Question

1. Which of the following has its area and perimeter numerically equal?
A) An equilateral triangle of side 1 cm
B) A rectangle of side 1 cm
C) A square of side 1cm
D) A regular pentagon of side 1cm​

Answer 1

A)an equilateral triangle of side 1cm

Answer 2

Answer:

None of the following has its area and perimeter numerically equal .

Step-by-step explanation:

To find : Which of the following has its area and perimeter numerically equal?

A) An equilateral triangle of side 1 cm

B) A rectangle of side 1 cm

C) A square of side 1cm

D) A regular pentagon of side 1cm​

Solution :

    A) An equilateral triangle of side 1 cm

• We know that, perimeter of triangle is sum of three side of triangle.
• If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are the sides of triangle then,
• We know that , all sides of equilateral triangle are equal .
• Perimeter of triangle $=a+b+c$
• Therefore, perimeter of triangle $=1 \mathrm{~cm}+1\ cm + 1\ cm$

                                                              $= 3\ cm$

• Perimeter of equilateral is $3$ cm
• Area of equilateral triangle = $\frac{\sqrt{3} }{4} a^{2}$

                                                     $= \frac{\sqrt{3} }{4} \times 1\\\\=\frac{\sqrt{3} }{4} \ cm ^{2}$

   B) A rectangle of side 1 cm .

• Here we need both length and breadth to find area perimeter of rectangle which is not given. Hence we cannot find area and perimeter of rectangle .

    C) A square of side 1cm

• We know that all sides of square are equal , then perimeter of square is given by ,
• Perimeter of square = $4a$

                                           $= 4\times 1 \\\\= 4\ cm$

• Area of square = $a^{2}$

                                  $=( 1 )^{2} \\\\= 1\ cm ^{2}$

     D) A regular pentagon of side 1cm​ .

• We know that all sides of regular pentagon are equal , then perimeter of regular pentagon is given by ,
• Perimeter of regular pentagon = $5a$

                                                              $= 5\times 1\\\\= 5\ cm$  

• Area of regular pentagon = $\frac{1}{4} \sqrt{5( 5+ 2\sqrt{5} } \ s^{2}$

                                                    $= \frac{1}{4} \sqrt{5( 5 + 2\sqrt{5} } ( 1) \\\\= \frac{1}{4} \sqrt{5( 5+ 2\sqrt{5}(1)} \\\\= 1.72 \ cm ^{2}$