Area with Herons Formula


 
 
Concept Explanation
 

Area with Herons Formula

Heron's Formula : In some cases, length of each side of the triangle are given but height of the triangle is neither given nor we are able to find in any way, then to find the area of such type of triangle, we use Heron's formula which is given below.

Area of triangle = large sqrt{s(s-a)(s-b)(s-c)},  where a,b,c are length of the sides of a triangle and s is the semi-perimeter of the triangle i.e. large s=frac{a+b+c}{2}

Example:  Find the area of a quadrilateral ABCD in which AB = 3 cm, BC = 4 cm, CD = 4 cm, DA = 5 cm and AC = 5 cm.

Solution :  We divide the quadrilateral  ABCD in two triangle ABC and ACD

For large Delta ABC, a = 3, b = 4, c = 5 and large s= frac{3+4+5}{2}=6

Area (large DeltaABC) = large sqrt{s(s-a)(s-b)(s-c)}

                                               =  large sqrt{6(6-3)(6-4)(6-5)}=sqrt{6times 3times 2times 1= 6 large ^{cm^{2}}

Similarly, for large Delta ACD,

a = 5 cm, b = 5 cm, c = 4 cm and s =large frac{5+5+4}{2}=7

Area (large Delta ACD ) = large {sqrt{s(s -a)(s - b)(s - c)} =large {sqrt{7(7 -5)(7- 5)(7- 4)}

  =large sqrt{7times 2times 2times 3}   =  large 2sqrt{21}cm^{2}

large therefore Area of quadrilateral ABCD = Area of triangle ABC+ Area of triangle ACD = large 6cm^{2} + large 2sqrt{21}cm^{2}

 

Sample Questions
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Question : 1

Assertoin: The sides of a triangle are 3 cm, 4 cm and 5 cm. Its area is large 6cm^2.

Reason: If 2s=(a+b+c), where a, b ,c  are the sides of a triangle, then area=large area=sqrt{(s-a)(s-b)(s-c)}

Right Option : C
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Explanation
Question : 2

If every side of a triangle is doubled then increase in area of triangle is :

Right Option : C
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Explanation
Question : 3

If the sides of a triangle are doubled, then its area ______________.

Right Option : D
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Explanation
 
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