Solve each of the following problems: 1. The longer side of a parallelogram is 8 cm. If the shorter side is 3/4 of the longer side, find the perimeter of the parallelogram. 2. Find... Solve each of the following problems: 1. The longer side of a parallelogram is 8 cm. If the shorter side is 3/4 of the longer side, find the perimeter of the parallelogram. 2. Find the area of a circle, whose radius is: (i) 2.1 cm (ii) 4.2 cm (iii) 6.3 cm (iv) 7.7 cm 3. The ratio of radii of two circles is 2:3. Find the ratio of their areas. 4. The ratio of circumferences of two circles is 3:4. Find the ratio of their areas. 5. The area of a circle is equal to its circumference. Find its radius. 6. A circular park of radius 8 m has a path of wide 2 m running around inside the boundary. Find the cost of paving the path at the rate of ₹18.50 per sq. m. 7. A wire bent in the form of a square encloses an area of 121 sq. cm. If the same wire is bent in the form of a circle, find the area it encloses. 8. A square and a circle both have a perimeter of 176 cm. Which one has more area, and how much? Read more

Steps to Solve

1. Parallelogram Perimeter The longer side of the parallelogram is given as 8 cm. The shorter side is $\frac{3}{4}$ of the longer side. Calculate the length of the shorter side: $ \text{Shorter side} = \frac{3}{4} \times 8 \text{ cm} = 6 \text{ cm} $ The perimeter of a parallelogram is given by $2 \times (\text{longer side} + \text{shorter side})$. $ \text{Perimeter} = 2 \times (8 \text{ cm} + 6 \text{ cm}) = 2 \times 14 \text{ cm} = 28 \text{ cm} $

2. Circle Area The area of a circle is given by the formula $A = \pi r^2$, where $r$ is the radius. We are given four different radii and need to calculate the area for each. We will use $\pi = \frac{22}{7}$. (i) $r = 2.1 \text{ cm}$: $ A = \frac{22}{7} \times (2.1 \text{ cm})^2 = \frac{22}{7} \times 4.41 \text{ cm}^2 = 13.86 \text{ cm}^2 $ (ii) $r = 4.2 \text{ cm}$: $ A = \frac{22}{7} \times (4.2 \text{ cm})^2 = \frac{22}{7} \times 17.64 \text{ cm}^2 = 55.44 \text{ cm}^2 $ (iii) $r = 6.3 \text{ cm}$: $ A = \frac{22}{7} \times (6.3 \text{ cm})^2 = \frac{22}{7} \times 39.69 \text{ cm}^2 = 124.74 \text{ cm}^2 $ (iv) $r = 7.7 \text{ cm}$: $ A = \frac{22}{7} \times (7.7 \text{ cm})^2 = \frac{22}{7} \times 59.29 \text{ cm}^2 = 186.34 \text{ cm}^2 $

3. Ratio of Circle Areas (Radii Ratio Given) The ratio of the radii of two circles is 2:3. Let the radii be $r_1 = 2x$ and $r_2 = 3x$. The areas will be $A_1 = \pi r_1^2 = \pi (2x)^2 = 4\pi x^2$ and $A_2 = \pi r_2^2 = \pi (3x)^2 = 9\pi x^2$. The ratio of the areas is $A_1 : A_2 = 4\pi x^2 : 9\pi x^2 = 4:9$.

4. Ratio of Circle Areas (Circumference Ratio Given) The ratio of the circumferences of two circles is 3:4. Let the circumferences be $C_1 = 3y$ and $C_2 = 4y$. Since $C = 2\pi r$, we have $r = \frac{C}{2\pi}$. Thus, $r_1 = \frac{3y}{2\pi}$ and $r_2 = \frac{4y}{2\pi}$. The areas are $A_1 = \pi r_1^2 = \pi (\frac{3y}{2\pi})^2 = \frac{9y^2}{4\pi}$ and $A_2 = \pi r_2^2 = \pi (\frac{4y}{2\pi})^2 = \frac{16y^2}{4\pi}$. The ratio of the areas is $A_1 : A_2 = \frac{9y^2}{4\pi} : \frac{16y^2}{4\pi} = 9:16$.

5. Area Equals Circumference The area of a circle is equal to its circumstance. $A = C \implies \pi r^2 = 2\pi r$. Divide both sides by $\pi r$ (since $r \ne 0$) to get $r = 2$.

6. Area of Circular Path The park has a radius of 8 m and the path is 2 m wide. The outer radius is $8 + 2 = 10$ m. The area of the path is the difference between the areas of the outer and inner circles. $ \text{Area of path} = \pi (10^2) - \pi (8^2) = \pi (100 - 64) = 36\pi $. Using $\pi = 3.14$, $\text{Area of path} = 36 \times 3.14 = 113.04 \text{ m}^2$. The cost of paving is ₹ 18.50 per sq. m. $ \text{Total cost} = 113.04 \text{ m}^2 \times ₹ 18.50/\text{m}^2 = ₹ 2091.24 $

7. Square to Circle Area A wire bent in the form of a square encloses an area of 121 sq. cm. Let $s$ be the side length of the square. Then $s^2 = 121$, so $s = 11$ cm. The perimeter of the square is $4s = 4 \times 11 = 44$ cm. The same wire is bent into a circle, so the circumference of the circle is also 44 cm. $C = 2\pi r = 44$, so $r = \frac{44}{2\pi} = \frac{22}{\pi}$. Using $\pi = \frac{22}{7}$, $r = \frac{22}{\frac{22}{7}} = 7$ cm. The area of the circle is $A = \pi r^2 = \frac{22}{7} \times 7^2 = \frac{22}{7} \times 49 = 22 \times 7 = 154 \text{ cm}^2$.

8. Square vs Circle Perimeter A square and a circle both have a perimeter of 176 cm. For the square, $4s = 176 \implies s = \frac{176}{4} = 44 \text{ cm}$. The area of the square is $s^2 = 44^2 = 1936 \text{ cm}^2$. For the circle, $2\pi r = 176 \implies r = \frac{176}{2\pi} = \frac{88}{\pi}$. Using $\pi = \frac{22}{7}$, $r = \frac{88}{\frac{22}{7}} = \frac{88 \times 7}{22} = 4 \times 7 = 28 \text{ cm}$. The area of the circle is $\pi r^2 = \frac{22}{7} \times 28^2 = \frac{22}{7} \times 784 = 22 \times 112 = 2464 \text{ cm}^2$. The circle has more area. The difference is $2464 - 1936 = 528 \text{ cm}^2$.