Question

Find area of triangle by Heron's formula having sides 60 cm, 100cm, and 140 cm.​

Answer 1

Given Information :

• Sides of the triangle :

• a = 60 cm
• b = 100 cm
• c = 140 cm

To calculate :

• Area of the triangle by Heron's formula.

Calculation :

Heron's formula :

$\boxed{ \sf { Area_{(\Delta) } = \sqrt{s(s-a)(s-b)(s-c)}} }$

• s = semi-perimeter
• a, b and c are sides.

Let's find out the Semi-perimeter of the ∆ first.

$\longmapsto$ Semi-perimeter = $\sf \dfrac{a+b+c}{2}$

$\longmapsto$ Semi-perimeter = $\sf \dfrac{(60+100+140) \: cm}{2}$

$\longmapsto$ Semi-perimeter = $\sf \dfrac{(160+140) \: cm}{2}$

$\longmapsto$ Semi-perimeter = $\sf \dfrac{300 \: cm}{2}$

$\implies \boxed { \sf \red { Semi-perimeter = 150 \: cm}}$

Now, substitute the value of Semi-perimeter and the sides in the formula .

$\longrightarrow \sf { Area_{(\Delta) } = \sqrt{s(s-a)(s-b)(s-c)} }$

$\longrightarrow \sf { Area_{(\Delta) } = \sqrt{150(150-60)(150-100)(150-140)} \: cm^2}$

$\longrightarrow \sf { Area_{(\Delta) } = \sqrt{150(90)(50)(10)} \: cm^2}$

$\longrightarrow \sf { Area_{(\Delta) } = \sqrt{150 \times 90 \times 50 \times 10} \: cm^2}$

$\longrightarrow \sf { Area_{(\Delta) } = \sqrt{15 \times 10 \times 3 \times 3 \times 10 \times 5 \times 10 \times 10} \: cm^2}$

$\longrightarrow \sf { Area_{(\Delta) } =10 \times 10 \times 3 \sqrt{15 \times 5 } \: cm^2}$

$\longrightarrow \sf { Area_{(\Delta) } = 300 \sqrt{5 \times 5 \times 3 } \: cm^2}$

$\longrightarrow \sf { Area_{(\Delta) } = 300 \times 5 \sqrt{3 } \: cm^2}$

$\implies \boxed { \sf \red { Area_{(\Delta) } = 1500 \sqrt{3} \: cm^2}}$

Therefore, area of the ∆ is 1500√3 $\sf cm^2$ .

$\Large {\underline { \sf {A \: little \: further !!}}}$

More about triangles :

• Sum of interior angles of a triangle = 180°
• Sum of two interior opposite angles of ∆ = Exterior angle of ∆
• Perimeter of triangle = Sum of all sides
• Area of an equilateral triangle = ‌$\sf { \dfrac{\sqrt{3}}{4} \times {Side}^{2} }$
• Area of ∆ = $\sf { \dfrac{1}{2} \times Base \times Height }$

Answer 2

Given :

• 1st side of Triangle = 60cm
• 2nd side of Triangle = 100cm
• 3rd side of Triangle = 140cm

To Find :

• Area of Triangle

Solution :

✰ As we know that, if three sides of a triangle are given then we have to use Heron's Formula to find the area of the Triangle. Now in this question, Three sides are given so firstly we will find the semi perimeter of the Triangle and then we will apply Heron's Formula to find the area of the Triangle.

⠀

⠀⠀⠀⟼⠀⠀⠀s = a + b + c/2

⠀⠀⠀⟼⠀⠀⠀s = 60 + 100 + 140/2

⠀⠀⠀⟼⠀⠀⠀s = 160 + 140/2

⠀⠀⠀⟼⠀⠀⠀s = 300/2

⠀⠀⠀⟼⠀⠀⠀s = 150cm

⠀

Thus semi perimeter of Triangle is 150cm

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⠀

✰ Now, We will find the area of the Triangle using Heron's Formula.

➟ √s (s - a) (s - b) (s - c)

➟ √150 (150 - 60) (150 - 100) (150 - 140)

➟ √150 × 90 × 50 × 10

➟ √13500 × 500

➟ √6750000

➟ 2,598.07cm²

⠀

Thus Area of Triangle is 2,598.07cm²

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