Solved Examples on Square Pyramid Formulas
Example 1: Determine the total surface area of a square pyramid if the base’s side length is 15 cm and the pyramid’s slant height is 21 cm.
Solution:
Given,
The side of the square base (s) = 15 cm
Slant height, (l) = 21 cm
The perimeter of the square base (P) = 4s = 4(15) = 60 cm
The lateral surface area of a square pyramid = (½) Pl
LSA = (½ ) × (60) × 21 = 630 sq. cm
Now, the total surface area = Area of the base + LSA
= s2 + LSA
= (15)2 + 630
= 225 + 630 = 855 sq. cm
Therefore, the total surface area of the given pyramid is 855 sq. cm.
Example 2: Determine the lateral surface area of a square pyramid if the side length of the base is 18 inches and the pyramid’s slant height is 22 inches.
Solution:
Given,
The side of the square base (s) = 18 inches
Slant height, (l) = 22 inches
The perimeter of the square base (P) = 4s = 4(18) = 72 inches
The lateral surface area of a square pyramid = (½) Pl
LSA = (½ ) × (72) × 22 = 792 sq. in.
Therefore, the lateral surface area of the given pyramid is 792 sq. in.
Example 3: What is the slant height of the square pyramid if its lateral surface area is 200 sq. in. and the side length of the base is 10 inches?
Solution:
Given data,
Length of the side of the base of a square pyramid (s) = 10 inches
The lateral surface area of a square pyramid = 200 sq. in
Slant height (l) = ?
We know that,
The lateral surface area of a square pyramid = (½) Pl
The perimeter of the square base (P) = 4s = 4(10) = 40 inches
⇒ 200 = ½ × 40 × l
⇒ l = 10 in
Hence, the slant height of the square pyramid is 10 inches.
Example 4: Calculate the side length of the base of the square pyramid if its lateral surface area is 480 sq. cm and the slant height is 24 cm.
Solution:
Given data,
The lateral surface area of a square pyramid = 480 sq. cm
Slant height (l) = 24 cm
Let the length of the side of the base of a square pyramid be “s”.
We know that,
The lateral surface area of a square pyramid = (½) Pl
The perimeter of the square base (P) = 4s
⇒ 480 = ½ × 4s × 24
⇒ s = 10 cm
Hence, the side length of the base of the square pyramid is 10 cm.
Example 5: Determine the total surface area of a square pyramid if the base’s side length is 14 cm and the pyramid’s height is 24 cm.
Solution:
Given,
The side of the square base (s) = 14 cm
The height of the square pyramid (h) = 24 cm
The slant height of the pyramid (l) = √[(s/2)2 + h2]
l = √[(14/2)2 + 242)] = √(49+576)
= √625 = 25 cm
The perimeter of the square base (P) = 4s = 4(14) = 56 cm
The lateral surface area of a square pyramid = (½) Pl
LSA = (½ ) × (56) × 25 = 700 sq. cm
Now, the total surface area = Area of the base + LSA
= s2 + LSA
= (14)2 + 700
= 196 + 700 = 896 sq. cm
Therefore, the total surface area of the given pyramid is 896 sq. cm.
Example 6: Determine the surface area of a square pyramid if the base’s side length is 10 cm and the pyramid’s slant height is 15 cm.
Solution:
Given,
The side of the square base (s) = 10 cm
Slant height (l) = 15 cm
We know that,
The total surface area of a square pyramid (TSA) = 2sl + s2 square units
= 2 × 10 × 15 + (10)2
= 300 + 100 = 400 sq. cm
Therefore, the surface area of the given pyramid is 400 sq. cm.
Surface Area of a Square Pyramid
Pyramid is a three-dimensional geometric structure with a polygonal base and triangular faces equal to the number of sides in the base. The triangular faces or lateral surfaces of a pyramid meet at a single point known as the apex or the vertex.
In a pyramid, the base is connected to all the faces of the pyramid. Pyramids are classified into different types, such as triangular pyramids, square pyramids, rectangular pyramids, pentagonal pyramids, hexagonal pyramid, etc., based on the shape of the polygonal base.
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