Exam Notes

Regular Polygons and Circles

Interior Angle, Perimeter, and Area of Regular Polygons

• Triangle: Interior angle = 60°, Perimeter = 48 in, Area = 27.71 cm^2
• Pentagon: Interior angle = 72°, Perimeter = 30 in, Area = 12.387 cm^2
• 14-sided Polygon: Interior angle = 25.714°, Perimeter = 84 in, Area = 39.432 cm^2
• Polygon with 28 Sides: Interior angle = 67.5°, Perimeter = 66.24 in, Area = 41.4 cm^2

Circumference of a Circle

• The perimeter of an n-gon approximates the circumference of a circle as n increases.
• Circumference formula: C = 2 \cdot r \cdot \pi or C = \pi \cdot d, where r is the radius and d is the diameter.
• Approximation of \pi as n approaches a large number: \frac{360}{n} \cdot sin(\frac{360}{n}) approaches 3.141592653.

Examples:

• If diameter (d) = 17 mm, C = \pi \cdot 17 \approx 53.4 mm.
• If radius (r) = 7 in, d = 14 in, C = \pi \cdot 14 \approx 44.0 in.

Area of a Circle

• Area formula: A = \pi \cdot r^2
• Area with diameter: A = \pi \cdot (\frac{d}{2})^2
• Area of a Half Circle: A = \frac{\pi \cdot r^2}{2}

Examples:

• If r = 7 mm, A = \pi \cdot (7)^2 \approx 153.9 mm^2
• If r = 12 yd, A = \pi \cdot (12)^2 \approx 452.4 yd^2
• If d = 22 cm, r = 11 cm, A = \frac{\pi \cdot (11)^2}{2} \approx 190.1 cm^2

Volume

• Rectangular Prism: Volume = l \cdot w \cdot h
• Cylinder: Volume = \pi \cdot r^2 \cdot h
• Cone: Volume = \frac{1}{3} \cdot \pi \cdot r^2 \cdot h

Volume Comparison

• Rectangular Box: Volume = l \cdot w \cdot h
• Cylinder: Volume = \pi r^2 h

Arc Length and Sector Area

• Arc Length Formula: Arc Length = \frac{central angle}{360} \cdot 2 \pi r
• Area of a Sector Formula: Area = \frac{central angle}{360} \cdot \pi r^2

Exponent Rules

• Product Rule: x^a \cdot x^b = x^{a+b}
• Quotient Rule: \frac{x^a}{x^b} = x^{a-b}
• Power Rule: (x^a)^b = x^{a \cdot b}
• Power of a Quotient: \frac{x^a}{y^a} = (\frac{x}{y})^a
• Power of a Product: (xy)^a = x^a y^a
• Exponent of 0 or 1: x^0 = 1 and x^1 = x
• Negative Exponent: x^{-a} = \frac{1}{x^a}

Linear Equations

• Slope-Intercept Form: f(x) = mx + b, where m is the slope and b is the y-intercept.
• The slopes of parallel lines are always the same.
• The slopes of perpendicular lines are always reciprocal.

Circle Properties

• Central Angle: An angle whose vertex is at the center of the circle.
• Inscribed Angle: An angle whose vertex is on the circle.
• Tangent Line: A line that intersects a circle at exactly one point.
• A tangent line is always perpendicular to the radius at the point of intersection.