Section 14.1: Surface Areas of Prisms

Property

The calculation of areas is based on these principles:

1. If you move a shape rigidly (without stretching or distorting it), then its area does not change.
2. If you combine shapes without overlapping them, then the area of the resulting shape is the sum of the areas of the individual shapes.

Formulas:

• Rectangle: With side lengths $a$ and $b$, the area is $A = ab$.
• Triangle: With base $b$ and height $h$, the area is $A = \frac{1}{2}bh$.

Examples

• A rectangular face of a box is 6 inches long and 4 inches wide. Its area is $6 \times 4 = 24$ square inches.
• A triangular face of a prism has a base of 8 centimeters and a height of 5 centimeters. Its area is $\frac{1}{2} \times 8 \times 5 = 20$ square centimeters.
• The rectangular base of a prism measures 12 feet by 7 feet. Its area is $12 \times 7 = 84$ square feet.

Explanation

Area measures the total space inside a 2D shape. When finding the surface area of prisms, we need to calculate the areas of rectangular and triangular faces. We can find the area of more complex polygon faces by breaking them into rectangles and triangles, then adding their areas together.

Property

Cube whose edge is of length $L$:
Surface Area $= 6L^2$
Rectangular Prism of length $L$, height $H$ and width $W$:
Surface Area $= 2(LW + LH + WH)$
Triangular Prism of height $H$ based on a triangle with side lengths $A$, $B$, $C$ and base area $B_{area}$:
Surface Area $= 2B_{area} + (A + B + C)H$

Examples

• A cube with an edge length of 4 cm has a Surface Area of $6(4^2) = 6 \times 16 = 96$ square cm.
• A rectangular prism is 5 in long, 3 in wide, and 6 in high. Its Surface Area is $2(5 \times 3 + 5 \times 6 + 3 \times 6) = 2(15 + 30 + 18) = 126$ square in.
• A triangular prism is 8 cm high and its base is a right triangle with legs 3 cm and 4 cm, and hypotenuse 5 cm. The base area is $\frac{1}{2}(3 \times 4) = 6$ square cm. Its Surface Area is $2(6) + (3+4+5)(8) = 12 + 96 = 108$ square cm.

Explanation

Surface area measures the total area of all the outside faces of a 3D shape. Think of it as how much wrapping paper you would need to completely cover the object. For prisms, we calculate the area of all faces and add them together using these formulas.

Property

Lateral surface area is the sum of the areas of only the side faces of a prism, excluding the top and bottom bases.

For rectangular prisms: $L = 2\ell h + 2wh$ where $\ell$ is length, $w$ is width, and $h$ is height.

Property

Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas $V=s^3$ and $A=6s^2$ to find the volume and surface area of a cube.

Examples

• Find the volume of a cube with side length $s = 5$ cm using the formula $V=s^3$. Substitute $s=5$ to get $V = 5^3 = 5 \cdot 5 \cdot 5 = 125$ cubic cm.
• Find the surface area of a cube with side length $s = 3$ ft using the formula $A=6s^2$. Substitute $s=3$ to get $A = 6 \cdot (3^2) = 6 \cdot 9 = 54$ square ft.
• The area of a square is $A=s^2$. If the side length $s$ is $1.2$ meters, the area is $A = (1.2)^2 = 1.44$ square meters.

Explanation

To evaluate an expression, you replace the variable with its given number and solve. It's like using a recipe: the formula is your instructions, the variable's value is your ingredient, and the final answer is your finished dish.