Geometry Unit Review: Area, Volume, Surface Area, and Similar Solids

Topic 1: Area of 2D Plane Figures

Area is defined as the calculation of the flat space contained inside a specific shape. This foundational concept relies on several core formulas for different geometric figures.

• Rectangle and Square

  • Formula for Rectangle: $A = l \times w$

  • Formula for Square: $A = s^2$

  • Variables: $l = \text{length}$, $w = \text{width}$, $s = \text{side}$.

• Parallelogram

  • Formula: $A = b \times h$

  • Variables: $b = \text{base}$, $h = \text{height}$.

  • Requirement: The height must be perpendicular ($\perp$) to the base.

• Triangle

  • Formula: $A = \frac{1}{2}bh$

  • Variables: $b = \text{base}$, $h = \text{perpendicular height}$.

• Trapezoid

  • Formula: $A = \frac{1}{2}h(b_1 + b_2)$

  • Variables: $b_1$, $b_2$ represent the parallel bases, and $h$ represents the height.

• Kite and Rhombus

  • Formula: $A = \frac{1}{2}d_1 d_2$

  • Variables: $d_1$ and $d_2$ represent the diagonals of the figure.

• Circle

  • Formula: $A = \pi r^2$

  • Variables: $r = \text{radius}$.

Essential Notes for 2D Shapes

• Orientation of Height: Height is ALWAYS perpendicular to the base. If a triangle or parallelogram leans (oblique), do not utilize the slanted side as the measurement for height. Always locate the right-angle square symbol to identify the true height.

• Radius vs. Diameter: In circle problems, be vigilant. If the diameter ($d$) is provided, you must calculate the radius by dividing the diameter by 2 ($r = \frac{d}{2}$) before inserting it into the area formula.

• Composite Figures: When tasked with finding the area of complex or unusual shapes, decompose them into standard geometric shapes. Calculate the area of each component individually and sum them. If the shape contains a "hole," calculate the area of the hole and subtract it from the total area.

Topic 2: Circles and Sectors

A sector is defined as a specific slice of a circle, similar to a slice of a pie. Its area represents a proportional fraction of the total area of the circle.

• Circumference (Perimeter):

  • $C = 2\pi r$

  • $C = \pi d$

• Area of a Sector:

  • $A = \frac{x}{360} \times \pi r^2$

  • Variable: $x$ is the central angle of the sector measured in degrees.

Topic 3: 3D Solids (Volume and Surface Area)

Calculations for three-dimensional solids involve two distinct measurements:

1. Volume ($V$): The measurement of the space inside the 3D shape, expressed in cubic units (e.g., $cm^3$).

2. Surface Area ($SA$): The total area encompassing all the external faces of the shape, expressed in square units (e.g., $cm^2$).

3. Lateral Area ($LA$): This is the area of the sides of the solid only, excluding the area of the bases.

Formula Table for 3D Solids

• General Prism

  • Volume: $V = Bh$

  • Lateral Area: $LA = hp$

  • Surface Area: $SA = hp + 2B$

• Rectangular Prism

  • Volume: $V = l \times w \times h$

  • Surface Area: $SA = 2lw + 2lh + 2wh$

• Cylinder

  • Volume: $V = \pi r^2 h$

  • Lateral Area: $LA = 2\pi rh$

  • Surface Area: $SA = 2\pi r^2 + 2\pi rh$

• Pyramid

  • Volume: $V = \frac{1}{3}Bh$

  • Lateral Area: $LA = \frac{1}{2}lp$

  • Surface Area: $SA = \frac{1}{2}lp + B$

• Cone

  • Volume: $V = \frac{1}{3}\pi r^2 h$

  • Lateral Area: $LA = \pi rl$

  • Surface Area: $SA = \pi r^2 + \pi rl$

• Sphere

  • Volume: $V = \frac{4}{3}\pi r^3$

  • Surface Area: $SA = 4\pi r^2$

• Hemisphere

  • Volume: $V = \frac{2}{3}\pi r^3$

  • Surface Area: $SA = 3\pi r^2$

Variable Decoder and 3D Principles

• $B$ (Capital B): This denotes the Area of the base shape. For example, if the base is a triangle, you must calculate $B = \frac{1}{2}bh$.

• $h$ (Lowercase h): The vertical distance representing the true height of the 3D solid, linking bases or the peak to the base.

• $p$ (Lowercase p): The perimeter surrounding the base shape.

• $l$ (Lowercase L): The slant height, which is the distance measured down the slanted face of a cone or pyramid.

Important Procedural Notes

• Base Identification: Always identify the "Base" first. For prisms, bases are the two parallel and identical faces (e.g., the triangles in a triangular prism).

• True vs. Slant Height: In pyramids and cones, use the true height ($h$) when calculating Volume. Use the slant height ($l$) when calculating Surface Area. If one is missing, use the Pythagorean theorem: $a^2 + b^2 = c^2$.

• Composite Solids: To determine the total volume of combined shapes, simply add the volumes of the individual components. For total surface area, add only the exposed surfaces; do not include faces that are hidden or touching where the shapes connect.

Topic 4: Similar Solids

Two 3D shapes are considered "similar" if they are proportional. Their relative measurements change according to strict mathematical scaling rules based on their dimensions.

Ratio Relationships

If the 1D ratio (Scale Factor) of two similar solids is expressed as $a : b$, the following rules apply:

• Linear Measurements: The ratio of perimeters, heights, or radii remains $a : b$.

• 2D Measurements (Area): The ratio of Surface Areas is determined by squaring the scale factor: $a^2 : b^2$.

• 3D Measurements (Volume): The ratio of Volumes is determined by cubing the scale factor: $a^3 : b^3$.

Practical Application Guidelines

• Simplification: Always reduce a given ratio or fraction to its simplest terms before attempting to square or cube it.

• Reversing the Process: If a problem provides the Volume Ratio first (e.g., $27 : 64$), you must take the cube root ($\sqrt[3]{x}$) to find the Scale Factor ($3 : 4$). Once you have the scale factor, you can square it to find the Surface Area ratio ($9 : 16$).