Question

The sides of a triangle are 20cm, 34cm, and 42cm. Find the area of the triangle using Heron's formula.

Answer 1

Side A ⇒ 20 cm

Side B ⇒ 34 cm

Side C ⇒ 42 cm

Heron's Formula ⇒ $\sf \sqrt{s(s-a)(s-b)(s-c)}$

Here the value of 's' is the half of the perimeter.

So to find the area we must find the perimeter first.

Perimeter of Triangle ⇒ Side A + Side B + Side C

Perimeter of Given Triangle ⇒ 20 + 34 + 42

Perimeter of Given Triangle ⇒ 96

Now we shall find the half of the perimeter which is 96. To do so we will divide 96 by 2.

⇒ 96 ÷ 2

⇒ 48

∴ The value of 's' would be 48 here.

Heron's Formula ⇒ $\sf \sqrt{s(s-a)(s-b)(s-c)}$

Here the value of 's' is the half of the perimeter. Here 's' is 48.

⇒ $\sf \sqrt{48(48-20)(48-34)(48-42)}$

⇒ $\sf \sqrt{48(28)(14)(6)}$

⇒ $\sf \sqrt{48(2352)}$

⇒ $\sqrt{112896}$

Now we shall find the square root of 112896. To do so we must do the prime factorization of 112896.

$\begin{array}{r | l} 2 & 112896 \\ \cline{2-2} 2 & 56448 \\ \cline{2-2} 2 & 28224\\ \cline{2-2} 2 & 14112\\ \cline{2-2} 2 & 7056 \\ \cline{2-2} 2 & 3528\\\cline{2-2} 2& 1764\\ \cline{2-2} 2&882 \\\cline{2-2} 3 &441 \\ \cline{2-2} 3 & 147 \\ \cline{2-2} 7&49 \\ \cline{2-2} &7 \end{array}$

112896 = 2×2×2×2×2×2×2×2×3×3×7×7

Now we will pair the product of primes.

112896 = (2×2) × (2×2) × (2×2) × (2×2) × (3×3) × (7×7)

Now we'll take one product of prime from each to find the square root.

√112896 = 2 × 2 × 2 × 2 × 3 × 7

√112896 = 336

∴ The area of the given triangle is 336 cm³.

Answer 2

Answer :-

• Area of the triangle is 336cm².

Given :-

• The sides of a triangle are 20cm, 34cm, and 42cm.

To Find :-

• Area of the triangle using Heron's formula.

Solution :-

→ Heron's formula = √{ s (s - a) (s - b) (s - c) }

Here

• a = 20
• b = 34
• c = 42

As we know that

Perimeter of a triangle is -

sum of all sides

⇒ 20 + 34 + 42

⇒ 96cm

Semi perimeter (S) = perimeter/2

⇒ 96/2

⇒ 48

Put the values in the formula

√ { 48 (48 - 20) (48 - 34) (48 - 42)

⇒ √ 48 × 28 × 14 × 6

⇒ √ 112896

⇒ √ 336 × 336

⇒ 336

Hence, the area of the triangle is 336cm²