

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

Find the area of the entire figure.

Possible Answers:

3612.06

3938.29

4510.20

3782.09

Correct answer:

3938.29

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.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3(36)^2}{4}=\frac{1296\sqrt3}{4}=324\sqrt3$

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

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

$\text{Area of Figure}=7(324\sqrt3)=3928.29$

Make sure to round to places after the decimal.

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Example Question #48 : How To Find The Area Of An Equilateral Triangle

Pyramid

The cube in the above diagram has edges of length 1. Give the area of $\bigtriangleup ABC$ .

Possible Answers:

$\frac{1}{2}$

$\frac{ 3\sqrt{3}}{4}$

$\frac{ \sqrt{3}}{3}$

$\frac{ \sqrt{3}}{2}$

$\frac{2}{3}$

Correct answer:

$\frac{ \sqrt{3}}{2}$

Explanation:

Each of the three triangles $\bigtriangleup ADB$ , $\bigtriangleup ADC$ , and $\bigtriangleup BDC$ has two legs of length 1; therefore, each is an isosceles right triangle, and each is a 45-45-90 triangle. By the 45-45-90 Triangle Theorem, each hypotenuse - $\overline{AB} , \overline{AC} , \overline{BC}$ - has length $\sqrt{2}$ times that of the legs, or simply $\sqrt{2}$ . This makes $\bigtriangleup ABC$ an equilateral triangle with common sidelength $s = \sqrt{2}$ . Using the area formula

$A = \frac{s^{2}\sqrt{3}}{4}$

and substituting, the area can be calculated as

$A = \frac{ (\sqrt{2})^{2}\sqrt{3}}{4} = \frac{2 \sqrt{3}}{4}= \frac{ \sqrt{3}}{2}$ .

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

ΔABC is an equilateral triangle with side 12.

Find the height of ΔABC (to the nearest tenth).

Possible Answers:

10.0

10.2

10.6

10.4

9.8

Correct answer:

10.4

Explanation:

Equilateral triangles have sides of all equal length and angles of 60°. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equal 30-60-90 triangles.

Now, the side of the original equilateral triangle (lets call it "a") is the hypotenuse of the 30-60-90 triangle. Because the 30-60-90 triange is a special triangle, we know that the sides are x, x $\sqrt{3}$ , and 2x, respectively.

Thus, a = 2x and x = a/2.
Height of the equilateral triangle = $\frac{a\sqrt{3}}{2}$

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

ΔABC is an equilateral triangle with side of length 8.

Find the height (to the nearest tenth).

Possible Answers:

6.9

7.3

6.7

7.5

7.1

Correct answer:

6.9

Explanation:

Equilateral triangles have sides of equal length, with angles of 60°. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equal 30-60-90 triangles.

Thus, a = 2x and x = a/2.
Height of the equilateral triangle = $\frac{a\sqrt{3}}{2}$

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

What is the height of an equilateral triangle with a side length of 8 in?

Possible Answers:

$4\sqrt{2}$

$6\sqrt{2}$

$6\sqrt{3}$

$4\sqrt{3}$

Correct answer:

$4\sqrt{3}$

Explanation:

An equilateral triangle has three congruent sides, and is also an equiangular triangle with three congruent angles that each meansure 60 degrees.

To find the height we divide the triangle into two special 30 - 60 - 90 right triangles by drawing a line from one corner to the center of the opposite side. This segment will be the height, and will be opposite from one of the 60 degree angles and adjacent to a 30 degree angle. The special right triangle gives side ratios of , $x\sqrt{3}$ , and . The hypoteneuse, the side opposite the 90 degree angle, is the full length of one side of the triangle and is equal to . Using this information, we can find the lengths of each side fo the special triangle.

$8=2x\rightarrow 4=x\rightarrow 4\sqrt{3}=x\sqrt{3}$

The side with length $x\sqrt{3}$ will be the height (opposite the 60 degree angle). The height is $4\sqrt{3}$ inches.

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

What is the height of a triangle with side lengths 4, 4, 4?

Possible Answers:

$\sqrt{12}$

$\sqrt{36}$

$\sqrt{20}$

Correct answer:

$\sqrt{12}$

Explanation:

To solve, it's easiest to first visualize the height's relationship with the rest of the triangle's sides:

Equilateral height

The height is one of the legs of a right triangle. The hypotenuse is 4, and the other leg is 2, or half of the base side, 4. To determine the height, use Pythagorean Theorem:

x^2 + 2^2 = 4^2

x^2 + 4 = 16 subtract 4 from both sides

x^2 = 12 take the square root of both sides

$x = \sqrt{12}$

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

If an equilateral triangle has a length of $6\;cm$ for all sides, what would be the heigh of the triangle? Round to the nearest tenth.

Possible Answers:

$4.2\;cm$

$5.2\;cm$

$4.5\;cm$

$3.9\;cm$

$6\;cm$

Correct answer:

$5.2\;cm$

Explanation:

The following formula can be used to determine the height of an equilateral triangle when we are given the length of the sides:

$\\h^2=s^2-\frac{s^2}{2} \\h^2=(6\;cm)^2-(\frac{1}{2}\cdot6\;cm)^2$

$\\h^2=36\;cm^2-9\;cm^2\\h^2=27\;cm^2\\h\approx5.2\;cm$

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Example Question #2 : How To Find The Height Of An Equilateral Triangle

The equilateral triangle shown below has a side length of $15\;cm$ . Given this information, solve for the height of the triangle.

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Possible Answers:

$15\;cm$

$12\;cm$

$14\;cm$

$13\;cm$

$11\;cm$

Correct answer:

$13\;cm$

Explanation:

To solve for the height of an equilateral triangle, we can divide the triangle into two right triangles. In the below image, the bisecting line represents the height, and we can solve for height by applying the Pythagorean Theorem:

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$\\z^2=x^2+y^2\\(15\;cm)^2=x^2+(7.5\;cm)^2\\225\;cm^2=x^2+56.25\;cm^2\\x^2=168.75\;cm^2\\x=\sqrt{168.75\;cm^2}\approx13\;cm$

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

Find the height of an equilateral triangle with a side length of $12\:cm$ . Round your answer to the nearest tenth.

Possible Answers:

$10.1\:cm$

$12.0\:cm$

$10.4\:cm$

$10.2\:cm$

$12.0\sqrt2\:cm$

Correct answer:

$10.4\:cm$

Explanation:

An equilateral triangle has three equal sides, as can be deduced from its name. This also means that as a result, the triangle is also equiangular. That is, all its interior angles are the same. Because the sum of internal angles of a triangle is $180^{\circ}$ , that means that each interior angle is $60^{\circ}$ .

Find_the_height_of_equilateral

The height (the dotted line in the diagram) is the length that bisects the apical angle while bisecting the base of the triangle. Drawing the dotted line splits the equilateral triangle into two congruent right triangles. Because they're the same, we only need to consider one of the two smaller triangles to obtain our answer. Keeping in mind that the base of the equilateral triangle has been bisected, this means that the base of the right triangle has the length of $6\:cm$ . With a base of $6\:cm$ and a hypotenuse of $12\:cm$ , we can easily solve for the height (the third side) with the Pythagorean Theorem.

This problem can also be solved using trigonometric functions or even revisiting 30-60-90 triangle rules.

Using the Pythagorean Theorem,

a^2+b^2=c^2

a^2+6^2=12^2

a^2+36=144

a^2+36-36= 144-36

a^2=108

$\sqrt{a^2}=\sqrt{108}$

$a=\sqrt{108}$

$a= 10.39 \approx10.4\:cm$

Therefore, the height of the equilateral triangle is $10.4\:cm$ .

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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #261 : Triangles

Example Question #62 : Equilateral Triangles

Example Question #48 : How To Find The Area Of An Equilateral Triangle

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

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

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

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

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

Example Question #2 : How To Find The Height Of An Equilateral Triangle

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

All Intermediate Geometry Resources

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