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ISEE Upper Level Math : Equilateral Triangles

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

Equilateral

In the above diagram, $\Delta ABC$ is equilateral. Give its area.

Possible Answers:

$1452 \sqrt{3}$

2,178

$726 \sqrt{3}$

$726 \sqrt{2}$

1452

Correct answer:

$1452 \sqrt{3}$

Explanation:

The interior angles of an equilateral triangle all measure 60 degrees, so, by the 30-60-90 Theorem,

$BM = \frac{AM}{\sqrt{3}} = \frac{66}{\sqrt{3}} = \frac{66\cdot \sqrt{3}}{\sqrt{3}\cdot \sqrt{3}} = \frac{66 \sqrt{3}}{3} =22 \sqrt{3}$

Also, is the midpoint of $\overline{BC}$ , so $BC = 2 \cdot BM = 44 \sqrt{3}$ ; this is the base.

The area of this triangle is half the product of the base and the height :

$A= \frac{1}{2} \cdot 66 \cdot 44 \sqrt{3}= 1,452 \sqrt{3}$

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

Icecreamcone

Refer to the above figure. The shaded region is a semicircle with area $24 \pi$ . Give the area of $\bigtriangleup ABC$ .

Possible Answers:

$24\sqrt{3}$

$48\sqrt{3}$

$36 \sqrt{3}$

$12 \sqrt{3}$

Correct answer:

$48\sqrt{3}$

Explanation:

Given the radius of a semicircle, its area can be calculated using the formula

$A = \frac{1}{2} \pi r ^{2}$ .

Substituting $A = 24 \pi$ :

$\frac{1}{2} \pi r ^{2} = 24 \pi$

$\frac{2}{\pi} \cdot \frac{1}{2} \pi r ^{2} = \frac{2}{\pi} \cdot 24 \pi$

$r ^{2} = 48$

$r = \sqrt{48} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}$

The diameter of this semicircle is twice this, which is $d = 2 \cdot 4\sqrt{3} = 8 \sqrt{3}$ ; this is also the length of $\overline{BC}$ .

$\bigtriangleup ABC$ has two angles of degree measure 60; its third angle must also have measure 60, making $\bigtriangleup ABC$ an equilateral triangle with sidelength $s = 8 \sqrt{3}$ . Substitute this in the area formula:

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

$= \frac{( 8 \sqrt{3})^{2}\sqrt{3}}{4}$

$= \frac{8^{2} \cdot ( \sqrt{3})^{2}\sqrt{3}}{4}$

$= \frac{64 \cdot 3 \cdot \sqrt{3}}{4}$

$= \frac{192 \sqrt{3}}{4}$

$= 48\sqrt{3}$

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

Icecreamcone

Refer to the above figure. The shaded region is a semicircle with area $16 \pi$ . Give the perimeter of $\bigtriangleup ABC$ .

Possible Answers:

$24 \sqrt{3}$

$24 \sqrt{2}$

$12 \sqrt{3}$

$12 \sqrt{2}$

Correct answer:

$24 \sqrt{2}$

Explanation:

Given the radius of a semicircle, its area can be calculated using the formula

$A = \frac{1}{2} \pi r ^{2}$ .

Substituting $A = 16 \pi$ :

$\frac{1}{2} \pi r ^{2} = 16 \pi$

$\frac{2}{\pi} \cdot \frac{1}{2} \pi r ^{2} = \frac{2}{\pi} \cdot 16 \pi$

$r ^{2} = 32$

$r = \sqrt{32} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}$

The diameter of this semicircle is twice this, which is $d = 2 \cdot 4\sqrt{2} = 8 \sqrt{2}$ ; this is also the length of $\overline{BC}$ .

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

Find the perimeter of an equilateral triangle with a base of length 8cm.

Possible Answers:

$24\text{cm}$

$64\text{cm}$

$\text{There is not enough information to solve the problem.}$

$16\text{cm}$

$32\text{cm}$

Correct answer:

$24\text{cm}$

Explanation:

To find the perimeter of a triangle, we will use the following formula:

P = a+b+c

where a, b, and c are the lengths of the sides of the triangle

Now, we know the base of the triangle is 8cm. Because it is an equilateral triangle, all sides are equal. Therefore, all sides are 8cm.

Knowing this, we can substitute into the formula. We get

$P = 8\text{cm} + 8\text{cm} + 8\text{cm}$

$P = 24\text{cm}$

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Example Question #3 : How To Find The Perimeter Of An Equilateral Triangle

Find the perimeter of an equilateral triangle with a base of 21in.

Possible Answers:

$48\text{in}$

$156\text{in}$

$42\text{in}$

$63\text{in}$

$128\text{in}$

Correct answer:

$63\text{in}$

Explanation:

To find the perimeter of a triangle, we will use the following formula:

P = a+b+c

where a, b, and c are the lengths of the sides of the triangle.

Now, we know the base of the triangle has a length of 21in. Because it is an equilateral triangle, all lengths are the same. Therefore, all lengths are 21in.

Knowing this, we can substitute into the formula. We get

$P = 21\text{in} +21\text{in} +21\text{in}$

$P = 63\text{in}$

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Example Question #4 : How To Find The Perimeter Of An Equilateral Triangle

Find the perimeter of an equilateral triangle with a base of 23in.

Possible Answers:

$58\text{in}$

$69\text{in}$

$46\text{in}$

$264\text{in}$

$128\text{in}$

Correct answer:

$69\text{in}$

Explanation:

To find the perimeter of a triangle, we will use the following formula:

P = a+b+c

where a, b, and c are the lengths of the sides of the triangle.

Now, we know the base of the triangle has a length of 23in. Because it is an equilateral triangle, all lengths are the same. Therefore, all lengths are 23in.

Knowing this, we can substitute into the formula. We get

$P = 23\text{in} +23\text{in} +23\text{in}$

$P = 69\text{in}$

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

Find the perimeter of an equilateral triangle with a base of 22in.

Possible Answers:

$66\text{in}$

$242\text{in}$

$121\text{in}$

$\text{There is not enough information to solve the problem.}$

$44\text{in}$

Correct answer:

$66\text{in}$

Explanation:

An equilateral triangle has 3 equal sides. So, we will use the following formula:

P = 3a

where a is the length of one side of the triangle.

Now, we know the base of the triangle is 22in. Because all sides are equal, all sides are 22in. So, we can substitute into the formula:

$P = 3 \cdot 22\text{in}$

$P = 66\text{in}$

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

Find the perimeter of an equilateral triangle with a base of 14in.

Possible Answers:

$48\text{in}$

$32\text{in}$

$52\text{in}$

$42\text{in}$

$36\text{in}$

Correct answer:

$42\text{in}$

Explanation:

An equilateral triangle has 3 equal sides. To find the perimeter of an equilateral triangle, we will use the following formula:

P = 3a

where a is the length of one side of the triangle.

Now, we know the base of the equilateral triangle is 14in. So, we get

$P = 3 \cdot 14\text{in}$

$P = 42\text{in}$

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

Find the perimeter of an equilateral triangle with a base of 19in.

Possible Answers:

$57\text{in}$

$48\text{in}$

$63\text{in}$

$51\text{in}$

$69\text{in}$

Correct answer:

$57\text{in}$

Explanation:

To find the perimeter of a triangle, we will use the following formula:

P = a+b+c

where a, b, and c are the lengths of the sides of the triangle.

Now, we know the base of the triangle is 19in. Because it is an equilateral triangle, all sides are equal. Therefore, all sides are 19in. So, we get

$P = 19\text{in} + 19\text{in} + 19\text{in}$

$P = 57\text{in}$

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

The length of one side of an equilateral triangle is 4 centimeters. Give the height (altitude) of the triangle.

Possible Answers:

$2\sqrt{3}\ cm$

$4\sqrt{3}\ cm$

$4\ cm$

$3\sqrt{3}\ cm$

$3\ cm$

Correct answer:

$2\sqrt{3}\ cm$

Explanation:

$Area=\frac{1}{2}absinC$ ,

where and are the lengths of two sides and is the corresponding angle.

In an equilateral triangle, all of the sides have the same length, and all three angles are $60^{\circ}$ .

$Area=\frac{1}{2}absinC=\frac{1}{2}\times 4\times 4\times sin60^{\circ}$

$sin60^{\circ}=\frac{\sqrt{3}}{2}\Rightarrow Area=\frac{1}{2}\times 4\times 4\times \frac{\sqrt{3}}{2}=4\sqrt{3}\ cm^2$

Now plug this area into the alternate formula for the area of the triangle and solve for the height:

$Area =\frac{bh}2{}\Rightarrow 4\sqrt{3}=\frac{4\times h}{2}\Rightarrow h=2\sqrt{3}\ cm$

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ISEE Upper Level Math : Equilateral Triangles

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

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

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

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

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

Example Question #3 : How To Find The Perimeter Of An Equilateral Triangle

Example Question #4 : 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 #1 : How To Find The Height Of An Eqilateral Triangle

All ISEE Upper Level Math Resources

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