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Maths / Mensuration / Area with Herons Formula

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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 ( $large Delta$ ABC) = $large sqrt{s(s-a)(s-b)(s-c)}$

= $large sqrt{6(6-3)(6-4)(6-5)}=sqrt{6times 3times 2times 1$ = 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 = $large 6cm^{2}$ + $large 2sqrt{21}cm^{2}$

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Area with Herons Formula

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