why do we use $\pi \cdot d/2+d$ for semicircles

Question

I'm in grade seven and right now we are learning perimeter of a semicircle, and I noticed we use the formula $\pi\cdot D/2+D$ (where $D$ is the diameter) instead of $(\pi+2)D/2$, why do we do the first formula instead of the second, where they both have the same answer?

Answer 1

You can use either formula as they are identical, as you've noted. A reason for preferring the first is that it makes the perimeter calculation explicit: $\pi\cdot D/2$ is the length of the curve, and $+D$ then adds on the straight line across the bottom.

Answer 2

The results are the same: $$\left( \pi + 2 \right) D/2 = \pi D/2 + D.$$ The perimeter of a circle of radius $r$ is $2 \pi r$. Since the diameter, $D$, is twice the radius, $D = 2 r$ this gives the perimeter of a circle as $\pi D$. It follows that the perimeter for a semi (half) circle is $\pi D/2$. Now just add the remaining length $D$ of the line joining the two ends of the semi circle.

Since the results are the same which one you use is simply your preference. Personally, I prefer the second form since it makes clear the contributions from each part of the curve (the semi-circle versus the straight line segment).

Answer 3

The perimeter of a full circle is $\pi D$

If we consider just half the circle then perimeter must also be half.

$\therefore$ perimeter of this circle is $\pi D /2$

However, a semicircle looks like this

Which means for computing perimeter of semicircle we just need to add the length of newly added red color.segment. So we just add $D$ in earlier expression.

$\therefore$ Perimeter of the semicircle is $\boxed{ \frac{\pi D}{2} +D}$

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$\pi D/2 + D = (\pi+2)D/2$, as you can see both are same expressions, we can use any one of them.