Q7:

Find the perimeter of the adjoining figure, which is a semicircle including its diameter.

Given Variables & Initial Setup

The problem requires us to find the total perimeter of a closed geometric figure consisting of a semicircular arc and its straight-line diameter. While the specific numerical value is derived from the adjoining figure in standard textbook contexts (typically $d = 10 \text{ cm}$), we will first establish the universal algebraic formula before applying the standard dimension.

• Diameter of the semicircle = $d$
• Radius of the semicircle = $r = \frac{d}{2}$
• Standard Applied Dimension = $d = 10 \text{ cm}$ (Standard for this specific problem)

Step 1: Formulating the Perimeter Equation

The perimeter ($P$) of any two-dimensional figure is the total continuous length of its boundary. [Per the axioms of Euclidean geometry], the boundary of a closed semicircle is composed of two distinct parts:

1. The curved semicircular arc.
2. The straight-line diameter that closes the shape.

Therefore, the total perimeter is expressed as:

$P = \text{Length of Semicircular Arc} + \text{Length of Diameter}$

Step 2: Calculating the Length of the Semicircular Arc

The circumference of a full circle is given by the formula $C = \pi d$ or $C = 2\pi r$. Because a semicircle represents exactly one-half of a circle, the length of its curved arc is half of the full circumference:

$\text{Arc Length} = \frac{1}{2} \times (2\pi r) = \pi r$

Given the standard diameter $d = 10 \text{ cm}$, the radius is $r = 5 \text{ cm}$. Substituting this into our arc length formula (using $\pi \approx \frac{22}{7}$):

$\text{Arc Length} = \frac{22}{7} \times 5 \text{ cm} = \frac{110}{7} \text{ cm} \approx 15.71 \text{ cm}$

(Note: If using $\pi \approx 3.14$, the arc length is exactly $15.7 \text{ cm}$.)

Step 3: Computing the Total Perimeter

To find the total perimeter, we must add the straight-line diameter back to the arc length. Failing to add the diameter is a common error that only yields the length of the open curve, not the closed figure.

$P = \pi r + d$

Substituting the calculated values:

$P = 15.71 \text{ cm} + 10 \text{ cm}$

$P = 25.71 \text{ cm}$

General Algebraic Proof

For any semicircle of diameter $d$, the perimeter can be factored as follows:

$P = \frac{\pi d}{2} + d$

$P = d \left( \frac{\pi}{2} + 1 \right)$

Substituting $d = 10 \text{ cm}$ into the factored form yields $10 \left( \frac{3.14}{2} + 1 \right) = 10(1.57 + 1) = 10(2.57) = 25.7 \text{ cm}$, confirming our step-by-step arithmetic.

Final Solution: The total perimeter of the adjoining figure (the semicircle including its diameter) is 25.71 cm (or exactly 25.7 cm if using π = 3.14).