The surface area of a frustum of a square pyramid is found by summing the areas of its two parallel square bases and the areas of its four trapezoidal lateral faces.
Understanding the Frustum of a Square Pyramid
A frustum of a square pyramid is a three-dimensional geometric shape created when a square pyramid is cut by a plane parallel to its base, effectively removing the top portion. What remains is a solid with two parallel square bases (one larger, one smaller) and four identical trapezoidal sides connecting these bases.
Components of Total Surface Area
To calculate the total surface area ($TSA$) of a frustum of a square pyramid, you must determine and add the areas of three distinct components:
- Area of the Top Base ($A_{top}$): This is the smaller square face at the top of the frustum.
- Area of the Bottom Base ($A_{bottom}$): This is the larger square face at the bottom of the frustum.
- Lateral Surface Area ($A_{lateral}$): This is the combined area of the four identical trapezoidal sides that connect the top and bottom bases.
This method involves taking the area of the top base, then the area of the bottom base, and finally the area of the lateral sides, which are primarily trapeziums.
Essential Variables
Before diving into the calculations, let's define the key variables:
| Variable | Description |
|---|---|
| $s_1$ | Side length of the top square base |
| $s_2$ | Side length of the bottom square base |
| $h$ | Perpendicular height of the frustum |
| $l$ | Slant height of the frustum (height of a trapezoidal face) |
Calculating Each Component Area
1. Area of the Top Base ($A_{top}$)
Since the top base is a square with side length $s_1$, its area is:
$$A_{top} = s_1^2$$
2. Area of the Bottom Base ($A_{bottom}$)
Similarly, the bottom base is a square with side length $s_2$. Its area is:
$$A_{bottom} = s_2^2$$
3. Lateral Surface Area ($A_{lateral}$)
The lateral surface consists of four identical trapezoidal faces. The area of a single trapezoid is given by the formula: $\frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$. In this context, the parallel sides are $s_1$ and $s_2$, and the "height" of the trapezoid is the slant height ($l$) of the frustum.
- Area of one trapezoidal face: $\frac{1}{2} (s_1 + s_2) \times l$
- Since there are four such faces, the total lateral surface area is:
$$A_{lateral} = 4 \times \frac{1}{2} (s_1 + s_2) \times l = 2l(s_1 + s_2)$$
Determining the Slant Height ($l$)
The slant height ($l$) is crucial for calculating the lateral surface area. If $l$ is not directly provided, it can be calculated using the perpendicular height ($h$) of the frustum and the side lengths of the bases. Consider a right-angled triangle formed by the frustum's perpendicular height, the frustum's slant height, and half the difference between the base side lengths.
Using the Pythagorean theorem, the formula for the slant height ($l$) is:
$$l = \sqrt{h^2 + \left(\frac{s_2 - s_1}{2}\right)^2}$$
Total Surface Area Formula
Once you have calculated the areas of the top base, bottom base, and the lateral surface, sum them up to obtain the total surface area ($TSA$):
$$TSA = A{top} + A{bottom} + A_{lateral}$$
$$TSA = s_1^2 + s_2^2 + 2l(s_1 + s_2)$$
Step-by-Step Guide to Calculation
Follow these steps to accurately find the surface area of a frustum of a square pyramid:
- Note Given Values: Identify the side length of the top base ($s_1$), the side length of the bottom base ($s_2$), and either the perpendicular height ($h$) or the slant height ($l$).
- Calculate Slant Height: If the slant height ($l$) is not given, use the formula $l = \sqrt{h^2 + \left(\frac{s_2 - s_1}{2}\right)^2}$ to determine it.
- Calculate Top Base Area: Compute $A_{top} = s_1^2$.
- Calculate Bottom Base Area: Compute $A_{bottom} = s_2^2$.
- Calculate Lateral Surface Area: Compute $A_{lateral} = 2l(s_1 + s_2)$.
- Sum for Total Surface Area: Add the three component areas: $TSA = A{top} + A{bottom} + A_{lateral}$.
Example Calculation
Let's find the exact surface area of a frustum of a square pyramid with the following dimensions:
- Side length of the top base ($s_1$) = 4 cm
- Side length of the bottom base ($s_2$) = 8 cm
- Perpendicular height ($h$) = 3 cm
Solution:
-
Identify Values:
- $s_1 = 4$ cm
- $s_2 = 8$ cm
- $h = 3$ cm
-
Calculate Slant Height ($l$):
- $l = \sqrt{h^2 + \left(\frac{s_2 - s_1}{2}\right)^2}$
- $l = \sqrt{3^2 + \left(\frac{8 - 4}{2}\right)^2}$
- $l = \sqrt{3^2 + \left(\frac{4}{2}\right)^2}$
- $l = \sqrt{3^2 + 2^2}$
- $l = \sqrt{9 + 4}$
- $l = \sqrt{13}$ cm
-
Calculate Top Base Area ($A_{top}$):
- $A_{top} = s_1^2 = 4^2 = 16$ cm$^2$
-
Calculate Bottom Base Area ($A_{bottom}$):
- $A_{bottom} = s_2^2 = 8^2 = 64$ cm$^2$
-
Calculate Lateral Surface Area ($A_{lateral}$):
- $A_{lateral} = 2l(s_1 + s_2)$
- $A_{lateral} = 2 \times \sqrt{13} \times (4 + 8)$
- $A_{lateral} = 2 \times \sqrt{13} \times 12$
- $A_{lateral} = 24\sqrt{13}$ cm$^2$
-
Calculate Total Surface Area ($TSA$):
- $TSA = A{top} + A{bottom} + A_{lateral}$
- $TSA = 16 + 64 + 24\sqrt{13}$
- $TSA = 80 + 24\sqrt{13}$ cm$^2$
The exact surface area of the frustum is $80 + 24\sqrt{13}$ cm$^2$.
Further Resources
For additional understanding and visualization of frustums and geometric surface areas, consider exploring these resources:
