A rectangle whose length and width are in the ratio of 3∶4 and the perimeter of the rectangle is 28 cm. If the rectangle is inscribed in a circle, the shaded area of circle is ______

Explanation

Perimeter of the rectangle = 28 cm. Ratio of length and width = 3 ∶ 4. Let length(l) = x cm, width(b) = y cm. By the given ratio, length/width = 3/4 ⇒ 4x = 3y ⇒ x = (3/4)y. Perimeter of the rectangle = 2(l + b) ⇒ 28 = 2(x + y) ⇒ 28 = (2/4)(7y) ⇒ 56 = 7y ⇒ y = 8 cm. Put the value of y in eq (I): x = (3/4) × 8 = 6 cm. For the diameter of the circle: each corner of rectangle is 90°; applying Pythagoras theorem, diagonal AD² = 8² + 6² = 64 + 36 = 100 ⇒ AD = √100 = 10 cm. Radius of the circle = 10/2 = 5 cm. Area of rectangle = l × b = 6 × 8 = 48 cm². Area of the circle = (22/7) × 25 = 78.57 cm². ∴ Shaded area = Area of the circle – Area of the rectangle = 78.57 – 48 = 30.57 cm².