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

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?

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.

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

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

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).

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