3D Shapes Surface Area Quiz

Cube:
- Formula: ( SA = 6a^2 )
- ( a ) = length of a side.
Rectangular Prism:
- Formula: ( SA = 2lw + 2lh + 2wh )
- ( l ) = length, ( w ) = width, ( h ) = height.
Cylinder:
- Formula: ( SA = 2\pi r(h + r) )
- ( r ) = radius, ( h ) = height.
Sphere:
- Formula: ( SA = 4\pi r^2 )
- ( r ) = radius.
Cone:
- Formula: ( SA = \pi r(l + r) )
- ( r ) = radius, ( l ) = slant height.

Measures of Central Tendency:
- Mean: Average value; sum of all values divided by count.
- Median: Middle value when data is ordered; if even number of observations, average the two middle values.
- Mode: Most frequently occurring value(s).
Measures of Dispersion:
- Range: Difference between highest and lowest values.
- Variance: Average of squared differences from the mean.
- Standard Deviation: Square root of variance; indicates data spread.
Skewness and Kurtosis:
- Skewness: Measure of asymmetry of the distribution.
- Kurtosis: Measure of the "tailedness" of the distribution.
Data Visualization:
- Histograms: Shows frequency distribution of data.
- Box Plots: Displays median, quartiles, and outliers.
- Scatter Plots: Shows relationship between two variables.

Cube:
- Surface area calculated using the formula ( SA = 6a^2 )
- ( a ) denotes the length of one side.
Rectangular Prism:
- Surface area determined by ( SA = 2lw + 2lh + 2wh )
- Variables represent: ( l ) = length, ( w ) = width, ( h ) = height.
Cylinder:
- Surface area calculated with ( SA = 2\pi r(h + r) )
- ( r ) indicates the radius, ( h ) the height of the cylinder.
Sphere:
- Surface area formula is ( SA = 4\pi r^2 )
- ( r ) is the radius of the sphere.
Cone:
- Surface area given by ( SA = \pi r(l + r) )
- ( r ) = radius, ( l ) = slant height.

Cube:
- Volume calculated using ( V = a^3 )
- ( a ) represents the side length.
Rectangular Prism:
- Volume determined by the formula ( V = l \cdot w \cdot h ).
Cylinder:
- Volume calculated with ( V = \pi r^2 h )
- ( r ) is the radius, ( h ) is height.
Sphere:
- Volume given by ( V = \frac{4}{3}\pi r^3 )
- ( r ) denotes the radius.
Cone:
- Volume expressed as ( V = \frac{1}{3}\pi r^2 h ).

Measures of Central Tendency:
- Mean: Average value calculated by summing all values and dividing by their count.
- Median: Middle value in ordered data; in cases of an even count, average the two middle values.
- Mode: Value that appears most frequently in the dataset.
Measures of Dispersion:
- Range: Difference between the maximum and minimum values in a dataset.
- Variance: Average of squared differences from the mean; indicates how data points spread out.
- Standard Deviation: Square root of variance; provides insight into the distribution's spread.
Skewness and Kurtosis:
- Skewness: Measures the asymmetry of the data distribution.
- Kurtosis: Measures the "tailedness" or extremity of distribution outcomes.
Data Visualization:
- Histograms: Visual representation showing frequency distribution of different data values.
- Box Plots: Displays median, quartiles, and highlights outliers in data.
- Scatter Plots: Illustrates the relationship between two variables, highlighting correlation.

Trigonometric functions relate the angles and sides of right triangles, essential in various applications including physics and engineering.
Primary functions include:
- Sine (sin): Ratio of the length of the opposite side to the hypotenuse.
- Cosine (cos): Ratio of the length of the adjacent side to the hypotenuse.
- Tangent (tan): Ratio of the length of the opposite side to the adjacent side.
Reciprocal functions enhance the primary functions:
- Cosecant (csc): Inversely related to sine (1/sin).
- Secant (sec): Inversely related to cosine (1/cos).
- Cotangent (cot): Inversely related to tangent (1/tan).
Domains and ranges define the behavior of functions:
- Sine and cosine have a domain of all real numbers and range from -1 to 1.
- Tangent has a domain excluding angles of the form (90° + k*180°), with a range of all real numbers.

Angles can be categorized based on their measurements:
- Acute: Less than 90°.
- Right: Exactly 90°.
- Obtuse: More than 90° but less than 180°.
- Straight: Exactly 180°.
Measurement of angles can be done in degrees or radians:
- Degrees: A full circle is divided into 360 degrees.
- Radians: A full circle is divided into 2π radians; thus, 180° equals π radians.
Conversion between degrees and radians involves:
- Degrees to radians: Multiply by π/180.
- Radians to degrees: Multiply by 180/π.

The unit circle is a circle of radius 1 centered at the origin of the coordinate plane, crucial for trigonometric interpretation.
Any point on the unit circle can be defined by its coordinates as (cos θ, sin θ).
Important angles and their corresponding coordinates include:
- 0° (0): (1, 0)
- 30° (π/6): (√3/2, 1/2)
- 45° (π/4): (√2/2, √2/2)
- 60° (π/3): (1/2, √3/2)
- 90° (π/2): (0, 1)
- 180° (π): (-1, 0)
- 270° (3π/2): (0, -1)
- 360° (2π): (1, 0)

Pythagorean identities are fundamental relationships among sine, cosine, and tangent:
- sin² θ + cos² θ = 1
- 1 + tan² θ = sec² θ
- 1 + cot² θ = csc² θ
Reciprocal identities connect trigonometric functions with their counterparts:
- sin θ = 1/csc θ
- cos θ = 1/sec θ
- tan θ = 1/cot θ
Co-function identities reveal relationships involving complementary angles:
- sin(90° - θ) = cos θ
- cos(90° - θ) = sin θ
- tan(90° - θ) = cot θ
Even-odd identities describe the symmetry of trigonometric functions:
- sin(-θ) = -sin θ (sine is odd)
- cos(-θ) = cos θ (cosine is even)
- tan(-θ) = -tan θ (tangent is odd)