G.1-G.4 Geometry and Scaling Notes

Circles: Definitions, Constants, and Fundamental Formulas

In the study of geometry, circles are defined by specific relationships between their radius, diameter, circumference, and area. The constant used to relate these dimensions is Pi ($\pi$), which is approximately $3.14$. According to the notes by Christian W. dated May 12, 2026, the following core variables and formulas are fundamental to circle geometry:

• Radius ($r$): Defined as half the length of the circle's diameter, extending from the center point to the edge.
• Diameter ($d$): The whole length across the circle through the center. To find the diameter from the radius, one must multiply the radius value by $2$; conversely, the radius is found by dividing the diameter by $2$.
• Circumference ($C$): The distance around the circle, calculated using the formula $C = 2 \times \pi \times r$.
• Area ($A$): The total square units contained within the circle, calculated using the formula $A = \pi \times r^2$.

Geometric Problem Solving: Comparative Area of Circles

A practical application of circle geometry involves comparing the sizes of circular objects, such as pizzas. Consider a scenario involving two individuals, Martin and Ricky, who order pizzas of different sizes. To find the approximate difference in area between the two, we apply the area formula $A = \pi \times r^2$ for each.

Martin ordered a pizza with a $16\text{-inch}$ diameter. For this pizza, the radius ($r$) is $8\,\text{inches}$. The calculation for the area is: $A = 3.14 \times 8^2$$A = 3.14 \times 64$$A = 200.96\,\text{square inches}$ Rounding to the nearest whole number, Martin's pizza is approximately $201\,\text{square inches}$.

Ricky ordered a pizza with a $20\text{-inch}$ diameter. For this pizza, the radius ($r$) is $10\,\text{inches}$. The calculation for the area is: $A = 3.14 \times 10^2$$A = 3.14 \times 100$$A = 314\,\text{square inches}$

To find the difference in area between the two pizzas: $314 - 200.96 = 113.04\,\text{square inches}$ Rounded to the nearest whole number, the approximate difference is $113\,\text{square inches}$.

Principles of Planar Geometry: The Triangle Inequality Theorem

The Triangle Inequality Theorem states that for any three segments to form a triangle, the sum of the lengths of any two sides must be strictly greater than the length of the third side ($a + b > c$). This rule determines which sets of measurements can successfully create a closed geometric shape. Evaluating the following side measurements:

• Option A ($8\,\text{cm}$, $9\,\text{cm}$, $13\,\text{cm}$): Applying the rule, $8 + 9 = 17$. Since $17 > 13$, these measurements will form a triangle.
• Option B ($8\,\text{cm}$, $9\,\text{cm}$, $17\,\text{cm}$): Applying the rule, $8 + 9 = 17$. Since $17$ is not greater than $17$, these segments cannot form a triangle.
• Option C ($7\,\text{cm}$, $7\,\text{cm}$, $14\,\text{cm}$): Applying the rule, $7 + 7 = 14$. Since $14$ is not greater than $14$, these segments cannot form a triangle.
• Option D ($6\,\text{cm}$, $8\,\text{cm}$, $15\,\text{cm}$): Applying the rule, $6 + 8 = 14$. Since $14$ is less than $15$, these segments cannot form a triangle.

Scale Drawings and Spatial Proportionality in Architecture

Scale drawings provide a proportional representation of a physical space. In a store diagram where the scale is defined as $1\,\text{inch} = 12\,\text{feet}$, the actual dimensions and area of the store can be calculated by converting the drawing measurements. The drawing shows a structure that can be broken into sections.

One section measures $4\,\text{inches}$ by $3\,\text{inches}$. Converting these to actual feet:

• Width: $4\,\text{inches} \times 12\,\text{ft/in} = 48\,\text{feet}$
• Height: $3\,\text{inches} \times 12\,\text{ft/in} = 36\,\text{feet}$
• Area: $48 \times 36 = 1728\,\text{square feet}$

A second section corresponds to a width of $4\,\text{inches}$ and an additional height segment of approximately $1.33\,\text{inches}$ (or $16\,\text{feet}$). Based on the calculations provided in the notes, the total area of the actual store, when rounded to the nearest $50\,\text{square feet}$, is approximately $2,450\,\text{square feet}$.

Linear Scaling and Geographic Measurement

Scale factors are essential for map creation and interpretation. Marie and Jeanine used different scales for their neighborhood and regional maps:

• Marie's Neighborhood Map: Marie used a scale where $1.5\,\text{inches} = 0.75\,\text{mile}$. To find the actual length of a road that is $6\,\text{inches}$ long on the map, we set up a proportion: $\frac{1.5\,\text{inches}}{0.75\,\text{mile}} = \frac{6\,\text{inches}}{x}$. Solving for $x$, we find that $1.5x = 4.5$, resulting in $x = 3\,\text{miles}$.
• Jeanine's Regional Map: Jeanine is drawing a map where the actual distance between her house and her cousin's house is $195\,\text{miles}$. The scale is $2\,\text{inches} = 15\,\text{miles}$. To find the distance on the map: $\frac{2\,\text{inches}}{15\,\text{miles}} = \frac{x\,\text{inches}}{195\,\text{miles}}$. Solving for $x$ ($15x = 390$), we find the houses will be separated by $26\,\text{inches}$ on the map.

Dimensional Conversions and Model Scaling

Scaling is also used in creating physical models. A model of a car was built using a scale of $1\,\text{inch} = 24\,\text{inches}$. If the length of the model is $5.5\,\text{inches}$, the actual length of the car in inches is calculated as: $5.5\,\text{inches} \times 24 = 132\,\text{inches}$ To convert the total length into feet, divide by $12$: $\frac{132\,\text{inches}}{12\,\text{inches/foot}} = 11\,\text{feet}$

Similarly, a scale drawing of a piano uses a scale of $\frac{1}{4}\,\text{inch} = 1\,\text{foot}$. If the piano measures $1\frac{3}{8}\,\text{inches}$ on the drawing, the actual length is found by determine how many $\frac{1}{4}\,\text{inch}$, or $0.25$, segments fit into $1.375\,\text{inches}$: $1.375 / 0.25 = 5.5\,\text{feet}$ This is equivalent to $5\frac{1}{2}\,\text{feet}$.

Ratio Analysis in Architectural Scale Drawings

Complex scale problems often involve relating area to linear distance. A rectangular room measuring $20\,\text{feet}$ by $12\,\text{feet}$ in reality has an actual area of: $A_{actual} = 20 \times 12 = 240\,\text{square feet}$ On a scale drawing, this room is represented by a rectangle with an area of $15\,\text{square inches}$. To determine the linear scale, we first find the ratio of areas: $\text{Area Ratio} = \frac{240\,\text{sq ft}}{15\,\text{sq in}} = 16\,\text{sq ft per sq in}$ To find the linear distance corresponding to $1\,\text{inch}$, we take the square root of the area ratio: $\sqrt{16} = 4\,\text{feet per inch}$ Therefore, a distance of $2\,\text{inches}$ on the scale drawing is equal to an actual distance of: $2\,\text{inches} \times 4\,\text{ft/in} = 8\,\text{feet}$