Although the tangent of an angle can be defined from the ratio of two sides in a right triangle, the tangent of any angle can be found from its sine and cosine since tangent=sine/cosine. All three trig ratios can be found from tables or a calculator. Another way of finding tangents of angles is to use the trig identities. Starting with a known angle and its tangent, it's possible to find the tangents of other angles.
For example, tan(45°)=tan(π/4 radians)=1. (From the dimensions of an isosceles right triangle.)
tan(45°)=2tan(22.5°)/(1-tan2(22.5°)). If we make x=tan(22.5°), then:
tan(45°)=1=2x/(1-x2) so 1-x2=2x, or x2+2x=1, x2+2x+1=2, (x+1)2=2, x+1=√2, x=tan(22.5°)=√2-1.
You can find tan(10°) from tan(30°)=√3/3 (from the dimensions of the right triangles formed by bisecting an equilateral triangle.
tan(30°)=√3/3=(tan(20°)+tan(10°))/(1-tan(10°)tan(20°)), and:
tan(20°)=tan(2×10°)=2tan(10°)/(1-tan2(10°)). Let t=tan(10°), then tan(20°)=2t/(1-t2).
tan(30°)=√3/3=1/√3=(2t/(1-t2)+t)/(1-2t2/(1-t2))=(3t-t3)/(1-3t2),
1-3t2=(3t-t3)√3, t3√3-3t2-3t√3+1=0. Solve for t to find tan(10°) (not easy!).
Another useful identity is tanθ=cot(90-θ)=1/tan(90-θ). For example, tan(60°)=cot(30°)=1/tan(30°)=√3. Another example, tan(67.5°)=1/tan(22.5°)=1/(√2-1)=1+√2.
Finally, there are easy formulae for sine and cosine:
sin(x)=x-x3/3!+x5/5!-...; cos(x)=1-x2/2!+x4/4!-... so tan(x)=sin(x)/cos(x). But x has to be in radians, not degrees. (! means factorial, so, for example, 3!=1×2×3=6; 5!=1×2×3×4×5=120, etc.)
Measuring tangents using the lengths of sides of a right triangle will only give you an approximate result because measuring will always involve some error.
You can understand why calculators and tables are easier to use in most cases to find the tangents of angles.