Figure 1
This question aims to find the area of the given Figure 1 with two semi-circles and a parallelogram attached together.
The question is based on the geometry of 2D shapes that are circles and a parallelogram. The area of the parallelogram can be calculated by taking the product of its height and base sides. The equation is given as:
\[ P = b \times h \]
The area of the circle can be calculated as $\pi$ times the square of the circle’s radius. The equation is given as:
\[ C = \pi \times r^2 \]
Expert Answer
The total area of Figure 1 can be calculated by adding areas of the different shapes in the figure. The area of the first semi-circle added to the area of the parallelogram, and their result added with the area of the second semi-circle will give us the total area of the Figure. The equation is given as:
\[ Area\ A = Area\ of\ Semi-Circle (C_1)\ + Area\ of\ Parallelogram (P)\ + Area\ of\ Semi-Circle (C_2) \]
\[ A = C_1 + P + C_2 \]
The values given in Figure 1 are as follows:
\[ Base\ of\ Parallelogram\ b = 40 cm \]
\[ Height\ of\ Parallelogram\ h = 18 cm \]
\[ Radius\ of\ Circles\ r_1 = r_2 = 9 cm \]
First of all, let us find the area of the first semi-circle. The equation for the area of the circle is given as:
\[ C = \pi \times r^2 \]
The area of the semi-circle can be calculated by dividing 2 from the area of the circle as the semi-circle is exactly half of the circle. The equation is given as:
\[ C_1 = \dfrac { \pi }{ 2 } \times r_1^2 \]
Substituting the values, we get:
\[ C_1 = \dfrac { \pi }{ 2 } \times (0.09)^2 \]
Solving the equation, we get:
\[ C_1 = 1.27 cm^2 \]
As both the semi-circles are identical, their areas will be the same. Hence, the area of the second semi-circle is given as:
\[ C_2 = 1.27 cm^2 \]
The area of the parallelogram is given as:
\[ P = b \times h \]
Substituting the values, we get:
\[ P = 40 \times 18 \]
\[ P = 720 cm^2 \]
The total area of the figure is given as:
\[ A = C_1 + P + C_2 \]
Substituting the values, we get:
\[ A = 1.27 + 720 + 1.27 \]
\[ A = 722.54 cm^2 \]
Numerical Result
The area of the given Figure 1 is calculated to be:
\[ A = 722.54 cm^2 \]
Example
Find the area of the figure given below.
Figure 2
The radius of the semi-circle is given as 5 cm.
The figure given has two different shapes a semi-circle and a square. The side of the square is the diameter of the circle. Knowing the circle’s radius, we can find its diameter, which is the square’s side.
\[ d = 2r \]
\[ d = 2 \times 5 \]
\[ d = 10 cm \]
The diameter of the circle is 10 cm, which is also the side of the square.
\[ l = 10 cm \]
The area of the semi-circle is given as:
\[ C = \dfrac { \pi }{ 2 } \times (0.10)^2 \]
\[ C = 1.6 cm^2 \]
The area of the square is given as:
\[ S = 10^2 \]
\[ S = 100 cm^2 \]
The total area of the figure is given as:
\[ A = C + S \]
\[ A = 1.6 + 100 \]
\[ A = 101.6 cm^2 \]