Question

The area of a triangle whose sides are 42 cm, 34 cm and 20, in length is

Answer 1

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Question :

Find the area of the triangle whose sides are 42cm , 34cm , 20cm ?

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Answer :-

Given :

Sides of the triangle = 42cm , 34 cm , 20cm

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Required to find :

• Area of the triangle ?

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Formulae used :

$\boxed{\tt{Semi \: perimeter = \dfrac{a + b + c}{2}}}$

Heron's Formula

$\boxed{\tt{ Area \; of \; the \; triangle = \sqrt{ s ( s - a )(s - b )(s - c)}}}$

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Solution :

Given measurement of the sides ;

42cm , 34cm & 20cm

Let's consider the sides as :-

a = 42cm

b = 34cm

c = 20cm

Here, we don't know what is base and what is height .

In this case , Heron's Formula is applicable .

In order to use Heron's Formula we should find the semi-perimeter

To find semi perimeter we have to divide the perimeter by 2 .

So,

$\boxed{\tt{Semi \: perimeter = \dfrac{a + b + c}{2}}}$

$\rightarrow{\tt{Semi \; perimeter = \dfrac{ 42 + 34 + 20}{2}}}$

$\rightarrow{\tt{Semi \; perimeter = \dfrac{ 96}{2}}}$

$\tt{\implies{s = 48 cm}}$

However,

$\tt{\boxed{ Area \; of \; the \; triangle = \sqrt{ s ( s - a )(s - b )(s - c)}}}$

here,

s = semi - perimeter

a,b,c = respective three sides of the triangle

Now, substitute the respective values

$\tt{Area = \sqrt{48(48-42)(48-34)(48-20)}}$

$\tt{Area = \sqrt{48(6)(14)(28)}}$

Now , split each number into it's factors such that leaving prime numbers

$\tt{Area = \sqrt{2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 2 \times 2 \times 2 \times 2 \times 7 \times 7}}$

$\tt{Area =\sqrt{{2}^{2} \times {2}^{2} \times {2}^{2} \times {2}^{2} \times {3}^{2} \times {7}^{2}}}$

$\tt{ Area = \sqrt{{2}^{2}} \times \sqrt{ {2}^{2} } \times \sqrt{ {2}^{2} } \times \sqrt{ {2}^{2} } \times \sqrt{ {3}^{2} } \times \sqrt{ {7}^{2} } }$

Here, squares and square roots get cancelled

$\tt{ Area = 2 \times 2 \times 2 \times 2 \times 3 \times 7}$

$\tt{Area = 16 \times 21}$

$\tt{\underline{\green{Area = 336 {cm}^{2}}}}$

Therefore,

$\red{\boxed{\large{\tt{Area \; of \; the \; triangle = 336 {cm}^{2}}}}}$

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Points to remember :-

The Heron's Formula is applicable ,

• When the measurements of the three sides are given .
• Similarly, when the measurement of the altitude is not given .

Why altitude is important ?

Altitude is the perpendicular height in a triangle .

According to the formula ,

Area of the triangle is half the product of base and height.

So, if height is not given we can find area using the general formula .

So, in such cases we have to use the Heron's Formula .

To find the area of the triangle with Heron's Formula we need to find the semi-perimeter of the triangle .

Semi-perimeter is the half of the perimeter .

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Answer 2

Given ,

The lengths of three sides of triangle are 42 cm, 34 cm and 20 cm

We know that , by Heron's formula

$\large \mathtt{ \fbox{Area \: of \: Δ</p><p> = \sqrt{s(s - a)(s - b)(s - c)} }}$

Where ,

a , b and c are the length of three sides of triangle

s is the semi perimeter of triangle i.e

$\large \mathtt{ \fbox{s = \frac{a + b + c}{2}} }$

Thus ,

$\sf \mapsto s = \frac{42 + 34+ 20}{2} \\ \\ \sf \mapsto s = \frac{96}{2} \\ \\ \sf \mapsto s = 48 \: \: cm$

and

$\sf \mapsto Area \: of \: Δ = \sqrt{48(48- 42)(48 - 34)(48 - 20)} \\ \\\sf \mapsto Area \: of \: Δ = \sqrt{48 \times 6 \times 14 \times 28} \\ \\\sf \mapsto Area \: of \: Δ = \sqrt{112896} \\ \\\sf \mapsto Area \: of \: Δ = </p><p>336 \: \: {cm}^{2}$

Hence , the area of triangle is 336 cm²