We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

ISEE Upper Level Math : How to find the area of an equilateral triangle

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

Create An Account

All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Equilateral Triangles

The length of one side of an equilateral triangle is 6 inches. Give the area of the triangle.

Possible Answers:

$9\sqrt{3}\ in^2$

$9\ in^2$

$9\sqrt{2}\ in^2$

$6\sqrt{3}\ in^2$

$6\ in^2$

Correct answer:

$9\sqrt{3}\ in^2$

Explanation:

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

where and are the lengths of two sides of the triangle and is the angle measure.

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

$Area=\frac{1}{2}absinC=\frac{1}{2}\times 6\times 6\times sin60^{\circ}$

$sin60^{\circ}=\frac{\sqrt{3}}{2}\Rightarrow Area=\frac{1}{2}\times 6\times 6\times \frac{\sqrt{3}}{2}=9\sqrt{3}\ in^2$

Report an Error

Example Question #2 : Equilateral Triangles

The perimeter of an equilateral triangle is $30\sqrt[4]{3}$ . Give its area.

Possible Answers:

150

$25\sqrt{3}$

$50\sqrt{3}$

$25\sqrt{6}$

Correct answer:

Explanation:

An equilateral triangle with perimeter $30\sqrt[4]{3}$ has three congruent sides of length

$s = 30\sqrt[4]{3} \div 3 = 10\sqrt[4]{3}$

The area of this triangle is

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{ \left ( 10\sqrt[4]{3} \right )^{2}\sqrt{3}}{4} = \frac{100 \sqrt[4]{9} \cdot \sqrt{3}}{4}$

$\sqrt[4]{9} = \sqrt[4]{3^{2}} = 3^{2/4} = 3^{1/2} = \sqrt{3}$ , so

$A = \frac{100 \sqrt{3} \cdot \sqrt{3}}{4} = \frac{100 \cdot 3 }{4}= 75$

Report an Error

Example Question #2 : Equilateral Triangles

The perimeter of an equilateral triangle is $18\sqrt{3}$ . Give its area.

Possible Answers:

$36\sqrt{3}$

$27\sqrt{3}$

The correct answer is not among the other four choices.

Correct answer:

$27\sqrt{3}$

Explanation:

An equilateral triangle with perimeter $18\sqrt{3}$ has three congruent sides of length

$s = 18\sqrt{3} \div 3 = 6\sqrt{3}$

The area of this triangle is

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{ \left ( 6\sqrt{3} \right )^{2}\sqrt{3}}{4} = \frac{36 \cdot 3 \cdot \sqrt{3}}{4} = 27\sqrt{3}$

Report an Error

Example Question #1 : Equilateral Triangles

The perimeter of an equilateral triangle is . Give its area in terms of .

Possible Answers:

$\frac{n^{2} \sqrt{3}}{18}$

$\frac{n^{2} \sqrt{3}}{12}$

$\frac{n^{2} \sqrt{3}}{36}$

$\frac{n^{2} }{12}$

$\frac{n^{2} }{6}$

Correct answer:

$\frac{n^{2} \sqrt{3}}{36}$

Explanation:

An equilateral triangle with perimeter has three congruent sides of length $\frac{n}{3}$ . Substitute this for in the following area formula:

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

$A = \left ( \frac{n}{3} \right )^{2} \cdot \frac{\sqrt{3}}{4}$

$A = \frac{n^{2} }{9} \cdot \frac{\sqrt{3}}{4}$

$A = \frac{n^{2} \sqrt{3}}{36}$

Report an Error

Example Question #2 : Equilateral Triangles

An equilateral triangle is inscribed inside a circle of radius 8. Give its area.

Possible Answers:

$64\sqrt{3}$

$48\sqrt{3}$

$48\sqrt{2}$

Correct answer:

$48\sqrt{3}$

Explanation:

The trick is to know that the circumscribed circle, or the circumcircle, has as its center the intersection of the three altitudes of the triangle, and that this center, or circumcenter, divides each altitude into two segments, one twice the length of the other - the longer one being a radius. Because of this, we can construct the following:

Circumcircle

Each of the six smaller triangles is a 30-60-90 triangle, and all six are congruent.

We will find the area of $\Delta COY$ , and multiply it by 6.

By the 30-60-90 Theorem, $CY = OY \cdot \sqrt{3} = 4\sqrt{3}$ , so the area of $\Delta COY$ is

$A = \frac{1}{2}\cdot OY \cdot CY = \frac{1}{2} \cdot 4\cdot 4\sqrt{3} = 8 \sqrt{3}$ .

Six times this - $6 \cdot 8 \sqrt{3} = 48 \sqrt{3}$ - is the area of $\Delta ABC$ .

Report an Error

Example Question #111 : Plane Geometry

The perimeter of an equilateral triangle is . Give its area.

Possible Answers:

$24\sqrt{3}$

$36\sqrt{3}$

$12\sqrt{3}$

Correct answer:

$36\sqrt{3}$

Explanation:

An equilateral triangle with perimeter 36 has three congruent sides of length

$s = 36 \div 3 = 12$

The area of this triangle is

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{12^{2}\sqrt{3}}{4} = \frac{144\sqrt{3}}{4} = 36\sqrt{3}$

Report an Error

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

2,178

$726 \sqrt{2}$

$1452 \sqrt{3}$

$726 \sqrt{3}$

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 :

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

Report an Error

Example Question #1 : 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:

$12 \sqrt{3}$

$24\sqrt{3}$

$36 \sqrt{3}$

$48\sqrt{3}$

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:

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

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

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

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

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

$= 48\sqrt{3}$

Report an Error

View Tutors

Evgenii
Certified Tutor

View Tutors

Pavan
Certified Tutor

University of Tennessee, Knoxville, Master's/Graduate, Mechanical Engineering. Pennsylvania State University, Master's/Gradua...

View Tutors

Omar
Certified Tutor

Arizona State University, Bachelors, Civil Engineering.

All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Computer Science Tutors in Houston, SSAT Tutors in Atlanta, Spanish Tutors in Los Angeles, SAT Tutors in New York City, SAT Tutors in Chicago, MCAT Tutors in Boston, GRE Tutors in Seattle, Spanish Tutors in San Francisco-Bay Area, Calculus Tutors in Philadelphia, MCAT Tutors in Seattle

Popular Courses & Classes

SAT Courses & Classes in Boston, GMAT Courses & Classes in San Diego, ISEE Courses & Classes in San Diego, GRE Courses & Classes in Phoenix, ISEE Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Seattle, GMAT Courses & Classes in Miami, SAT Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Boston, GMAT Courses & Classes in New York City

Popular Test Prep

SSAT Test Prep in Denver, LSAT Test Prep in San Francisco-Bay Area, LSAT Test Prep in Boston, MCAT Test Prep in Philadelphia, GRE Test Prep in Phoenix, ISEE Test Prep in Boston, LSAT Test Prep in Dallas Fort Worth, LSAT Test Prep in Miami, LSAT Test Prep in Los Angeles, LSAT Test Prep in Seattle

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

ISEE Upper Level Math : How to find the area of an equilateral triangle

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : Equilateral Triangles

Example Question #2 : Equilateral Triangles

Example Question #2 : Equilateral Triangles

Example Question #1 : Equilateral Triangles

Example Question #2 : Equilateral Triangles

Example Question #111 : Plane Geometry

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

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

All ISEE Upper Level Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: