Approximation to solve an oblique triangle

The previous post gave a simple and accurate approximation for the smaller angle of a right triangle. Given a right triangle with sides a, b, and c, where a is the shortest side and c is the hypotenuse, the angle opposite side a is approximately

$A \approx \frac{3a}{b + 2c}$

in radians. The previous post worked in degrees, but here we’ll use radians.

If the triangle is oblique rather than a right triangle, there an approximation for the angle A that doesn’t require inverse trig functions, though it does require square roots. The approximation is derived in [1] using the same series that is the basis of the approximation in the earlier post, the power series for 2 csc(x) + cot(x).

For an oblique triangle, the approximation is $A \approx \frac{6 \sqrt{(s - b)(s - c)}}{2\sqrt{bc} + \sqrt{s(s-a)}}$

where s is the semiperimeter.

$s = \frac{a + b + c}{2}$

For comparison, we can find the exact value of A using the law of cosines.

$a^2 = b^2 + c^2 - 2 bc \cos A$

and so

$A = \cos^{-1}\left(\frac{b^2 + c^2 - a^2}{2bc}\right)$

Here’s a little Python script to see how accurate the approximation is.

from math import sqrt, acos

def approx(a, b, c):
    "approximate the angle opposite a"
    s = (a + b + c)/2
    return 6*sqrt((s - b)*(s - c)) / (2*sqrt(b*c) + sqrt(s*(s - a)))

def exact(a, b, c):
    "exact value of the angle opposite a"    
    return acos((b**2 + c**2 - a**2)/(2*b*c))

a, b, c = 6, 7, 12
print( approx(a, b, c) )
print( exact(a, b, c) )

This prints

0.36387538476776243
0.36387760856668505

showing that in our example the approximation is good to five decimal places.

[1] H. E. Stelson. Note on the approximate solution of an oblique triangle without tables. American Mathematical Monthly. Vol 56, No. 2 (February, 1949), pp. 84–95.

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