Given Variables & Initial Setup

We are tasked with determining the circumference of a circle given its radius. The fundamental parameters provided for this geometric system are:

• Radius of the circle, $r = 28 \text{ mm}$
• Approximation for Pi, $\pi = \frac{22}{7}$

Theoretical Foundation: Circumference of a Circle

The circumference ($C$) of a circle is defined as the continuous line forming the boundary of the closed geometric figure. [Per the geometric definition of a circle's perimeter in Euclidean space], the circumference is directly proportional to its radius, with the constant of proportionality being $2\pi$. The governing formula is:

$C = 2\pi r$

Step 1: Substitution of Values

We substitute the given values of $r$ and $\pi$ into the standard circumference formula:

$C = 2 \times \left(\frac{22}{7}\right) \times 28$

Step 2: Algebraic Simplification

To optimize the calculation, we first analyze the terms for common factors. The radius ($28$) is a multiple of the denominator ($7$) in our fractional value of $\pi$. [By the fundamental properties of rational numbers], we can divide $28$ by $7$ to simplify the expression before proceeding with multiplication:

$\frac{28}{7} = 4$

Substituting this simplified integer back into the equation yields:

$C = 2 \times 22 \times 4$

Step 3: Final Calculation

We now perform the sequential multiplication of the remaining scalar quantities:

First, multiply the constants:

$2 \times 22 = 44$

Next, multiply the result by the simplified radius factor:

$C = 44 \times 4$

$C = 176$

Since the radius was provided in millimeters ($\text{mm}$), the circumference, being a one-dimensional measure of length, must carry the identical unit.

Final Solution: The circumference of the circle is $176 \text{ mm}$.

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).

Initial Setup & Given Variables

Based on the standard geometric configuration for this specific problem, we extract the primary dimensions of the given parallelogram from the provided visual data:

• Base ($b$): $2.5 \text{ cm}$
• Corresponding Altitude/Height ($h$): $3.5 \text{ cm}$

Step 1: Theoretical Foundation

The area of a parallelogram is determined by the product of its base and its corresponding altitude. [Per the geometric principle of area equivalence, a parallelogram can be orthogonally decomposed and rearranged into a rectangle of identical base and height without any loss of area].

The governing mathematical formula is:

$\text{Area} = b \times h$

Step 2: Substitution of Dimensional Values

We substitute the given scalar quantities into the area formula. It is critical to ensure that both measurements share the same unit ($\text{cm}$) before proceeding with the multiplication.

$\text{Area} = 2.5 \text{ cm} \times 3.5 \text{ cm}$

Step 3: Arithmetic Computation & Analytical Breakdown

To ensure absolute precision and avoid floating-point errors, we convert the decimal values into their fractional equivalents prior to multiplication:

$2.5 = \frac{25}{10}$

$3.5 = \frac{35}{10}$

Multiplying the fractions algebraically:

$\text{Area} = \left( \frac{25}{10} \right) \times \left( \frac{35}{10} \right)$

$\text{Area} = \frac{25 \times 35}{10 \times 10}$

Calculating the numerator ($25 \times 35$):

$25 \times 30 = 750$

$25 \times 5 = 125$

$750 + 125 = 875$

Substituting the numerator back into the fraction:

$\text{Area} = \frac{875}{100}$

Converting the fraction back to a standard decimal format by shifting the decimal point two places to the left:

$\text{Area} = 8.75 \text{ cm}^2$

Final Solution: The area of the parallelogram is $8.75 \text{ cm}^2$.

Given Variables & Initial Setup

Based on the standard geometric parameters provided in the visual data for this specific problem, we extract the following dimensions for the triangle:

• Base ($b$): $3\text{ cm}$
• Height ($h$): $2\text{ cm}$

Below is the precise geometric reconstruction of the given figure. Note that for an obtuse-angled triangle, the altitude (height) dropped from the top vertex intersects the extended base outside the boundary of the triangle.

Step 1: Geometric Analysis of the Figure

The figure represents an obtuse-angled triangle $\triangle ABC$. [By definition, an obtuse triangle contains one interior angle strictly greater than $90^\circ$]. When calculating the area of an obtuse triangle using a base that forms one of the sides of the obtuse angle, the corresponding altitude (perpendicular height) must be drawn from the opposite vertex to the line containing the base. This altitude falls outside the triangle, meeting the extended base at a right angle (point $D$).

Step 2: Formulating the Area Equation

The area ($A$) of any triangle in Euclidean geometry is determined by the product of its base and its corresponding altitude, halved. [Per Euclidean geometry principles, the area of a triangle is exactly half the area of a parallelogram constructed on the same base and between the same parallels].

The governing formula is:

$A = \frac{1}{2} \times b \times h$

Where:

• $b$ is the length of the base segment ($BC$).
• $h$ is the length of the perpendicular altitude ($AD$).

Step 3: Substitution and Algebraic Calculation

We substitute the identified scalar values into the area formula. It is critical to include units during the calculation to ensure dimensional consistency.

$A = \frac{1}{2} \times (3\text{ cm}) \times (2\text{ cm})$

First, multiply the scalar magnitudes and the units:

$A = \frac{1}{2} \times 6\text{ cm}^2$

Next, apply the scalar multiplication by $\frac{1}{2}$:

$A = 3\text{ cm}^2$

Final Solution: The area of the given triangle is $3\text{ cm}^2$.

Given Variables & Initial Setup

Let us define the fundamental parameters of the circle based on the provided data:

• Circumference ($C$) = $31.4 \text{ cm}$
• Mathematical constant ($\pi$) $\approx 3.14$
• Radius ($r$) = ?
• Area ($A$) = ?

Step 1: Deriving the Radius from the Circumference

[Per the geometric definition of a circle's circumference], the total arc length (perimeter) of a circle is directly proportional to its radius. The governing equation is:

$C = 2\pi r$

By substituting the given values into the equation, we can isolate the unknown variable, $r$:

$31.4 = 2 \times 3.14 \times r$

$31.4 = 6.28 \times r$

Dividing both sides by $6.28$ to solve for $r$:

$r = \frac{31.4}{6.28}$

$r = 5 \text{ cm}$

Step 2: Geometric Visualization

The following diagram illustrates the spatial relationship between the center, the calculated radius, and the given circumference of the circle.

Step 3: Calculating the Area of the Circle

[Per the fundamental theorem of circle geometry], the area $A$ enclosed by a circle is proportional to the square of its radius. The formula is expressed as:

$A = \pi r^2$

Substitute the derived radius ($r = 5 \text{ cm}$) and the given value of $\pi$ ($3.14$) into the area formula:

$A = 3.14 \times (5)^2$

First, evaluate the exponent:

$A = 3.14 \times 25$

Next, perform the multiplication to find the final area:

$A = 78.5 \text{ cm}^2$

Final Solution: The radius of the circle is $5 \text{ cm}$ and the area of the circle is $78.5 \text{ cm}^2$.

Initial Setup & Given Variables

Based on the standard geometrical parameters provided in the corresponding exercise figure, we are analyzing a right-angled triangle with the following dimensions:

• Base ($b$): $3\text{ cm}$
• Height ($h$): $4\text{ cm}$

[Because the triangle features a $90^\circ$ angle between these two segments, the side measuring $4\text{ cm}$ acts as the exact perpendicular altitude to the $3\text{ cm}$ base.]

Step 1: Theoretical Foundation

To determine the two-dimensional space enclosed by the triangle, we apply the standard area theorem for triangles [derived from the area of a rectangle, where a diagonal bisects the rectangle into two congruent right triangles].

The formula is given by:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Step 2: Geometrical Representation

Below is the precise, scaled geometrical construction of the given triangle. The base is scaled to 150 units (representing $3\text{ cm}$) and the height is scaled to 200 units (representing $4\text{ cm}$) to maintain strict proportional accuracy.

Step 3: Algebraic Substitution and Calculation

We substitute the given scalar values into the area formula. [By the properties of dimensional analysis, multiplying two lengths in centimeters ($\text{cm}$) will yield an area in square centimeters ($\text{cm}^2$)].

$\text{Area} = \frac{1}{2} \times 3\text{ cm} \times 4\text{ cm}$

First, compute the product of the base and the height:

$3 \times 4 = 12\text{ cm}^2$

Next, apply the $\frac{1}{2}$ multiplier [as the triangle represents exactly half the area of the bounding $3\text{ cm} \times 4\text{ cm}$ rectangle]:

$\text{Area} = \frac{1}{2} \times 12\text{ cm}^2$

$\text{Area} = 6\text{ cm}^2$

Final Solution: The area of the given triangle is $6\text{ cm}^2$.