Probable error (Area)

Question

Calculate the area of the triangular tract of land and its most

Answer 1

I'll add my answer because the question is requesting the probable error. To calculate the area A (essentially the same as kamran's answer but in line with the OP symbols) as:

$$A= \sqrt{k(k-a)(k_a-b)(k-c)}$$

where:

• $k =\frac{a+b+c}{2}$
• a, b,c are respectively 180.21, 275.26, 156.31

This simplifies to: $$\sqrt{(a+b) (a+c) (b+c) (a+b+c)}$$

In order to calculate the error you can use the following relationship:

$$ dA = \frac{\partial A}{\partial a}da+\frac{\partial A}{\partial b}db+\frac{\partial A}{\partial c}dc$$

where:

• $\frac{\partial A}{\partial a}= \frac{(b+c) \left(3 a^2+4 a (b+c)+b^2+3 b c+c^2\right)}{2 \sqrt{(a+b) (a+c) (b+c) (a+b+c)}}$
• $\frac{\partial A}{\partial b}=\frac{(a+c) \left(a^2+4 a b+3 a c+3 b^2+4 b c+c^2\right)}{2 \sqrt{(a+b) (a+c) (b+c) (a+b+c)}}$
• $\frac{\partial A}{\partial c}= \frac{(a+b) \left(a^2+3 a b+4 a c+b^2+4 b c+3 c^2\right)}{2 \sqrt{(a+b) (a+c) (b+c) (a+b+c)}}$
• $da, db, dc$ are respectively $0.05, 0.02, 0.04$

Since you probably want the relative error you'd need to divide by A:

$$\frac{dA }{A} =\frac{1}{A} \left(\frac{\partial A}{\partial a}da+\frac{\partial A}{\partial b}db+\frac{\partial A}{\partial c}dc\right)\Rightarrow$$

$$\frac{dA }{A} =\frac{1}{A}\frac{\partial A}{\partial a}da+\frac{1}{A}\frac{\partial A}{\partial b}db+\frac{1}{A}\frac{\partial A}{\partial c}dc$$

where:

• $\frac{1}{A}\frac{\partial A}{\partial a}=\frac{3 a^2+4 a (b+c)+b^2+3 b c+c^2}{2 (a+b) (a+c) (a+b+c)}$
• $\frac{1}{A}\frac{\partial A}{\partial b}=\frac {a^2 + 4 a b + 3 a c + 3 b^2 + 4 b c + c^2} {2 (a + b) (b + c) (a + b + c)}$
• $\frac{1}{A}\frac{\partial A}{\partial c}=\frac{a^2+3 a b+4 a c+b^2+4 b c+3 c^2}{2 (a+c) (b+c) (a+b+c)}$

A simpler example for the error (for the rectangle case) can be seen here.

Answer 2

Famous ancient mathematician Heron of Alexandria, Greco- Egyptian mathematician has this formula.

Say S1 and S2 and S3 are the sides.

And we call $$ S_a =\frac{S1+S2+S3}{2}, \\ then \ Area= \sqrt{S_a(S_a-S1)(S_a-S2)(S_a-S3)} $$

I let you do the calcs.

Wikipedia link to Heron