<< Chapter < Page Chapter >> Page >

Heron’s formula

Heron’s formula finds the area of oblique triangles in which sides a , b , and c are known.

Area = s ( s a ) ( s b ) ( s c )

where s = ( a + b + c ) 2 is one half of the perimeter of the triangle, sometimes called the semi-perimeter.

Using heron’s formula to find the area of a given triangle

Find the area of the triangle in [link] using Heron’s formula.

A triangle with angles A, B, and C and opposite sides a, b, and c, respectively. Side a = 10, side b - 15, and side c = 7.

First, we calculate s .

s = ( a + b + c ) 2 s = ( 10 + 15 + 7 ) 2 = 16

Then we apply the formula.

Area = s ( s a ) ( s b ) ( s c ) Area = 16 ( 16 10 ) ( 16 15 ) ( 16 7 ) Area 29.4

The area is approximately 29.4 square units.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use Heron’s formula to find the area of a triangle with sides of lengths a = 29.7 ft , b = 42.3 ft , and c = 38.4 ft .

Area = 552 square feet

Got questions? Get instant answers now!

Applying heron’s formula to a real-world problem

A Chicago city developer wants to construct a building consisting of artist’s lofts on a triangular lot bordered by Rush Street, Wabash Avenue, and Pearson Street. The frontage along Rush Street is approximately 62.4 meters, along Wabash Avenue it is approximately 43.5 meters, and along Pearson Street it is approximately 34.1 meters. How many square meters are available to the developer? See [link] for a view of the city property.

A triangle formed by sides Rush Street, N. Wabash Ave, and E. Pearson Street with lengths 62.4, 43.5, and 34.1, respectively.

Find the measurement for s , which is one-half of the perimeter.

s = ( 62.4 + 43.5 + 34.1 ) 2 s = 70 m

Apply Heron’s formula.

Area = 70 ( 70 62.4 ) ( 70 43.5 ) ( 70 34.1 ) Area = 506,118.2 Area 711.4

The developer has about 711.4 square meters.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the area of a triangle given a = 4.38 ft , b = 3.79 ft, and c = 5.22 ft .

about 8.15 square feet

Got questions? Get instant answers now!

Access these online resources for additional instruction and practice with the Law of Cosines.

Key equations

Law of Cosines a 2 = b 2 + c 2 2 b c cos α b 2 = a 2 + c 2 2 a c cos β c 2 = a 2 + b 2 2 a b c o s γ
Heron’s formula     Area = s ( s a ) ( s b ) ( s c ) where  s = ( a + b + c ) 2

Key concepts

  • The Law of Cosines defines the relationship among angle measurements and lengths of sides in oblique triangles.
  • The Generalized Pythagorean Theorem is the Law of Cosines for two cases of oblique triangles: SAS and SSS. Dropping an imaginary perpendicular splits the oblique triangle into two right triangles or forms one right triangle, which allows sides to be related and measurements to be calculated. See [link] and [link] .
  • The Law of Cosines is useful for many types of applied problems. The first step in solving such problems is generally to draw a sketch of the problem presented. If the information given fits one of the three models (the three equations), then apply the Law of Cosines to find a solution. See [link] and [link] .
  • Heron’s formula allows the calculation of area in oblique triangles. All three sides must be known to apply Heron’s formula. See [link] and See [link] .

Section exercises

Verbal

If you are looking for a missing side of a triangle, what do you need to know when using the Law of Cosines?

two sides and the angle opposite the missing side.

Got questions? Get instant answers now!

If you are looking for a missing angle of a triangle, what do you need to know when using the Law of Cosines?

Got questions? Get instant answers now!

Explain what s represents in Heron’s formula.

s is the semi-perimeter, which is half the perimeter of the triangle.

Got questions? Get instant answers now!

Questions & Answers

what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
What are the treatment for autism?
Magret Reply
hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities. before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these
Jharna
continue.. sometimes due to these physical problems..the diagnosis may be misdiagnosed. treatment for autism. well it depends on the severity. since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child
Jharna
child interact with other kids under doc supervision. play therapy. speech therapy. Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game. string and beads game. the more you interact with the child the more effective
Jharna
results you'll get.. please consult a therapist to know what suits best on your child. and last as a parent. I know sometimes it's overwhelming to guide a special kid. but trust the process and be strong and patient as a parent.
Jharna
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask