Call Now to Set Up Tutoring:

(888) 888-0446

Advanced Geometry : Rhombuses

Study concepts, example questions & explanations for Advanced Geometry

Create An Account

All Advanced Geometry Resources

6 Diagnostic Tests 57 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #11 : Rhombuses

Find the area of a rhombus with diagonal lengths of $(2x+2)\:in$ and $(4x+4)\:in$ .

Possible Answers:

$(8x+6)\:in^2$

$(3x+3)\:in^2$

$(4x^2+8x+4)\:in^2$

$(4x+3)\:in^2$

$(8x^2+16x+8)\:in^2$

Correct answer:

$(4x^2+8x+4)\:in^2$

Explanation:

Write the formula for the area of a rhombus.

$A=\frac{d_1\cdot d_2}{2}$

Substitute the given diagonal lengths:

$A=\frac{(2x+2)(4x+4)}{2}$

Use FOIL to multiply the two parentheticals in the numerator:

First: $2x\cdot4x=8x^2$

Outer: $2x\cdot4=8x$

Inner: $2\cdot4x=8x$

Last: $2\cdot 4=8$

Add your results together:

$A=\frac{8x^2+16x+8}{2}$

Divide all elements in the numerator by two to arrive at the correct answer:

$A=(4x^2+8x+4) \:in^2$

Report an Error

Example Question #11 : Calculating The Area Of A Quadrilateral

Rhombus_1

The above figure shows a rhombus RHOM . Give its area.

Possible Answers:

120

$40 \sqrt{11}$

$40\sqrt{61}$

Correct answer:

$40 \sqrt{11}$

Explanation:

Construct the other diagonal of the rhombus, which, along with the first one, form a pair of mutual perpendicular bisectors.

Rhombus_1

By the Pythagorean Theorem,

$n = \sqrt{12^{2}- 10^{2}} =\sqrt{144- 100} = \sqrt{44}= \sqrt{4}\cdot \sqrt{11} = 2 \sqrt{11}$

The rhombus can be seen as the composite of four congruent right triangles, each with legs 10 and $2 \sqrt{11}$ , so the area of the rhombus is

$A = 4 \cdot \frac{1}{2}\cdot 10 \cdot 2\sqrt{11} =40 \sqrt{11}$ .

Report an Error

Example Question #12 : Calculating The Area Of A Quadrilateral

Rhombus ABCD has perimeter 48; AC = 12 . What is the area of Rhombus ABCD ?

Possible Answers:

$72 \sqrt{3}$

$144 \sqrt{3}$

$72 \sqrt{2}$

144

Correct answer:

$72 \sqrt{3}$

Explanation:

Each side of a rhombus is congruent, so if it has perimeter 48, it has sidelength 12. Also, the diagonals of a rhombus are each other's perpendicular bisectors, so if they are both constructed, and their point of intersection is called , then AM = 6 . The following figure is formed by the rhombus and its diagonals.

Untitled

$\bigtriangleup AMB$ is a right triangle with its short leg half the length of its hypotenuse, so it is a 30-60-90 triangle, and its long leg measures $6\sqrt{3}$ by the 30-60-90 Theorem. Therefore, $BD = 12\sqrt{3}$ . The area of a rhombus is half the product of the lengths of its diagonals:

$\frac{1}{2} \times 12 \times 12 \sqrt{3} = 72 \sqrt{3}$

Report an Error

Example Question #12 : Rhombuses

A rhombus contains diagonals with the length $12 \:cm$ and $6 \:cm$ . Find the area of the rhombus.

Possible Answers:

$72\:cm^2$

$36\:cm^2$

$54\:cm^2$

$88\:cm^2$

$96\:cm^2$

Correct answer:

$36\:cm^2$

Explanation:

The equation for the area of a rhombus is given by:

$A= \frac{p \cdot q}{2}$

where and are the two diagonal lengths.

This problem very quickly becomes one of the "plug and chug" type, where the given values just need to be substituted into the equation and the equation then solved. By plugging in the values given, we get:

$A = \frac{12\:cm \cdot 6\:cm}{2}$

$A = \frac{72\:cm^2}2{}$

$A= 36\:cm^2$

Report an Error

Example Question #11 : How To Find The Area Of A Rhombus

Find the area of a rhombus if the diagonals lengths are $20\:cm$ and $40\:cm$ .

Possible Answers:

$600\:cm^2$

$200\:cm^2$

$400\:cm^2$

$800\:cm^2$

$900\:cm^2$

Correct answer:

$400\:cm^2$

Explanation:

Write the formula for the area of a rhombus:

$A=\frac{d_1 \cdot d_2}{2}$

Substitute the given lengths of the diagonals and solve:

$A=\frac{d1 \cdot d2}{2} = \frac{20\:cm \cdot 40\:cm}{2} =\frac{800\:cm^2}{2} = 400\:cm^2$

Report an Error

Example Question #11 : How To Find The Area Of A Rhombus

Find the area of a rhombus if the diagonals lengths are and 5a^2 .

Possible Answers:

10a^3

5a^3

5a^2

2a+5a^2

7a^3

Correct answer:

5a^3

Explanation:

Write the formula for finding the area of a rhombus. Substitute the diagonals and evaluate.

$A=\frac{d1\cdot d2}{2}= \frac{2a \cdot 5a^2}{2}= 5a^3$

Report an Error

Example Question #1 : How To Find The Area Of A Rhombus

A rhombus has a side length of 5. Which of the following is NOT a possible value for its area?

Possible Answers:

Correct answer:

Explanation:

The area of a rhombus will vary as the angles made by its sides change. The "flatter" the rhombus is (with two very small angles and two very large angles, say 2, 178, 2, and 178 degrees), the smaller the area is. There is, of course, a lower bound of zero for the area, but the area can get arbitrarily small. This implies that the correct answer would be the largest choice. In fact, the largest area of a rhombus occurs when all four angles are equal, i.e. when the rhombus is a square. The area of a square of side length 5 is 25, so any value bigger than 25 is impossible to acheive.

Report an Error

Example Question #13 : Rhombuses

Find the area of the rhombus shown below. You will have to find the lengths of the sides as well.

Rhombus area

The rhombus shown has the following coordinates:

(-2,5)(4,5)(3,-3) (-3,-3)

Round to the nearest hundredth.

Possible Answers:

24.18

48.36

Correct answer:

24.18

Explanation:

Finding the area of a rhombus follows the formula:

$\frac{p\times q}{2}$

In this rhombus, you will find that p=4-(-2) , which are the two x coordinates.

The length of q is a more involved process. You can find q by using the Pythagorean Theorem.

1^2+8^2=65

$\sqrt{65}=8.06$ .

Therefore, the area is

$\frac{6\times 8.06}{2}=24.18$ .

Report an Error

Example Question #16 : Rhombuses

Show algebraically how the formula for the area of a rhombus is developed.

Varsity4

Possible Answers:

$\frac{1}{4}d_{1}+\frac{1}{4}d_{2}$

$= (\frac{1}{4}+\frac{1}{4})d_{1}d_{2}$

$=(\frac{1}{2})d_{1}d_{2}$

$(\frac{1}{2}d_{1}+\frac{1}{2}d_{1})(\frac{1}{2}d_{2}+\frac{1}{2}d_{2})$

$=\frac{1}{2}(d_{1})\frac{1}{2}(d_{2})$

$=(\frac{1}{2})d_{1}d_{2}$

$\frac{1}{2}(\frac{1}{2}d_{1})(d_{2}) + \frac{1}{2}(\frac{1}{2}d_{1})(d_{2})$

$= \frac{1}{4}d_{1}d_{2} +\frac{1}{4}d_{1}d_{2}$

$=\frac{1}{2}d_{1}d_{2}$

$\frac{1}{4}d_{1}\cdot \frac{1}{4}d_{2}$

$=\frac{1}{2}d_{1}d_{2}$

$\frac{1}{2}\cdot (d_{1}\cdot d_{2})$

$= \frac{1}{2}d_{1}d_{2}$

Correct answer:

$\frac{1}{2}(\frac{1}{2}d_{1})(d_{2}) + \frac{1}{2}(\frac{1}{2}d_{1})(d_{2})$

$= \frac{1}{4}d_{1}d_{2} +\frac{1}{4}d_{1}d_{2}$

$=\frac{1}{2}d_{1}d_{2}$

Explanation:

1) The given rhombus is divded into two congruent isosceles triangles.

2) Each isosceles triangle has a height $h = \frac{1}{2}d_{1}$ and a base $b = d_{2}$ .

3) The area of each isosceles triangle is $A = \frac{1}{2}bh$ .

4) The areas of the two isosceles triangles are added together,

A + A

$= \frac{1}{2}(\frac{1}{2}d_{1})(d_{2}) + \frac{1}{2}(\frac{1}{2}d_{1})(d_{2})$

$= \frac{1}{4}d_{1}d_{2} +\frac{1}{4}d_{1}d_{2}$

$=\frac{1}{2}d_{1}d_{2}$

Report an Error

Example Question #17 : Rhombuses

Find the area of the rhombus below.

Possible Answers:

120

100

Correct answer:

120

Explanation:

Recall that the diagonals of the rhombus are perpendicular bisectors. From the given side and the given diagonal, we can find the length of the second diagonal by using the Pythagorean Theorem.

$\text{side}^2=(\frac{\text{diagonal 1}}{2})^2-(\frac{\text{diagonal 2}}{2})^2$

Let the given diagonal be diagonal 1, and rearrange the equation to solve for diagonal 2.

$(\frac{\text{diagonal 2}}{2})^2=\text{side}^2-(\frac{\text{diagonal 1}}{2})^2$

$(\frac{\text{diagonal 2}}{2})=\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

$\text{diagonal 2}=2\sqrt{\text{side}^2-(\frac{\text{diagonal 1}}{2})^2}$

Plug in the given side and diagonal to find the length of diagonal 2.

$\text{diagonal 2}=2\sqrt{13^2-(\frac{10}{2})^2}=2\sqrt{144}=24$

Now, recall how to find the area of a rhombus:

$\text{Area of Rhombus}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Plug in the two diagonals to find the area.

$\text{Area of Rhombus}=\frac{(10)(24)}{2}=120$

Report an Error

View Tutors

Alice
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Accounting.

View Tutors

Alyce
Certified Tutor

The University of Alabama, Master of Arts, German.

View Tutors

Joseph
Certified Tutor

Bank Street College of Education , Master's/Graduate, Early Childhood and Childhood General Education.

All Advanced Geometry Resources

6 Diagnostic Tests 57 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

GMAT Tutors in San Diego, LSAT Tutors in Washington DC, LSAT Tutors in Houston, GMAT Tutors in Denver, GRE Tutors in Atlanta, Chemistry Tutors in Phoenix, French Tutors in Phoenix, Reading Tutors in Los Angeles, Math Tutors in Denver, French Tutors in Philadelphia

Popular Courses & Classes

LSAT Courses & Classes in Chicago, GRE Courses & Classes in Chicago, GRE Courses & Classes in Seattle, GMAT Courses & Classes in Phoenix, GRE Courses & Classes in Miami, ACT Courses & Classes in New York City, SSAT Courses & Classes in Chicago, LSAT Courses & Classes in Philadelphia, Spanish Courses & Classes in Washington DC, SSAT Courses & Classes in Seattle

Popular Test Prep

ACT Test Prep in New York City, SAT Test Prep in Seattle, ISEE Test Prep in Washington DC, SSAT Test Prep in Dallas Fort Worth, MCAT Test Prep in Washington DC, ISEE Test Prep in Seattle, GRE Test Prep in Miami, MCAT Test Prep in Atlanta, SSAT Test Prep in Washington DC, SAT Test Prep in Miami

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

Advanced Geometry : Rhombuses

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #11 : Rhombuses

Example Question #11 : Calculating The Area Of A Quadrilateral

Example Question #12 : Calculating The Area Of A Quadrilateral

Example Question #12 : Rhombuses

Example Question #11 : How To Find The Area Of A Rhombus

Example Question #11 : How To Find The Area Of A Rhombus

Example Question #1 : How To Find The Area Of A Rhombus

Example Question #13 : Rhombuses

Example Question #16 : Rhombuses

Example Question #17 : Rhombuses

All Advanced Geometry Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: