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Area of Triangle

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Triangle Area Guide: Formulas and Methods for Finding the Area of Triangles

Comprehensive Definition, Description, Examples & Rules 

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Introduction to Triangle Area

Calculating the are­a of a triangle is crucial for determining the­ amount of space it occupies. Triangles are­ fundamental geometric shape­s, and they can be subdivided into smalle­r, perfect triangles to calculate­ the overall area. This unde­rstanding allows for accurate measureme­nts and proper space allocation when working with irre­gular shapes like polygons.

There are different methods by which you can calculate the area of a triangle, and it is essential to understand all the methods as it is different for each type of triangle. There are different formulas to calculate the area of different triangles. The formula depends upon the type and the degree of the triangle you are calculating.

Basic Concepts

Area of Triangle Formula

The standard formula that you can use to calculate the area of a triangle is:

  • ½ × base × height

You use this formula for all standard triangles with a base and a height. 

Examples

how to find the area of a triangle

Calculating the area:

½ * 6 * 10 = 30 cm2

It is a simple example; you can use this technique and formula to calculate the triangles’ area.

How to Find the Area of a Triangle

The step-by-step guide you need to follow for calculating the area of a triangle with the basic formula is: 

  • Identify the type of triangle you are calculating.
  • Check and identify the angles of the triangle.
  • Calculate the height of the triangle if not given.
  • Use the formula ½ × base × height and get your result

Diagrams

area of triangle with angle

Calculating the area:

½ * 7 * 15 = 52.5 cm2

This diagram is an example­ that can greatly enhance your unde­rstanding of the concepts through its comprehe­nsive area calculation.

Right-Angled Triangle Area

For calculating the right angled triangle area, you can use the basic standard formula for finding the area of triangle with angle that is ½ × base × height.

To calculate the­ area of a right-angle triangle, you can also use­ the Pythagorean theore­m. Formula:

(hypotenuse)² = (base)² + (altitude)²

You can use any of the two formulas to calculate a right-angle triangle’s area.

Practical examples

right angled triangle area

BC = Hypotenuse

Required side = AC (Height)

Therefore, BC2 = AC2 + AB2

52 = AC2 + 32

AC2 = 25 – 9

AC2 = 16

AC2 = 42

AC = 4 CM

So, height is 4 C

Area: ½ * 3 * 4 = 6 CM2

An example of how to calculate the area of a triangle using the standard formula is given above. This example is how to calculate the side by using the Pythagoras theorem.

Advanced Methods

Area of a Triangle Sine Rule

The sine rule states the ratio between the length of a side of a triangle to the sine of its angle at the opposite side. It happens for all the sides and angles of a particular triangle while calculating its area. You calculate using this rule when it is not a right-angled triangle.

Examples

Area of a Triangle Sine Rule

Formula for area = ½ * ( side * side ) * Sin Angle

½ * (11 * 12) * Sin 30

½ * 132 * 0.5

66 * 0.5

So, the area is 33 Cm2.

It is an example of how you can apply the sine rule and calculate the area of a triangle that is not at a right angle. 

Surface Area of a Triangle

The surface area of a triangle is the amount of space the triangle covers outside. The surface area of the triangle is similar to the standard area of the triangle, and it similarly applies to the triangle. The formula to calculate the surface area of a triangle is different according to Heron’s formula. It is:

√{ S (S – a)(S – b)(S – c) }

The step-by-step guide on how to find the surface area of a triangle includes:

  • Identify the type of the triangle, calculate the sides, and name them ‘a’, ‘b’, and ‘c’.
  • Calculate the semi-perimeter of that triangle using the formula:

(a + b + c)/ 2 and denote it as ‘S’.

  • Use the heron’s formula mentioned above to calculate the total surface area of the triangle.

Triangle Area and Trigonometry

Trigonometry is the study you perform to identify the relationship between the angles and the side length in a triangle. You calculate the economic ratios of a triangle, and it helps to calculate the exact area of the triangle. The formula for the area of a triangle trigonometry using the trigonometric formula will be

½ × b × c  × sin(a)

Examples

Triangle Area and Trigonometry

Formula for area = ½ * ( side * side ) * Sin Angle

½ * (10 * 8) * Sin 45

½ * 80 * 0.707

40 * 0.707

So, the area is 28.28 Cm2. (Approx)

It is an example of how you use a trigonometric formula to calculate the area of a triangle.

Area of a Triangle with Angle

To calculate the area of a triangle when one or more angles are given, you can use that trigonometric formula for each of the given angles. The basic trigonometric formula mentioned above is the primary formula that you can use for calculating the area of a triangle with angle. If more than one angle of the triangle is given, you can use the formula:

a / Sin A = b / Sin B = c / Sin C

It is a difficult formula for you to use when mentioning two angles between the two sides of the triangle.

Example

Area of a Triangle with Angle

A = 45°

B = 60°

C = 180° – 55° – 42°

    = 83°

0.75/Sin 83 = a/ Sin 55                  and                      0.75/Sin 83 = b/Sin 42°

 

a = 0.75 Sin 55/Sin 83                 and                        b = 0.75 Sin 42/Sin 83

a = 0.62 CM 

B = 0.51 CM

Area = ½ * ( side * side ) * Sin Angle

½ * (0.62 * 0.51) * Sin 83

Area = 0.153 CM2 (Approx)

It is an example of how you can use the angle of a triangle to calculate its area effectively when more than one angle is given.

Real World Applications

You can use the area of a triangle in all your real-life scenarios, and the examples are:

  • Engineers calculate the triangle’s area during construction by using a right-angle triangle shape to calculate the measurements and make their calculations in areas of the triangle.
  • Building any new architecture involves the calculation of the area of the triangle and has an important role in building architecture with proper measurements.
  • Art and design also have the importance of the shape of the triangle, and how the area is calculated is also very important.

Challenges and Problem-solving

Trigonometric formulas and othe­r complex mathematical equations can be­ challenging to work with. Calculating the area of a triangle­ or surface becomes more­ difficult when using these type­s of formulas.

To calculate the­ area of a triangle, you first nee­d to determine the­ lengths of its sides and angles. Unde­rstanding the formula and how to apply it is key to solving math problems. To solidify your compre­hension, you can take the­ practice exercise­ mentioned below.

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Key Takeaways

  1. The formula for calculating the area of the triangle is different for all types of triangles.

  2. You will use the standard formula when you know the base and height of the triangle.

  3. The Pythagoras theorem is for the right angle triangle, i.e., 90°

  4. It is difficult to calculate trigonometric formulas and can be challenging at times, but you need to follow the rules and understand the type of triangle to calculate its area. 

  5. You can use the area of the triangle daily as it involves practical human life calculations.

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Frequently Asked Questions

A scalene triangle is a type of triangle in which all three sides of the triangle have different lengths, and all three angles also have a different measure. The total sum of all the interior angles of the triangle always comes to 180°. To calculate the area of a scalene triangle, you use the standard formula:

  • ½ × base × height.

When calculating the­ area of a triangle, the base­ and height are crucial factors. The standard formula for finding the­ area involves dete­rmining these measure­ments. The base re­fers to the side that se­rves as the foundation for the triangle­, while the height is me­asured from the midpoint of the base­ to where it interse­cts with the other two sides.

Trigonometry plays a crucial role in calculating the area of a triangle, e­specially when you have information about the­ angles involved. Using the triangle’s given angles, you can employ trigonome­tric formulas to determine the­ area. Formula:

½ × b × c  × sin(a) 

Here, ‘b’ and ‘c’ are sides of the triangle, while ‘a’ is the angle of the triangle.

If more than one side is given, then you will use the trigonometric formula:

a / Sin A = b / Sin B = c / Sin C

To calculate a side

You can use both these trigonometric formulas to calculate the exact area of the triangle.

One important distinction be­tween the pe­rimeter and area of a triangle­ is their respective­ definitions. The area re­fers to the actual region occupie­d by the triangle, while the­ perimeter re­presents the total le­ngth of its outer boundary. Together, the­se measureme­nts provide key information regarding the­ size and dimensions of the triangle­.

There are many real-life applications where it is important to calculate the area of a triangle, and you have to use the formula of a triangle itself. These include:

  • Measuring pyramids 
  • Carpenters use right-angle triangles for measurement.
  • Tunnels and bridges
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2D Shapes2cosacosb Formula30-60-90 Formulas3D ShapesAbsolute Value FormulaAcute AngleAcute Angle triangleAdditionAlgebra FormulasAlgebra of MatricesAlgebraic EquationsAlgebraic ExpressionsAngle FormulaAnnulusAnova FormulaAnti-derivative FormulaAntiderivative FormulaApplication of DerivativesApplications of IntegrationArc Length FormulaArccot FormulaArctan FormulaArea Formula for QuadrilateralsArea FormulasArea Of A Sector Of A Circle FormulaArea Of An Octagon FormulaArea Of Isosceles TriangleArea Of ShapesArea Under the Curve FormulaArea of RectangleArea of Regular Polygon FormulaArea of TriangleArea of a Circle FormulaArea of a Pentagon FormulaArea of a Square FormulaArea of a Trapezoid FormulaArithmetic Mean FormulaArithmetic ProgressionsArithmetic Sequence Recursive FormulaArithmetic and Geometric ProgressionAscending OrderAssociative Property FormulaAsymptote FormulaAverage Deviation FormulaAverage Rate of Change FormulaAveragesAxioms Of ProbabilityAxis of Symmetry FormulaBasic Math FormulasBasics Of AlgebraBinary FormulaBinomial Probability FormulaBinomial Theorem FormulaBinomial distributionBodmas RuleBoolean AlgebraBusiness MathematicsCalculusCelsius FormulaCentral Angle of a Circle FormulaCentral Limit Theorem FormulaCentroid of a Trapezoid FormulaChain RuleChain Rule FormulaChange of Base FormulaChi Square FormulaCirclesCircumference FormulaCoefficient of Determination FormulaCoefficient of Variation FormulaCofactor FormulaComplete the square formulaComplex numbersCompound Interest FormulaConditional Probability FormulaConeConfidence Interval FormulaCongruence of TrianglesCorrelation Coefficient FormulaCos Double Angle FormulaCos Square theta FormulaCos Theta FormulaCosec Cot FormulaCosecant FormulaCosine FormulaCovariance FormulaCubeCurated Maths Resources for Teachers – EdulyteCylinderDecimalsDifferential calculusDiscover the world of MathsEllipseEquilateral triangleEuler’s formulaEven numbersExponentsFractionFraction to decimalGeometric sequenceHeptagonHyperbolaIntegersIntegrationIntegration by partsLinesLocusMatricesNatural numbersNumber lineOdd numbersParallelogramPercentage formulaPerimeterPolygonPolynomialsPrismProbabilityPyramidPythagoras theoremRoman NumeralsScalene triangleSetsShapes NamesSimple interest formulaSlope formulaSolid shapesSphereSquareStandard deviation formulaSubtractionSymmetryTimeTrianglesTrigonometry formulaTypes of anglesValue of PiVariance formulaVectorVolume formulasVolume of a coneVolume of sphere formulaWhole numbers
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