Practise Problems based on Regular Square Pyramid Formula
Problem 1: Calculate a square pyramid’s total surface area if the base’s side length is 20 inches and the pyramid’s slant height is 25 inches.
Solution:
Given,
The side of the square base (a) = 20 inches, and
Slant height, l = 25 inches
The perimeter of the square base (P) = 4a = 4(20) = 80 inches
The lateral surface area of a regular square pyramid = (½) Pl
LSA = (½ ) × (80) × 25 = 1000 sq. in
Now, the total surface area = Area of the base + LSA
= a2 + LSA
= (20)2 + 1000
= 400 + 1000 = 1400 sq. in
Hence, the total surface area of the given pyramid is 1400 sq. in.
Problem 2: Calculate the slant height of the regular square pyramid if its lateral surface area is 192 sq. cm and the side length of the base is 8 cm.
Solution:
Given data,
Length of the side of the base (a) = 8 cm
The lateral surface area of a regular square pyramid = 192 sq. cm
Slant height (l) = ?
We know that,
The lateral surface area of a regular square pyramid = (½) Pl
The perimeter of the square base (P) = 4a = 4(8) = 32 cm
⇒ 192 = ½ × 32 × l
⇒ l = 12 cm
Hence, the slant height of the square pyramid is 12 cm.
Problem 3: What is the volume of a regular square pyramid if the sides of a base are 10 cm each and the height of the pyramid is 15 cm?
Solution:
Given data,
Length of the side of the base (a)= 10 cm
Height of the pyramid (h) = 15 cm.
The volume of a regular square pyramid (V) = 1/3 × Area of square base × Height
Area of square base = a2 = (10)2 = 100 sq. cm
V = 1/3 × (100) ×15 = 500 cu. cm
Hence, the volume of the given square pyramid is 500 cu. cm.
Problem 4: Calculate the lateral surface area of a regular square pyramid if the side length of the base is 7 cm and the pyramid’s slant height is 12 cm.
Solution:
Given,
The side of the square base (a) = 7 cm
Slant height, l = 12 cm
The perimeter of the square base (P) = 4a = 4(7) = 28 cm
The lateral surface area of a regular square pyramid = (½) Pl
LSA = (½ ) × (28) × 12 = 168 sq. cm
Hence, the lateral surface area of the given pyramid is 168 sq. cm.
Problem 5: Calculate the height of the regular square pyramid if its volume is 720 cu. in. and the side length of the base is 12 inches.
Solution:
Given,
The side of the square base (a) = 12 cm
Volume = 720 cu. in
Height (H) =?
We know that,
The volume of a regular square pyramid (V) = 1/3 × Area of square base × Height
Area of square base = a2 = (12)2 = 144 sq. in
⇒ 720 = 1/3 × 144 × H
⇒ 48H = 720
⇒ H = 720/48 = 15 inches
Hence, the height of the square pyramid is 15 inches.
Problem 6: Calculate the volume of a regular square pyramid if the base’s side length is 8 inches and the pyramid’s height is 14 inches.
Solution:
Given data,
Length of the side of the base of a square pyramid = 8 inches
Height of the pyramid = 14 inches.
The volume of a regular square pyramid (V) = (1/3)a2h cubic units
V = (1/3) × (8)2 ×14
= (1/3) × 64 × 14
= 298.67 cu. in
Hence, the volume of the given square pyramid is 298.67 cu. in.
Problem 7: Find the surface area of a regular square pyramid if the base’s side length is 15 units and the pyramid’s slant height is 22 units.
Solution:
Given,
The side of the square base (a) = 15 units, and
Slant height, l = 22 units.
We know that,
The total surface area of a regular square pyramid (TSA) = 2al + a2 square units
= 2 × 15 × 22 + (15)2
= 660 + 225= 885 sq. units
Hence, the total surface area of the given pyramid is 885 sq. units.
Square Pyramid Formula
In geometry, a pyramid is a three-dimensional shape with a polygonal base and three or more triangular faces that meet at a common point above the base, known as the apex or vertex. A pyramid is a polyhedron that is classified according to the shape of its polygonal base, such as
The apex is the meeting point of a pyramid’s lateral surfaces or side faces. The perpendicular distance from the center of the base to the apex is called the height of a pyramid, while the slant height of a pyramid is defined as the perpendicular distance between the apex and the base of a lateral surface.
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