False, unless the figures are the same type.
Imagine a cross-section of a staircase with 5 steps, each with a tread and drop of 1 unit. So the height of the staircase is 5 units and its length is also 5 units. The perimeter of the cross-section is 5 units up (height) and 5 units along (length). There are 5 steps so each step counts for 2 units, making 10 units (imagine carpeting the steps). The total perimeter is 5+5+10=20 units. The angle of the staircase is 45º.
Now consider a right isosceles triangle, with equal sides x units long. The hypotenuse is x√2 units, so the perimeter is 2x+x√2=x(2+√2)=3.414x units approx.
The area of the triangle is x²/2 square units.
The area of the cross-section of the staircase is 1+2+3+4+5=15 square units.
If the perimeters are made equal, 3.414x=20, so x=20/3.414=5.858 units approx and the area of the triangle is 17.16 square units approx, bigger than the area of the staircase cross-section, which is only 15 square units.
So, although the perimeters are the same the triangle has a larger area. It follows then that the figure with the smaller area (staircase) doesn’t have a smaller perimeter—in this case they have the same perimeter length. If we made the areas the same, the perimeter of the staircase would be greater than the perimeter of the triangle. So the statement is false, in general.