How to Find the Area of a Square Pyramid?
Are you interested in learning how to find the area of a square pyramid? Let’s first understand what an area is. The area of a square is the quantity of square units required to cover its surface entirely.
In a broader context, area refers to the space enclosed within the perimeter of a flat object or 2D figure. Measurement is typically expressed in square units, with the standard unit being square meters (m2).
In this segment, we will delve into the concept of determining the surface area of a square pyramid. A pyramid, a three-dimensional entity, features congruent triangular side faces, with its base potentially forming any polygon.
One side of each triangular face aligns with a side of the base polygon. When the base is a square, the pyramid is aptly termed a square pyramid.
The nomenclature of pyramids is based on the shapes of their bases. Much like other 3D shapes, a square pyramid encompasses two distinct types of areas: Total Surface Area (TSA) and Lateral Surface Area (LSA).
Let’s now go into understanding the surface area of a square pyramid, exploring the formulas, and examining a few solved examples. Practice questions are also available towards the end for hands-on application.
How to Find the Area of a Square Pyramid (Easy Hacks)
The area of a square pyramid is not difficult to calculate. You need to know the step by step on how to calculate.
These easy hacks below will help you to calculate the area of a square pyramid easily:
1. Base Area Calculation
You will start by finding the area of the square base. If the side length is ‘a’, then the base area (A_base) is a². This sets the foundation for our area calculation.
2. Embrace Symmetry
Since the triangular sides are congruent, you only need to calculate the area of one triangular side and then multiply it by four.
Each triangular face has a base (side of the square) and a height (slant height or ‘l’). The area of one triangular face is 1/2×a×l. Multiply this by four to get the total lateral area.
3. Combine for Total Surface Area (TSA)
Now, combine the base area (A_base) and the total lateral area to get the total surface area (TSA). TSA = Abase+Alateral.
Substituting the values, TSA = a²+2al.
4. Shortcut for Slant Height
If the slant height (‘l’) isn’t given, don’t fret. Use the Pythagorean theorem: l= √a²/4+h2, where ‘h’ is the pyramid’s height. This avoids any hassle in finding the slant height.
Putting it all Together
Base Area: Abase = a²
Lateral Area (One Triangle): Alateral, one = 1/2×a×l
Total Lateral Area: Alateral = 4×Alateral, one
Total Surface Area (TSA): TSA = Abase+Alateral = a²+2al
The Formula for Surface Area of a Square Pyramid
Consider a square pyramid with a base side length of ‘a’ and a slant height (‘l’) for each side face. The base area (a²) is paired with the area of each triangular face (1/2 * a * l).
Combining the areas of all four triangular faces yields the lateral surface area (2al). The formula for the total surface area of a square pyramid is given by:
Surface Area = a²+2al
Surface Area Using Slant Height
Expressing the slant height (‘l’) in the formula, the total surface area can also be represented as:
Surface Area = a²+2al
Surface Area Using Height
If the height of the pyramid (‘h’) is considered, the surface area formula becomes:
Surface Area = a²+a√a²/4+h2
Lateral Surface Area
The lateral surface area of a square pyramid, representing the space covered by the four triangular faces, can be calculated using the formula:
Lateral Surface Area = 2al
Various Practical Examples to Learn From
These detailed examples and workings will provide you with step-by-step solutions that will help you understand the application of formulas in calculating different aspects of a square pyramid.
Example 1
Consider a square pyramid with a base side length of 15 cm and a slant height of 21 cm. The perimeter of the base is 60 cm, and the lateral surface area is what?
Calculating Total Surface Area
Given:
- Base side length (s): 15 cm
- Slant height (l): 21 cm
The formula for Lateral Surface Area (LSA): 1/2×Perimeter×Slant Height
Perimeter (P) = 4s = 4(15) = 60cm
LSA = 1/2×60×21 = 630 sq. cm
Total Surface Area (TSA): Base Area+LSA
Base Area = S² = 15²= 225sq. cm
TSA = 225+630 = 855 sq. cm
The final answer for the total surface area of the given square pyramid is 855 sq. cm.
Example 2
In another scenario, let’s examine a square pyramid with a base side length of 18 inches and a slant height of 22 inches.
The perimeter of the base is 72 inches, and the lateral surface area is what?
Determining Lateral Surface Area
Given:
- Base side length (s): 18 inches
- Slant height (l): 22 inches
Perimeter (P) = 4s = 4(18) = 72inches
LSA = 1/2×P×l
LSA=1/2×72×22 = 792 sq. in
The final answer of the lateral surface area of the given square pyramid is 792 sq. in.
Example 3
Imagine a square pyramid with a lateral surface area of 200 sq. in. and a base side length of 10 inches. Find the slant height.
Finding Slant Height
Given:
- Base side length (s): 10 inches
- Lateral Surface Area (LSA): 200 sq. in.
Perimeter (P) = 4s = 4(10) = 40inches
200 = 1/2×40×Slant Height
Slant Height = 200/20 = 10 inches
The final answer for the slant height of the given square pyramid is 10 inches
Example 4
Consider a square pyramid that has a slant height of 24cm and a lateral surface area of 480 sq. cm. Using the formula for lateral surface area, determine the base side length.
Calculating Base Side Length
Given:
- Lateral Surface Area (LSA): 480 sq. cm
- Slant height (l): 24 cm
Perimeter (P) = 4s
LSA = 1/2×P×l
480 = 1/2×4s×24
s = 480/80 = 10cm
The final answer for the base side length (s) of the given square pyramid is 10 cm.
You’ve successfully navigated the realm of finding the area of a square pyramid. Armed with these simple steps and practical tips, you can confidently tackle this mathematical challenge.
Keep practicing, and soon, you’ll be a master at unraveling the mysteries of geometry!
