Surface Area and Volume of Solid Figures: Exhaustive Study Guide

Driving Question: The central inquiry for this unit is: How can we use our knowledge of 3D shapes to calculate Surface Area and Volume?
Polyhedron: This is defined as a solid figure whose faces are all polygons.
Surface Area: This term refers to the sum of the areas of all the surfaces (faces) of a 3D figure.
Volume: This is a measure of the amount of space a container holds or the capacity of a solid figure.
Radius: This is defined as half the diameter of a circle, appearing in figures with circular bases such as cylinders.

Definition: A Rectangular Prism is a polyhedron that has a congruent pair of parallel rectangular bases and four additional faces that are also rectangles.
Volume Formula: The volume ( $V$ ) of a rectangular prism is found by multiplying the length ( $l$ ), width ( $w$ ), and height ( $h$ ).
- $V = l \times w \times h$
Net of a Rectangular Prism: A 2D representation (net) of a rectangular prism shows six rectangles that, when folded, form the 3D solid.
Example Calculation 1:
- Dimensions: Length = $10\,\text{in}$ , Width = $2\,\text{in}$ , Height = $7\,\text{in}$ .
- Formula: $V = (10) \times (2) \times (7)$
- Result: $V = 140\,\text{in}^3$
Practice Calculation 1:
- Dimensions: Length = $6\,\text{cm}$ , Width = $12\,\text{cm}$ , Height = $8.9\,\text{cm}$ .
- Formula: $V = (6) \times (12) \times (8.9)$
- Result: $V = 640.8\,\text{cm}^3$ (rounded to the tenths place).

Definition: A cylinder is a solid figure formed by two congruent parallel faces called bases, which are joined by a curved surface.
Volume Formula: The volume ( $V$ ) of a cylinder can be calculated using the area of the base ( $B$ ) multiplied by the height ( $h$ ), or specifically using the radius ( $r$ ):
- $V = Bh$
- $V = \text{π} \times r^2 \times h$
Net of a Cylinder: The 2D net of a cylinder consists of two congruent circles (the bases) and one rectangle (the lateral surface area that wraps around the bases).
Constant for Pi: Throughout these calculations, the value $3.14$ is used for $\text{π}$ .
Example Calculation 2:
- Dimensions: Diameter = $10\,\text{cm}$ (Radius $r = 5\,\text{cm}$ ), Height = $9\,\text{cm}$ .
- Formula: $V = (3.14) \times (5^2) \times (9)$
- Step-by-step: $V = (3.14) \times (25) \times (9)$
- Result: $V = 706.5\,\text{cm}^3$
Practice Calculation 2:
- Dimensions: Radius = $7\,\text{in}$ , Height = $13\,\text{in}$ .
- Formula: $V = (3.14) \times (7^2) \times (13)$
- Step-by-step: $V = (3.14) \times (49) \times (13)$
- Result: $V = 2000.18\,\text{in}^3$

Solving for Unknowns (General): When given the total volume, one must rearrange the formulas to isolate the unknown variable, such as length, width, radius, or height.
Water Tank Scenario:
- The base of a water tank has an area of $28\,\text{ft}^2$ .
- The total volume ( $V$ ) when full is $140\,\text{ft}^3$ .
- To find height: $h = \frac{V}{B}$
- Calculation: $h = \frac{140}{28}$
- Result: $h = 5\,\text{ft}$ .
Scaling Dimensions in Rectangular Prisms:
- Height Scaling: If the height of a rectangular prism is increased by a factor of $2$ , the volume increases linearly by the same factor (assuming length and width remain constant).
- Width Scaling: If the width of a rectangular prism is increased by a factor of $3$ , the new volume will be three times the original volume.
Specific Unknown Variable Problems:
- Missing variable calculation where Volume = $646\,\text{in}^3$ .
- Missing variable calculation where Volume = $936\,\text{in}^3$ .
- Missing variable calculation for a cylinder where Volume = $32,000\,\text{in}^3$ (using $\text{π} = 3.14$ and rounding to the hundredths place).

Conceptual Distinction:
- Surface Area: Covers the exterior of a 3D shape. Analogies include wrapping paper for a gift or the label for a soup can.
- Volume: Measures how much of a substance can fit inside a 3D shape. Analogies include water in a fish tank, oxygen in a chamber, or sugar cubes in a box.
Real-World Application Scenarios:
- Juice Box Branding: To determine how many fluid ounces of apple juice will fit in a box, a designer must calculate the Volume.
- Watch Box Finishing: To calculate how much gold paint is needed to cover the entire outside of a wooden rectangular box, the shop must calculate the Surface Area.
- Shed Construction: To determine the amount of metal required to build the walls and the roof of a cylindrical storage shed, an engineer must calculate the Surface Area.

Surface Area Formula: The surface area ( $SA$ ) of a rectangular prism is the sum of the areas of its six faces, calculated as:
- $SA = 2lw + 2lh + 2wh$
Example Calculation 3:
- Dimensions: Length ( $l$ ) = $10\,\text{in}$ , Width ( $w$ ) = $2\,\text{in}$ , Height ( $h$ ) = $7\,\text{in}$ .
- Formula: $SA = 2(10)(2) + 2(10)(7) + 2(2)(7)$
- Step-by-step: $SA = 40 + 140 + 28$
- Result: $SA = 208\,\text{in}^2$
Practice Calculation 3:
- Dimensions: $l = 12\,\text{cm}$ , $w = 8.9\,\text{cm}$ , $h = 6\,\text{cm}$ .
- Formula: $SA = 2(12)(8.9) + 2(12)(6) + 2(8.9)(6)$
- Result: $SA = 464.4\,\text{cm}^2$
Scaling Effects on Surface Area:
- If the height of a rectangular prism increases by a factor of $2$ , the new surface area must be recalculated using the updated height dimension in the formula ( $2lh$ and $2wh$ terms), while the base term ( $2lw$ ) remains constant.
- If the width of a rectangular prism increases by a factor of $3$ , the new surface area is calculated by tripling the width value within the standard formula.

Formula Components: The surface area of a cylinder involves the area of the two circular bases added to the area of the rectangular lateral surface (the side).
- $SA = 2 \times \text{Area of Base} + \text{Circumference} \times \text{Height}$
- $SA = 2\text{π}r^2 + 2\text{π}rh$
Practice Context: When calculating, use $3.14$ for $\text{π}$ and round results to the hundredths place where necessary.