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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.

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

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.

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

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.

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

Explain what s represents in Heron’s formula.

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

Questions & Answers

Calculate price elasticity of demand and comment on the shape of the demand curve of a good ,when its price rises by 20 percentage, quantity demanded falls from 150 units to 120 units.
Helen Reply
5 %fall in price of good x leads to a 10 % rise in its quantity demanded. A 20 % rise in price of good y leads to do a 10 % fall in its quantity demanded. calculate price elasticity of demand of good x and good y. Out of the two goods which one is more elastic.
Helen
what is labor
Grace Reply
labor is any physical or mental effort that helps in the production of goods and services
Kwabena
what is profit maximizing level of out put for above hypothetical firm TC = Q3 - 21Q2 + 600 + 1800 P = 600 MC = 3Q2 - 42Q + 600
Sosna Reply
consider two goods X and Y. When the price of Y changes from 10 to 20. The quantity demanded of X changes from 40 to 35. Calculate cross elasticity of demand for X.
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Sosna
consider two goods X and Y. When the price of Y changes from 10 to 20. The quantity demanded of X changes from 40 to 35. Calculate cross elasticity of demand for X.
Sosna
The formula for calculation income elasticity of demand is the percent change in quantity demanded divided by the percent change in income.
Sosna
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Lizzy Reply
if the demand function is q=25-4p+p² 1.find elasticity of demand at the point p=5?
Puja Reply
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Obeng Reply
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Fosua
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it is the degree of responsiveness to a percentage change in the price of the commodity
Obeng
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John Reply
what is monopoly
Peter Reply
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Peter
is the compulsory transfer of wealth from the private sector to the public sector
Jonna
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Adeola Reply
because monopoly have no competitor on the market and they are price makers,therefore,they can easily increase the princes and produce small quantity of goods but still consumers will still buy....
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Adeola Reply
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jimmy
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investment is the purchase of good that are not consumed today but are used in the future to create wealth.
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 Supply represents how much the market can offer.
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explain how government taxes and government producer subsidies affect supply
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macroeconomics,microeconomics,positive economics and negative economics
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process of production
Mutia
Basically factors of production are four (4) namely: 1. Entrepreneur 2. Capital 3. Labour and; 4. Land but there has been a new argument to include an addition one to the the numbers to 5 which is "Technology"
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what is Economic
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economics is how individuals bussiness and governments make the best decisions to get what they want and how these choices interact in the market
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Economics as a social science, which studies human behaviour as a relationship between ends and scarce means, which have alternative uses.
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Economics is a science which study human behaviour as a relationship between ends and scarce means
John
Economics is a social sciences which studies human behavior as a relationship between ends and scarce mean, which have alternative uses.....
Pintu
types of demand elasticity
Farouq Reply
what is competitive market?
Shantal Reply
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
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The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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