The lateral area of an object refers to the surface area of its sides, but not the base. In the instance of a square pyramid, this refers to the four triangles that make up the top and side portions of the figure around a square base. Like any three-dimensional object, the best approach is to break it down into individual, two-dimensional figures.
Find the Perimeter
To find the lateral area of a square pyramid, you need to find the perimeter of the base and the lateral slant. The perimeter is the sum of the measurements that run along the square base of the pyramid. Since all four sides of a square are equal, multiplying the length of one edge by four will give you the perimeter.
For example:
The side of a square pyramid measures 4 inches. The perimeter of the pyramid is 4 times 4.
4 x 4 = 16 inches
Calculate the Lateral Slant
The lateral slant is the measurement from the top of the pyramid to the edge of the base that bisects one of the triangle faces.
To find the lateral slant, first picture the pyramid from the front and divide it down the middle. This creates two right triangles. The height of the pyramid is the height of one edge. The length of the right triangle is equal to half the length of the base. Finding the lateral slant is the same as finding the hypotenuse and involves using the Pythagorean Theorem: a^2 + b^2 = c^2.
If a square pyramid has a base of 4 inches and a height of 7 inches, start by dividing the base length in half.
4 / 2 = 2 inches
Next, use the Pythagorean Theorem.
2^2 + 7^2 = 53
Find the square root of the sum to find the lateral slant. The square root is represented by sq rt.
sq rt 53 = 7.28 inches
The lateral slant is 7.28 inches
Calculate the Area
Once you have the perimeter and the lateral slant, you can plug those numbers into the equation, LA = ½Ph to calculate the lateral area. LA represents the lateral area, P is the measurement of the perimeter, and h is the height of the pyramid.
For example:
Using the base length and height from before, you know that the perimeter is 16 inches and the lateral slant is 7.28 inches.
½(16 x 7.28) = 58.24 square inches
More Examples
If a square pyramid has a base length of 128 feet and a height of 102 feet, first find the perimeter.
128 x 4 = 512 feet
Next, calculate the lateral slant. Start by finding the dimensions of the right triangle, then use the Pythagorean Theorem.
128 / 2 = 64 feet
64^2 + 102^2 = 14,500
sq rt14,500 = 120.42 feet
With a perimeter of 512 feet and a lateral slant of 120.42 feet, calculate the lateral area.
½(512 x 120.42) = 30,827.52 square feet.
If a square pyramid has a base length of 72 inches and a height of 27 inches, find the perimeter.
72 x 4 = 288 inches
Now find the lateral slant.
72 / 2 = 36 inches
36^2 + 27^2 = 2025 inches
sq rt 2025 = 45 inches
Finally, calculate the lateral area.
½(288 x 45) = 6,480 square inches
