A square pyramid is a three-dimensional geometric shape with a square base and four triangular faces that meet at a single point called the apex. Several formulas are associated with a square pyramid, including those for the surface area, volume, and slant height.
In this article, we will learn about the square pyramid definition, related formulas, examples and others in detail.
What is a Square Pyramid?
A square pyramid is also known as a pentahedron as it has five faces, including a square base and four triangular faces that meet at a point at the top called the apex. The Great Pyramid of Giza is the best example of a square pyramid. A square pyramid is a pyramid that has a square base and four triangular faces (lateral faces). It has five (5) faces four (4) triangular faces, a square base, five (5) vertices, and eight (8) edges.
Square pyramids can be distinguished depending on the lengths of their edges, the position of the apex, and so on. The different types of square pyramids are equilateral square pyramids, right square pyramids, and oblique square pyramids.
- Equilateral Square Pyramid: An equilateral square pyramid is a square pyramid where all the triangular faces of a square pyramid have equal edges.
- Right Square Pyramid: A right square pyramid is a square pyramid with the apex just above the centre of the base, such that a straight line from the apex cuts the base perpendicularly.
- Oblique Square Pyramid: An oblique square pyramid is a square pyramid where the apex is not aligned right above the centre of the base.
Square PyramidThere are two types of surface areas, namely, the lateral surface area and the total surface area. A pyramid's total surface area is calculated by adding the areas of its base, side faces, and lateral surfaces, while the lateral surface area of a pyramid is calculated by adding the sum of its lateral surfaces, or side faces.
Regular Square PyramidSurface Area of Square Pyramid
Surface Area of Square Pyramid is calculated using the formula:
Lateral Surface Area of Regular Square Pyramid (LSA)= 2al square units
Total Surface Area of Regular Square Pyramid(TSA) = 2al + a2 square units
where,
- "a" is the base edge
- "l" is the slant height
Slant height of the pyramid (l) = √[(a/2)2 + h2]
Now,
Lateral Surface Area of Square Pyramid (LSA)= 2a√(a2/4 + h2) square units
Total Surface Area of Square Pyramid (TSA) = a2 + 2a√(a2/4 + h2) square units
Volume of Regular Square Pyramid
Formula to determine the volume of a pyramid is given as,
Volume of a Pyramid = 1/3×Ah cubic units
where,
- h is the height of the Pyramid
- A is the area of the Base
Here, as the base of the pyramid is a square,
Base Area = a2
Now,
Volume of Regular Square Pyramid (V)= (1/3)a2h cubic units
where,
- "a" is the Base Edge
- "h" is the Height of Pyramid
Also Check:
Example 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,
- Side of the square base (a) = 20 inches
- Slant height, l = 25 inches
Perimeter of the square base (P) = 4a = 4(20) = 80 inches
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.
Example 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
- Lateral surface area of a regular square pyramid = 192 sq. cm
Slant height (l) = ?
We know that,
Lateral surface area of a regular square pyramid = (½) Pl
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.
Example 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.
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.
Example 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,
- Side of the square base (a) = 7 cm
- Slant height, l = 12 cm
Perimeter of the square base (P) = 4a = 4(7) = 28 cm
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.
Example 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,
- Side of the square base (a) = 12 cm
- Volume = 720 cu. in
Height (H) =?
We know that,
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.
Example 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
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.
Example 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,
- Side of the square base (a) = 15 units
- Slant height, l = 22 units
We know that,
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.