Step 1: Given Variables & Initial Setup

To determine the area of the remaining aluminium sheet, we must first establish the dimensions of the primary geometric figures involved in the system. Let $s$ represent the side length of the square aluminium sheet, and $r$ represent the radius of the circular cutout.

• Side of the square sheet, $s = 6 \text{ cm}$
• Radius of the circular cutout, $r = 2 \text{ cm}$
• Constant approximation, $\pi = 3.14$

Step 2: Geometric Visualization

The following scale diagram illustrates the spatial relationship between the square sheet and the circular cutout. The dimensions are mapped to a precise coordinate system where $1 \text{ cm} = 40 \text{ units}$.

Step 3: Calculating the Area of the Square Sheet

The total area of the initial aluminium sheet is calculated using the standard area formula for a regular quadrilateral [Per Euclidean Geometry principles].

$A_{\text{square}} = s^2$
$A_{\text{square}} = (6 \text{ cm})^2$
$A_{\text{square}} = 36 \text{ cm}^2$

Step 4: Calculating the Area of the Circular Cutout

The area of the removed material is defined by the area of the circle. We apply the area formula for a circle, substituting the given approximation for $\pi$.

$A_{\text{circle}} = \pi r^2$
$A_{\text{circle}} = 3.14 \times (2 \text{ cm})^2$
$A_{\text{circle}} = 3.14 \times 4 \text{ cm}^2$
$A_{\text{circle}} = 12.56 \text{ cm}^2$

Step 5: Determining the Area of the Leftover Sheet

To find the area of the remaining aluminium, we subtract the area of the circular cutout from the total area of the square sheet [By the Additive Property of Area, which states that the area of a whole is equal to the sum of the areas of its non-overlapping parts].

$A_{\text{leftover}} = A_{\text{square}} - A_{\text{circle}}$
$A_{\text{leftover}} = 36 \text{ cm}^2 - 12.56 \text{ cm}^2$
$A_{\text{leftover}} = 23.44 \text{ cm}^2$

Summary of Area Calculations

Final Solution: The area of the left over aluminium sheet is $23.44 \text{ cm}^2$.

Initial Setup & Visual Data Extraction

Based on the standard geometric parameters provided in the visual figure for this specific problem, we extract the following fundamental dimensions of the triangle. The altitude is dropped perpendicularly to the chosen base.

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

Step 1: Establishing the Theoretical Foundation

To find the area of any triangle when the base and its corresponding perpendicular height (altitude) are known, we utilize the standard Euclidean area postulate for triangles. The area of a triangle is exactly half the area of a parallelogram that shares the same base and height.

The governing formula is:

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

Symbolically represented as:

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

Step 2: Dimensional Substitution

Substitute the given scalar values into the area formula. It is critical to include the units ($\text{cm}$) during substitution to ensure dimensional homogeneity [Area must result in square units].

$A = \frac{1}{2} \times (4 \text{ cm}) \times (3 \text{ cm})$

Step 3: Algebraic Execution & Simplification

First, multiply the scalar magnitudes of the base and the height, and apply the product rule to the units ($\text{cm} \times \text{cm} = \text{cm}^2$):

$A = \frac{1}{2} \times (12 \text{ cm}^2)$

Next, multiply by the scalar fraction $\frac{1}{2}$ (which is equivalent to dividing by 2):

$A = \frac{12}{2} \text{ cm}^2$

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

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

Given Variables & Initial Setup

We are given a right-angled triangle, $\triangle ABC$, with the right angle located at vertex $A$. The dimensions of the sides are provided as follows:

• Length of leg $AB = 5 \text{ cm}$
• Length of leg $AC = 12 \text{ cm}$
• Length of hypotenuse $BC = 13 \text{ cm}$
• $AD$ is the altitude dropped from the right angle $A$ to the hypotenuse $BC$, meaning $AD \perp BC$.

Step 1: Calculating the Area of $\triangle ABC$

The area of any triangle is given by the standard formula:

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

[Because $\triangle ABC$ is right-angled at $A$, the two legs forming the right angle ($AB$ and $AC$) are mutually perpendicular. Therefore, one leg can act as the base while the other acts as the corresponding height.]

Let the base be $AC = 12 \text{ cm}$ and the height be $AB = 5 \text{ cm}$. Substituting these values into the area formula yields:

$ \text{Area of } \triangle ABC = \frac{1}{2} \times AC \times AB $

$ \text{Area of } \triangle ABC = \frac{1}{2} \times 12 \text{ cm} \times 5 \text{ cm} $

$ \text{Area of } \triangle ABC = 6 \times 5 = 30 \text{ cm}^2 $

Step 2: Formulating the Equation for Altitude $AD$

[The area of a geometric figure is invariant; it remains constant regardless of which side is chosen as the base for the calculation.]

We can recalculate the area of $\triangle ABC$ by choosing the hypotenuse $BC$ as the new base. The corresponding height for this base is the perpendicular altitude $AD$.

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

$ \text{Area of } \triangle ABC = \frac{1}{2} \times BC \times AD $

Step 3: Solving for the Length of $AD$

We already established from Step 1 that the area of the triangle is $30 \text{ cm}^2$. We also know the length of the hypotenuse $BC = 13 \text{ cm}$. Substituting these knowns into our new area equation:

$ 30 = \frac{1}{2} \times 13 \times AD $

To isolate $AD$, multiply both sides of the equation by 2:

$ 60 = 13 \times AD $

Divide both sides by 13:

$ AD = \frac{60}{13} \text{ cm} $

Converting this fraction to a decimal provides the approximate length:

$ AD \approx 4.615 \text{ cm} $

Final Solution: The area of $\triangle ABC$ is $30 \text{ cm}^2$, and the length of the altitude $AD$ is $\frac{60}{13} \text{ cm}$ (approximately $4.62 \text{ cm}$).

Given Variables & Initial Setup

We are tasked with determining the area of a circle given a specific one-dimensional metric. The provided parameter is:

• Radius of the circle, $r = 5 \text{ cm}$

Step 1: Theoretical Foundation & Coordinate Geometry Context

In the context of coordinate geometry, a circle centered at the origin $(0,0)$ with a radius $r$ is defined by the locus of points $(x, y)$ that satisfy the Cartesian equation:

$x^2 + y^2 = r^2$

Substituting our given radius, the equation of this specific circle is:

$x^2 + y^2 = 25$

[Per the principles of Euclidean planar geometry and integral calculus], the area $A$ enclosed by this locus of points can be derived using double integration in polar coordinates ($A = \int_{0}^{2\pi} \int_{0}^{r} \rho \, d\rho \, d\theta$), which simplifies to the standard geometric formula:

$A = \pi r^2$

Step 2: Geometric Visualization

Below is a precise coordinate plane representation of the circle $x^2 + y^2 = 25$. The radius vector is drawn from the origin to a point on the circumference, demonstrating the constant distance $r = 5 \text{ cm}$.

Step 3: Algebraic Substitution and Calculation

We substitute the given radius $r = 5 \text{ cm}$ into the area formula. [By the exponentiation axiom, we must square the radius before multiplying by the constant $\pi$].

$A = \pi \cdot (5 \text{ cm})^2$

$A = \pi \cdot (25 \text{ cm}^2)$

$A = 25\pi \text{ cm}^2$

This is the exact area expressed in terms of the transcendental number $\pi$.

Step 4: Decimal Approximation

For practical applications, it is often necessary to compute the numerical value of the area. We evaluate this using standard approximations for $\pi$:

Final Solution: The exact area of the circle is $25\pi \text{ cm}^2$, which is approximately $78.5 \text{ cm}^2$ (or $78.57 \text{ cm}^2$ depending on the chosen approximation for $\pi$).

Given Variables & Initial Setup

Based on the principles of Coordinate Geometry, when a geometric figure is presented on a Cartesian plane, its dimensions must be extracted by identifying the coordinates of its vertices. For this analysis, we extract the coordinates of the triangle's vertices from the provided two-dimensional grid.

• Vertex $A$ (Apex): $(0, 5)$
• Vertex $B$ (Left base endpoint): $(-4, 0)$
• Vertex $C$ (Right base endpoint): $(3, 0)$

[Theoretical Justification: In a Cartesian system, the distance between two points lying on the same horizontal or vertical axis can be determined using the absolute difference of their respective non-zero coordinates, per the 1D Distance Formula $d = |a_2 - a_1|$].

Step 1: Determining the Length of the Base ($b$)

The base of the triangle, segment $BC$, lies entirely on the x-axis. Because both points share the same y-coordinate ($y = 0$), the length of the base is the absolute difference between their x-coordinates.

Let $x_B = -4$ and $x_C = 3$.

$ \text{Base } (b) = |x_C - x_B| $

$ b = |3 - (-4)| $

$ b = |3 + 4| = 7 \text{ units} $

Step 2: Determining the Altitude / Height ($h$)

The height of a triangle is the perpendicular distance from the apex to the line containing the base. Since the base lies on the x-axis, the perpendicular distance from vertex $A(0, 5)$ to the x-axis is simply the absolute value of its y-coordinate.

Let $y_A = 5$ and the y-coordinate of the base $y_{base} = 0$.

$ \text{Height } (h) = |y_A - y_{base}| $

$ h = |5 - 0| = 5 \text{ units} $

Step 3: Calculating the Area of the Triangle

According to Euclidean geometry, the area ($A$) of a triangle is given by half the product of its base and its corresponding altitude.

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

Substituting the derived values:

$ A = \frac{1}{2} \times 7 \times 5 $

$ A = \frac{1}{2} \times 35 $

$ A = 17.5 \text{ square units} $

Step 4: Rigorous Verification via Coordinate Geometry Formula

To ensure absolute precision, we verify the result using the determinant-based area formula for a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$:

$ A = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| $

Assigning the coordinates: $(x_1, y_1) = (0, 5)$, $(x_2, y_2) = (-4, 0)$, $(x_3, y_3) = (3, 0)$.

$ A = \frac{1}{2} |0(0 - 0) + (-4)(0 - 5) + 3(5 - 0)| $

$ A = \frac{1}{2} |0 + (-4)(-5) + 3(5)| $

$ A = \frac{1}{2} |0 + 20 + 15| $

$ A = \frac{1}{2} |35| = 17.5 \text{ square units} $

[The identical result confirms the geometric extraction is mathematically sound and verified across multiple theorems].

Final Solution: The area of the given triangle is 17.5 square units.

Given Variables & Initial Setup

To determine the distance covered by the tip of the minute hand, we must first establish the geometric parameters of the system based on the provided values:

• Length of the minute hand (Radius, $r$) = $15$ cm
• Time elapsed ($t$) = $1$ hour
• Value of Pi ($\pi$) = $3.14$

Step 1: Analyzing the Kinematics of the Minute Hand

The minute hand of a clock is anchored at the center of the clock face. As time progresses, the tip of the minute hand traces a perfect circle. [Per the definition of a circle: the locus of all points in a plane that are at a given distance from a given point, the center].

In exactly $1$ hour ($60$ minutes), the minute hand completes one full revolution around the clock face. Therefore, the angular displacement is $360^\circ$ or $2\pi$ radians. The total linear distance moved by the tip of the minute hand is exactly equal to the circumference of the circle it traces.

Step 2: Formulating the Geometric Equation

The circumference ($C$) of a circle is given by the standard geometric formula:

$C = 2\pi r$

Where:

• $C$ represents the total distance moved by the tip in one revolution.
• $\pi$ is the mathematical constant (given as $3.14$).
• $r$ is the radius of the circular path, which corresponds to the length of the minute hand ($15$ cm).

Step 3: Visualizing the Circular Path

The following diagram illustrates the circular path traced by the tip of the minute hand over the course of $1$ hour.

Step 4: Execution and Algebraic Calculation

Substitute the given values into the circumference formula:

$C = 2 \times 3.14 \times 15$

To simplify the calculation, use the commutative property of multiplication to group the integers first:

$C = (2 \times 15) \times 3.14$

$C = 30 \times 3.14$

Perform the final multiplication:

$C = 94.2$ cm

Final Solution: The tip of the minute hand moves a total distance of $94.2$ cm in $1$ hour.

Given Variables & Initial Setup

We are tasked with determining the area of a two-dimensional circle given its diameter. The primary parameter provided is:

• Diameter ($d$) = $49 \text{ m}$

Step 1: Derivation of the Radius

The radius ($r$) of a circle is defined as the linear distance from the center point to any point on its circumference. [Per Euclidean geometry, the diameter is the longest chord of the circle passing directly through the center, making it exactly twice the length of the radius]. Therefore, we establish the fundamental relationship:

$r = \frac{d}{2}$

Substituting the given diameter into the equation:

$r = \frac{49}{2} \text{ m} = 24.5 \text{ m}$

Geometric Visualization

Below is a scaled geometric representation of the circle, illustrating the relationship between the diameter and the radius.

Step 2: Application of the Area Formula

The area ($A$) of a circle is calculated using the standard formula:

$A = \pi r^2$

[Derived historically via the method of exhaustion by Archimedes, where the area of a circle is proven to be the limit of the areas of inscribed regular polygons as the number of sides approaches infinity].

For this computation, we will utilize the rational approximation $\pi \approx \frac{22}{7}$. This specific approximation is strategically chosen because the radius ($\frac{49}{2}$) contains a numerator that is a multiple of $7$, which will elegantly simplify the subsequent arithmetic.

Step 3: Algebraic Computation

Substitute $r = \frac{49}{2} \text{ m}$ and $\pi = \frac{22}{7}$ into the area formula:

$A = \frac{22}{7} \times \left(\frac{49}{2}\right)^2$

Expand the squared term to prepare for cross-cancellation:

$A = \frac{22}{7} \times \frac{49}{2} \times \frac{49}{2}$

Perform cross-cancellation to simplify the fractional expression. Divide $22$ by $2$ to yield $11$, and divide $49$ by $7$ to yield $7$:

$A = 11 \times 7 \times \frac{49}{2}$

Multiply the integer terms:

$A = 77 \times \frac{49}{2}$

Convert the remaining fraction to a decimal for final multiplication ($ \frac{49}{2} = 24.5 $):

$A = 77 \times 24.5$

Execute the final multiplication:

$A = 1886.5 \text{ m}^2$

Final Solution: The area of the circle is $1886.5 \text{ m}^2$.

Step 1: Given Variables & Initial Setup

Based on the standard geometric parameters provided in the referenced figure for this problem, we extract the primary dimensions of the parallelogram. Let the parallelogram be denoted as $ABCD$.

• Base ($b$): The length of the bottom edge of the parallelogram is given as $5\text{ cm}$.
• Height ($h$): The perpendicular distance (altitude) from the opposite parallel side to the base is given as $3\text{ cm}$.

Step 2: Theoretical Foundation & Formula

The area of a parallelogram is defined as the total two-dimensional space enclosed within its four sides. It is calculated by taking the product of its base and its corresponding perpendicular height.

Formula:

$\text{Area of a Parallelogram} = \text{Base} \times \text{Height}$

[Justification: By the principle of geometric decomposition, a right-angled triangle can be conceptually "cut" from one end of the parallelogram and translated to the opposite end. This transformation converts the parallelogram into a rectangle of identical base and height, proving that they share the same area formula.]

Step 3: Geometric Visualization

Below is a mathematically scaled representation of the parallelogram $ABCD$, where the base $AB = 5\text{ cm}$ and the altitude $DE = 3\text{ cm}$. The drawing maintains a strict $40:1$ coordinate ratio to ensure absolute visual accuracy.

Step 4: Execution of Area Calculation

We substitute the given scalar values into the area formula. Ensure that both measurements are in the same units (centimeters) before multiplying to yield an area in square centimeters ($\text{cm}^2$).

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

$\text{Area} = 5\text{ cm} \times 3\text{ cm}$

$\text{Area} = (5 \times 3) \text{ cm}^{1+1}$

$\text{Area} = 15\text{ cm}^2$

Final Solution: The area of the given parallelogram is $15\text{ cm}^2$.

Step 1: Given Variables & Initial Setup

Based on the standard geometric parameters provided in the referenced figure for this specific problem, we extract the following dimensions for the parallelogram:

• Base ($b$): $2\text{ cm}$
• Corresponding Height/Altitude ($h$): $4.4\text{ cm}$

[Note: In a parallelogram, any of the four sides can be chosen as the base. The corresponding height is the perpendicular distance from the chosen base to the opposite parallel side.]

Step 2: Geometric Visualization

Below is a mathematically scaled, high-precision vector representation of the parallelogram, illustrating the relationship between the base and its corresponding altitude.

Step 3: Mathematical Formulation

By the fundamental axioms of Euclidean geometry, a parallelogram can be transformed into a rectangle of equal area by translating the right-angled triangle formed by the altitude to the opposite side. Therefore, the area ($A$) of a parallelogram is strictly defined as the product of its base and its corresponding height.

The governing formula is:

$\text{Area} = \text{Base} \times \text{Height}$

$A = b \times h$

Step 4: Execution & Algebraic Calculation

Substitute the given scalar values into the area formula:

$A = 2\text{ cm} \times 4.4\text{ cm}$

To ensure absolute precision, we can perform the multiplication by converting the decimal to a fraction [Per standard arithmetic operations]:

$A = 2 \times \left(\frac{44}{10}\right)$

$A = \frac{88}{10}$

$A = 8.8$

Step 5: Dimensional Analysis

We must verify the units of the final result. Multiplying a one-dimensional length by another one-dimensional length yields a two-dimensional area:

$[\text{cm}] \times [\text{cm}] = [\text{cm}^2]$

The magnitude is $8.8$ and the unit is $\text{cm}^2$. The calculation is dimensionally consistent and mathematically sound.

Final Solution: The area of the parallelogram is $8.8\text{ cm}^2$.

Step 1: Given Variables & Initial Setup

To determine the number of rotations a wheel must make to cover a specific distance, we must first establish the geometric properties of the wheel and the total distance to be traversed. The fundamental principle governing this motion is that the linear distance covered by a wheel in exactly one full rotation is equal to its perimeter, known as the circumference.

• Radius of the wheel, $r = 28 \text{ cm}$
• Total distance to be covered, $D = 352 \text{ m}$
• Constant, $\pi = \frac{22}{7}$

Step 2: Dimensional Homogeneity (Unit Conversion)

To perform valid algebraic operations, all physical quantities must be expressed in the same units. The radius is given in centimeters ($\text{cm}$), while the total distance is given in meters ($\text{m}$). We will convert the total distance into centimeters.

[Per the standard metric conversion: $1 \text{ m} = 100 \text{ cm}$]

$D = 352 \text{ m}$
$D = 352 \times 100 \text{ cm}$
$D = 352,000 \text{ cm}$

Step 3: Calculating the Circumference of the Wheel

The distance covered by the wheel in one complete revolution is exactly equal to its circumference ($C$).

[By the geometric definition of a circle's perimeter: $C = 2\pi r$]

Substituting the given values into the formula:

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

We simplify the expression by dividing $28$ by $7$:

$C = 2 \times 22 \times 4$
$C = 44 \times 4$
$C = 176 \text{ cm}$

Thus, the wheel covers a linear distance of $176 \text{ cm}$ in exactly one rotation.

Step 4: Determining the Number of Rotations

Let $n$ represent the total number of rotations required to cover the distance $D$. The relationship between total distance, circumference, and the number of rotations is given by the linear equation:

$D = n \times C$

Isolating $n$, we obtain:

$n = \frac{D}{C}$

Substituting the calculated values into the equation:

$n = \frac{352,000 \text{ cm}}{176 \text{ cm}}$

To simplify the division, observe the relationship between the significant digits: $176 \times 2 = 352$. Therefore:

$n = 2000$

(Self-Correction/Refinement: Wait, $352 \times 100 = 35,200$, not $352,000$. Let us rigorously re-evaluate the arithmetic in Step 2.)

Correction in arithmetic:
$D = 352 \text{ m} \times 100 \text{ cm/m} = 35,200 \text{ cm}$.

Recalculating $n$ with the corrected magnitude:

$n = \frac{35,200}{176}$
$n = 200$

Final Solution: The wheel must rotate exactly 200 times to cover a distance of 352 meters.

Given Variables & Initial Setup

We are given the following geometric parameters for a circle:

• Radius ($r$) = $14 \text{ mm}$
• Mathematical constant ($\pi$) = $\frac{22}{7}$

Theoretical Foundation & Formula

The area ($A$) of a circle represents the total two-dimensional space enclosed within its boundary (circumference). [Per the geometric principles of Euclidean space], the area of a circle is directly proportional to the square of its radius. The formula is given by:

$A = \pi r^2$

Step 1: Substituting the Values into the Equation

To find the area, we substitute the given radius $r = 14 \text{ mm}$ and the fractional approximation of $\pi = \frac{22}{7}$ into the standard area formula:

$A = \left(\frac{22}{7}\right) \times (14 \text{ mm})^2$

Step 2: Algebraic Manipulation and Simplification

First, expand the squared term to separate the numerical values from the units:

$A = \frac{22}{7} \times (14 \times 14) \text{ mm}^2$

$A = \frac{22}{7} \times 196 \text{ mm}^2$

To optimize the calculation without dealing with large numbers, we can simplify the expression by dividing one of the $14$ factors by the denominator $7$ [By the fundamental property of fractions and associative multiplication]:

$A = 22 \times \left(\frac{14}{7}\right) \times 14 \text{ mm}^2$

$A = 22 \times 2 \times 14 \text{ mm}^2$

Now, proceed with sequential multiplication:

$A = (22 \times 2) \times 14 \text{ mm}^2$

$A = 44 \times 14 \text{ mm}^2$

Perform the final multiplication step:

$A = 44 \times (10 + 4) \text{ mm}^2$

$A = 440 + 176 \text{ mm}^2$

$A = 616 \text{ mm}^2$

Step 3: Dimensional Analysis

The radius is given in millimeters ($\text{mm}$). When the radius is squared in the formula ($r^2$), the unit is also squared ($\text{mm} \times \text{mm} = \text{mm}^2$). This confirms that our final unit correctly represents a two-dimensional area.

Final Solution: The area of the circle is $616 \text{ mm}^2$.