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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?

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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.

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

enter image description here

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

enter image description here

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

However, a semicircle looks like this

enter image description here

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 $\boxed{ \frac{\pi D}{2} +D}$


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

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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).

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    $\begingroup$ Sorry but this doesn't address the question of the OP. $\endgroup$ – Jean Marie Jun 14 at 8:45
  • $\begingroup$ They asked why, I’m sure (s)he knows they are the same $\endgroup$ – Saketh Malyala Jun 14 at 8:54

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