The perimeter of a right-angled isosceles triangle is 20 units. If α is the area of the triangle, then

Let the equal legs be x; the hypotenuse is x√2. Perimeter: 2x + x√2 = 20, so x(2 + √2) = 20, giving x = 20/(2 + √2) = 20(2 − √2)/[(2 + √2)(2 − √2)] = 20(2 − √2)/2 = 10(2 − √2) = 20 − 10√2 ≈ 20 − 14.14 = 5.86. Area α = (1/2)x² = (1/2)(20 − 10√2)² = (1/2)(400 − 400√2 + 200) = (1/2)(600 − 400√2) = 300 − 200√2 ≈ 300 − 282.84 = 17.16. So 17 < α < 18.