6.4 Coordinate Geometry, Perimeter, Area & Volume Formulas

6.4 Coordinate Geometry, Perimeter, Area & Volume Formulas

Coordinate geometry merges algebraic equations with geometric space, while measurement formulas allow you to calculate perimeter, area, surface area, and volume of two- and three-dimensional figures. These topics build broad Mathematics Knowledge skills.

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The Cartesian Coordinate Plane

The Cartesian coordinate plane is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis, intersecting at the origin $(0,0)$. The axes divide the plane into four quadrants:

• Quadrant I: $(+x, +y)$
• Quadrant II: $(-x, +y)$
• Quadrant III: $(-x, -y)$
• Quadrant IV: $(+x, -y)$

Essential Coordinate Formulas

For any two points $A(x_1, y_1)$ and $B(x_2, y_2)$:

1. Distance Formula: Derived directly from the Pythagorean Theorem. $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
2. Midpoint Formula: Calculates the average of the x-coordinates and y-coordinates. $$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$
3. Slope Formula ($m$): Measures the steepness and direction of a line ($\,\text{rise} / \text{run}\,$). $$m = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \neq x_2)$$

Slope Classifications & Line Relationships

• Positive Slope ($m > 0$): Line rises from left to right.
• Negative Slope ($m < 0$): Line falls from left to right.
• Zero Slope ($m = 0$): Horizontal line ($y = c$).
• Undefined Slope: Vertical line ($x = c$).
• Parallel Lines ($L_1 \parallel L_2$): Slopes are identical ($m_1 = m_2$).
• Perpendicular Lines ($L_1 \perp L_2$): Slopes are negative reciprocals ($m_1 \cdot m_2 = -1 \implies m_2 = -\frac{1}{m_1}$).
  Example: If $m_1 = \frac{3}{4}$, then $m_2 = -\frac{4}{3}$.

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Linear Equations of Lines

Worked Example: Perpendicular Line Equation

Find the equation of the line passing through point $(4, -2)$ that is perpendicular to the line $y = 2x + 5$.

1. Identify given slope: $m_1 = 2$.
2. Determine perpendicular slope: $m_2 = -\frac{1}{2}$.
3. Use Point-Slope form with point $(4, -2)$: $$y - (-2) = -\frac{1}{2}(x - 4) \implies y + 2 = -\frac{1}{2}x + 2$$
4. Subtract $2$ from both sides: $$y = -\frac{1}{2}x$$

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2D Perimeter & Area Formulas

Perimeter ($P$) measures total distance around a 2D boundary (linear units). Area ($A$) measures total surface enclosed (square units).

Heron's Formula for Triangle Area: Given three side lengths $a, b, c$, calculate semi-perimeter $s = \frac{a+b+c}{2}$. Then Area $A = \sqrt{s(s-a)(s-b)(s-c)}$.

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3D Geometry: Volume & Surface Area Formulas

Volume ($V$) measures space enclosed within a 3D object (cubic units). Surface Area ($SA$) measures total outer surface area (square units).

3D Geometric Formulas

• Rectangular Prism (Box):
  • Volume: $V = l \cdot w \cdot h$
  • Surface Area: $SA = 2(lw + lh + wh)$
• Cube:
  • Volume: $V = s^3$
  • Surface Area: $SA = 6s^2$
• Right Circular Cylinder:
  • Volume: $V = \pi r^2 h$
  • Lateral Area: $LA = 2\pi r h$
  • Total Surface Area: $SA = 2\pi r h + 2\pi r^2$
• Right Circular Cone:
  • Volume: $V = \frac{1}{3}\pi r^2 h$
  • Slant Height: $l = \sqrt{r^2 + h^2}$
  • Total Surface Area: $SA = \pi r l + \pi r^2$
• Sphere:
  • Volume: $V = \frac{4}{3}\pi r^3$
  • Surface Area: $SA = 4\pi r^2$
• Square Pyramid:
  • Volume: $V = \frac{1}{3} s^2 h$

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Composite Figures & Strategy

Many high-level practice questions present composite 2D or 3D figures combining multiple basic shapes (e.g., a rectangle surmounted by a semicircle, or a cylinder topped by a cone).

Strategy for Composite Figures

1. Deconstruct: Break the figure into standard geometric components.
2. Calculate: Find individual areas or volumes using standard formulas.
3. Combine: Add component measurements (or subtract cutouts) to find total area or volume.