Question

find the area of triangle whose sides are 10 cm 12cm and 14 cm​

Answer 1

Answer:

The sides of a triangle are 10 cm, 12 cm and 14 cm.

Semi Perimeter of Triangle :

$\sf \dfrac{10 + 12 + 14}{2} = \dfrac{36}{2} = 18 \: cm$

Now, By using herons formula we get,

$\sf Area = \sqrt{s(s-a) (s-b) (s-c)}$

$\sf Area = \sqrt{18(18-10) (18-12) (18-14)}$

$\sf Area = \sqrt{18 \times 8 \times 6 \times 4}$

$\sf Area = \sqrt{2 \times 3 \times 3 \times 2 \times 2 \times 2 \times 2 \times 3 \times 2 \times 2}$

$\sf Area = 24\sqrt{ 6} \: cm^{2}$

Answer 2

To Find :

The Area of the Triangle .

Given :

• a = 10 cm

• b = 12 cm

• c = 14 cm

Where a , b and c are the sides of the triangle.

We Know :

Heron's Formula :

$\blue{\sf{\underline{\boxed{A = \sqrt{s(s - a)(s - b)(s - c)}}}}}$

Where a , b , c are the sides of the triangle and s is the semi-perimeter .

Semi-Perimeter :

$\sf{s = \dfrac{a + b + c}{2}}$

Concept :

Since all it's sides are unequal it's a Scalene triangle .

So by using the formula for Scalene triangle , we can find the value of it's Area.

Solution :

Semi-Perimeter :

• a = 10 cm
• b = 12 cm
• c = 14 cm

Using the formula and substituting the values in it ,we get :

$\implies \sf{s = \dfrac{a + b + c}{2}} \\ \\ \\ \implies \sf{s = \dfrac{10 + 12 + 14}{2}} \\ \\ \\ \implies \sf{s = \dfrac{36}{2}} \\ \\ \\ \implies \sf{s = 18} \\ \\ \\ \therefore \purple{\sf{s = 18}}$

Hence, the Semi-Perimeter is 18 cm.

Area of the Triangle :

• a = 10 cm
• b = 12 cm
• c = 14 cm
• s = 18 cm

Using the Heron's Formula and Substituting the values in it , we get :

$\implies \sf{A = \sqrt{s(s - a)(s - b)(s - c)}} \\ \\ \\ \implies \sf{A = \sqrt{18(18 - 10)(18 - 12)(18 - 14)}} \\ \\ \\ \implies \sf{A = \sqrt{18 \times 8 \times 6 \times 4}} \\ \\ \\ \implies \sf{A = \sqrt{3456}} \\ \\ \\ \implies \sf{A = 24\sqrt{6} cm^{2}} \\ \\ \\ \therefore \purple{\sf{A = 24\sqrt{6} cm^{2}}}$

Hence , the area of the triangle is 24√6 cm².