Calculator Techniques for Quadrilaterals in Plane Geometry

Updated on December 15, 2019
JR Cuevas profile image

Ray is a Licensed Civil Engineer and specializes in Structural Engineering. He loves to write anything about education.

What Is a Quadrilateral?

A quadrilateral also called as "Tetragon" or "Quadrangle", is a general term for any polygon with four sides. There are six types of quadrilaterals. They are square, rectangle, parallelogram, trapezoid, rhombus, ad trapezium.

Common Parts of a Quadrilateral

The common parts of a quadrilateral are described as follows.

1. Side - A side is a line segment which joins any two adjacent vertices.

2. Interior Angle - An interior angle is the angle formed between two adjacent sides.

3. Height or Altitude - It is the distance between two parallel sides of a quadrilateral

4. Base - This is the side that is parallel to the altitude.

5. Diagonal - This is the line segment joining any two non-adjacent vertices

Calculator Techniques for Quadrilaterals in Plane Geometry
Calculator Techniques for Quadrilaterals in Plane Geometry | Source
Quadrilateral
Description
Area
Perimeter
Square
A special type of a rectangle in which all the sides are equal.
A = a^2
P = 4a
Rectangle
A parallelogram in which the interior angles are all right angles.
A = ab
P = 2b + 2a
Rhombus
A parallelogram in which all sides are equal.
A = (1/2) (d1) (d2)
P = 4a
Trapezoid
A quadrilateral with one pair of parallel sides.
A = (1/2) (a + b) (h)
P = a + b + c + d
Trapezium
A quadrilateral with no parallel sides.
 
P = a + b + c + d
Parallelogram
A quadrilateral in which the opposite sides are parallel.
A = bh OR A = absin(theta)
P= 2a + 2b

Properties of a Parallelogram

1. The opposite sides of a parallelogram are equal, and so also the opposite angles.

2. The diagonals of a parallelogram bisect each other.

3. If two sides of a quadrilateral are equal and parallel, then the other two sides are equal and parallel, and the figure is therefore a parallelogram.

Cyclic Quadrilateral

A cyclic quadrilateral is a quadrilateral whose vertices lie on the circumference of a circle. According to Ptolemy's theorem, the product of the diagonals of any cyclic quadrilateral is equal to the sum of the products of the opposite sides.

d1d2 = (a) (c) + (b) (d)

Cyclic Quadrilateral
Cyclic Quadrilateral | Source

Problem 1: Area of a Rhombus

A rhombus has diagonals of 32 and 20 inches. What is the area?

Calculator Technique for Quadrilaterals: Area of a Rhombus
Calculator Technique for Quadrilaterals: Area of a Rhombus | Source

Calculator Technique

a. Given the length of the rhombus' diagonals, use the formula in solving for the area.

d1 = 32 inches
d2 = 20 inches
Area of rhombus = 1/2 (d1) (d2)
Area of rhombus = 1/2 (32) (20)
Area of rhombus = 320 square inches

Final Answer: The area of the rhombus is 320 square inches.

Problem 2: Square Inscribed in a Semicircle

Find the difference of the area of the square inscribed in a semicircle having a radius of 15 centimeters.

Calculator Techniques for Quadrilaterals: Square Inscribed in a Circle
Calculator Techniques for Quadrilaterals: Square Inscribed in a Circle | Source

Calculator Technique

a. Calculate the side length of the square inside the semicircle.

a^2 + b^2 = c^2
x^2 + (x/2)^2 = 15^2
x^2 + (x^2/4) = 225
5/4 x^2 = 225
x = 180 centimeters
x = 13.42 centimeters

b. Solve for the area of the semicircle.

Area of semicircle = (pi) (r)^2 / 2
Area of semicircle = (pi) (15)^2 / 2
Area of semicircle = 353.49 square centimeters

c. Solve for the area of the square inscribed.

Area of square = x^2
Area of square = (13.42)^2
Area of square = 180.10 square centimeters

d. Get the difference of areas.

Area difference = Area of semicircle - Area of square
Area difference = 353.49 - 180.10
Area difference = 173.39 square centimeters

Final Answer: The difference of the area of the square inscribed in a semicircle is 173.39 square centimeters.

Problem 3: Lengths of Trapezoid's Bases

A trapezoid has an area of 36 square centimeters and a height of 2 centimeters. Its two bases have a ratio of 4: 5. What are the lengths?

Calculator Technique for Quadrilaterals: Lengths of Trapezoid's Bases
Calculator Technique for Quadrilaterals: Lengths of Trapezoid's Bases | Source

Calculator Technique

a. Solve for the area of the trapezoid.

Altitude of trapezoid (h) = 2 centimeters
Area of trapezoid = 36 square centimeters
Area of trapezoid = (1/2) (b1 + b2) (h)
36 square centimeters = (1/2) (b1 + b2) (2)

b. Let b1 = 4x and b2 = 5x. Solve for the value of x.

36 square centimeters = (1/2) (b1 + b2) (2)
36 square centimeters = (1/2) (4x + 5x) (2)
36 square centimeters = 4x + 5x
36 square centimeters = 9x
x = 4 centimeters

c. Solve for b1 and b2 by substituting the value of x.

4x = 4 (4)
4x = 16 centimeters
5x = 5 (4)
5x = 20 centimeters

Final Answer: The lengths of the trapezoid's bases are 16 and 20 centimeters.

Problem 4: Diagonals and Angles of a Rhombus

One of the diagonals of the rhombus is 12 inches. If the area of the rhombus is 132 square inches, determine the following.

a. length of the other diagonal
b. the measure of acute angles in degrees between the two sides

Calculator Techniques for Quadrilaterals: Diagonals and Angles of a Rhombus
Calculator Techniques for Quadrilaterals: Diagonals and Angles of a Rhombus | Source

Calculator Technique

a. Solve for the length of other diagonal using the formula for area.

Area of rhombus = 1/2 (d1) (d2)
132 square inches = 1/2 (12 inches) (d2)
d2 = 22 inches

b. Solve for the length of one side of the rhombus using the Pythagorean theorem.

X^2 = 11^2 + 6^2
X = 12.53 inches

c. Solve for the angle θ.

tan θ = opposite / adjacent
tan θ = 6/11
θ = 28.61 degrees

Final Answer: The length of the other diagonal is 22 inches, and the angle between the two sides is 28.61 degrees.

Problem 5: Interior Angles of Parallelogram

Two sides of parallelogram measure 68 centimeters and 83 centimeters and its shorter diagonal is 42 centimeters. Calculate the hugest interior angle.

Calculator Techniques for Quadrilaterals: Interior Angles of Parallelogram
Calculator Techniques for Quadrilaterals: Interior Angles of Parallelogram | Source

Calculator Technique

a. Using the cosine law, solve for the θ between the two known sides.

42^2 = 68^2 + 83^2 – 2(68)(83) cosine (θ)
θ = 30.27 degrees

b. Solve for the supplementary angle of θ.

Largest interior angle = 180 – 30.27 degrees
Largest interior angle = 149.73 degrees

Final Answer: The interior angle of the rhombus with the highest value is 149.73 degrees.

Which calculator techniques would you like me to cover?

See results

Questions & Answers

    © 2019 Ray

    Comments

      0 of 8192 characters used
      Post Comment

      No comments yet.

      working

      This website uses cookies

      As a user in the EEA, your approval is needed on a few things. To provide a better website experience, owlcation.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

      For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: https://owlcation.com/privacy-policy#gdpr

      Show Details
      Necessary
      HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
      LoginThis is necessary to sign in to the HubPages Service.
      Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
      AkismetThis is used to detect comment spam. (Privacy Policy)
      HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
      HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
      Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
      CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
      Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
      Features
      Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
      Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
      Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
      Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
      Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
      VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
      PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
      Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
      MavenThis supports the Maven widget and search functionality. (Privacy Policy)
      Marketing
      Google AdSenseThis is an ad network. (Privacy Policy)
      Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
      Index ExchangeThis is an ad network. (Privacy Policy)
      SovrnThis is an ad network. (Privacy Policy)
      Facebook AdsThis is an ad network. (Privacy Policy)
      Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
      AppNexusThis is an ad network. (Privacy Policy)
      OpenxThis is an ad network. (Privacy Policy)
      Rubicon ProjectThis is an ad network. (Privacy Policy)
      TripleLiftThis is an ad network. (Privacy Policy)
      Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
      Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
      Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
      Statistics
      Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
      ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
      Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)
      ClickscoThis is a data management platform studying reader behavior (Privacy Policy)