# What is the area of a triangle with sides 2, 5 and 7

### 3 Answers | Add Yours

There is no triangle with sides 2, 5, and 7, because the sum of two sides equals the third side.

Then we can not form a triangle using sides 2, 5, and 7.

The area then equals 0.

Let us try an prove it using Heron's rule.

A = sqrt[s*(s-a)(s-b)(s-c)]

s = p/2 = (2+5+7)/2 = 14/2 = 7

==> A = sqrt[7*(7-2)(7-5)(7-7)]

= sqrt(7*5*2*0) = sqrt0 = 0

**Then the area (A) = 0**

If you look at the sides given you can see that 2+5=7. Now it is not possible to form a triangle when the sum of two sides is equal to the third side. Hence there is no need to attempt to find the area. You can see the lengths of the sides given and infer that the result cannot be determined.

Even if we do try to find the area, it can be done using the formula that Area= sqrt [s* (s-a)* (s-b)*(s-c)]

where s is the semi- perimeter equal to (a+b+c)/2

Now we have a=2, b=5 and c=7.

So the semi-perimeter is (2+5+7)/2= 7

Area = sqrt [ 7*(7-2)*(7-5)*(7-7)] =0

As expected you can see that the area is 0.

We use Heron's formula to find the area when all 3 sides a,b,c of a triangle is known.

But here 2+5 = 0. When the sum of the 2 sides are equal to the third side, there is no formation of a triangle. It becomes a triangle of zero area.

Ina triangle the sum of any two sides should be greter than the third side.

But heron's formula still holds good.

Heron's formula:

Area of the triangle = sqrt{(s(s-a)(s-b)(s-c)}. Where a, b and c are the length of the sides of the triangtle. s = (a+b+c)/2.

Here a = 2 , b = 5 and c = 7. Therefore s = (2+5+7)/2 = 7.

Area of triangle = sqrt{7(7-1)(7-5)(7-7)} = sqrt{7(5*7*0)} = 0.