What Is The Smallest Perimeter For A Rectangle With An Area Of 16

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Answer: What could be the least perimeter of a rectangle of area 16 sq cm? Here is the smallest perimeter for a given area calculator and the smallest perimeter for a rectangle calculator. The minimum perimeter of a rectangle with an area of 16 inch² is 16 inches for equal sides of 4 inches.

What could be the least perimeter of a rectangle of area 16 sq cm?

The question is ‘What Is The Smallest Perimeter For A Rectangle With An Area Of 16?’. If we denote one side of the rectangle with ‘a’ and the other side with ‘b’, we can write that:

a x b = 16

so we can agree that b = 16 ÷ a

Now we can write perimeter ‘P’ as a function of ‘a’ as follows:

P = 2 ⋅ (a + 16/a)

We are looking for the smallest perimeter for the rectangle. So we have to calculate the derivative as follows:

P(a) = 2a + 32/a

P'(a) = 2 + (-32/a²)

P'(a) = 2 – 32/a²

= (2a² – 32)/a²

The extreme values can only be found in points where
P'(a) = 0

So P'(a) = 0 ⇔ 2a² – 32 = 0

2a² – 32 = 0

xa² – 16 = 0

X x . . a² = 16

X X xa = -4 or a = 4

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Since length is a scalar quantity, therefore it cannot be negative.

When a = 4,

b = 16 ÷ 4

b = 4

You may be lost in thoughts that since both sides of the rectangle are equal in length hence, the shape should be a square instead of a rectangle.

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However, the answer is ‘NO’ because you can recall that the properties of a rectangle are as follows:

The opposite sides of a rectangle are parallel.
The opposite sides of a rectangle are congruent.
The diagonals of a rectangle bisect each other.
The diagonals of a rectangle are congruent.
Each of the interior angles of a rectangle must be 90°.
Since there is no rule that states a rectangle cannot have all sides of equal length, all squares are rectangles, but not rectangles are squares.

Hence, the minimum perimeter of a rectangle with an area of 16 is 16 inches with equal sides of
4 inches.

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