Surface Area of a Pyramid

The surface area of a pyramid is the sum of the areas of the lateral surfaces, or side faces, and the base of a pyramid. The surface area of a pyramid has basically two types of surface areas: the lateral surface area and the total surface area, and they are measured in terms of m², cm², in², ft², etc.

Lateral surface area of a pyramid (LSA) = Sum of areas of the lateral surfaces (triangles) of the pyramid

Total surface area of a pyramid (TSA) = Lateral surface area of the pyramid + Area of the base

To determine the lateral surface of a pyramid, the areas of the side faces (triangles) are calculated and the results are added, whereas the total surface area of a pyramid is calculated by adding the area of the pyramid’s base and the lateral surface area.

Read More about Pyramid.

Surface Area of a Triangular Pyramid

A triangular pyramid is a pyramid having a triangular base, where the triangular base can be equilateral, isosceles, or a scalar triangle. It has three lateral (triangular) faces and a triangular base. 

We know that,

The total surface area of a pyramid (TSA) = Lateral surface area of the pyramid + Area of the base

Lateral surface area (LSA) = ½ × perimeter × slant height

So, TSA = ½ × perimeter × slant height + ½ × base × height

Total surface area (TSA) of a triangular pyramid = ½ ×  P ×  l + ½ bh

Where,

• P is perimeter of base,
• l is slant height of pyramid,
• b is base of base triangle, and
• h is height of pyramid.

Surface Area of a Square Pyramid

A square pyramid is a pyramid having a square base. It has four lateral (triangular) faces and a square base.

We know that,

The total surface area of a pyramid (TSA) = Lateral surface area of the pyramid + Area of the base

The slant height of the pyramid (l) = √[(a/2)² + h²]

LSA = 4 × [½ × a × l] = 2al

Lateral surface area of the square pyramid (LSA)= 2al 

So, TSA = 2al + a²

Total surface area of a square pyramid (TSA) = 2al + a²

Where,

• a is side of square base, and
• l is slant height of pyramid.

Surface Area of a Rectangular Pyramid

A rectangular pyramid is a pyramid having a rectangular base. It has four lateral (triangular) faces and a rectangular base.

We know that,

The total surface area of a pyramid (TSA) = Lateral surface area of the pyramid + Area of the base

The slant height of length face of the pyramid = √[h² + (l/2)²]

The slant height of width face of the pyramid = √[h² + (w/2)²]

Lateral surface area of a rectangular pyramid = 2 × {½ × l ×√[h² + (l/2)²]} + 2 × {½ × w ×√[h² + (w/2)²]

So,

Total surface area of the rectangular pyramid = 2 × {½ × l ×√[h² + (l/2)²]} + 2 × {½ × w ×√[h² + (w/2)²] + l × w  

Where,

• l is length of rectangle base,
• w is breadth of rectangle base, and
• h is height of pyramid.

Surface Area of a Pentagonal Pyramid

A pentagonal pyramid is a pyramid having a pentagonal base. It has five lateral (triangular) faces and a pentagonal base.

We know that,

The total surface area of a pyramid (TSA) = Lateral surface area of the pyramid + Area of the base

Apothem length of the base = a

Side length of the base = s

Slant height of the pyramid = l

Area of the pentagonal base = 5⁄2 (a × s)

Now,

LSA = 5 × [½ × base × height] = 5/2 × s × l

Lateral surface area of the pentagonal pyramid = 5⁄2 (s × l)

Total surface area of the pentagonal pyramid = 5⁄2 (s × l) + 5⁄2 (a × s)

Where,

• l is  slant height of the pyramid,
• a apothem length of the base, and
• s side length of the base.

Surface Area of a Hexagonal Pyramid

A hexagonal pyramid is a pyramid having a hexagonal base. It has six lateral (triangular) faces and a hexagonal base.

We know that,

The total surface area of a pyramid (TSA) = Lateral surface area of the pyramid + Area of the base

Side length of the base = s

Slant height of the pyramid = l

Area of the hexagonal base = 3√3/2 (s)²

Now, 

LSA = The sum of areas of the lateral surfaces (triangles) of the pyramid

⇒ LSA = 6 × [½ × base × height] =3(s × l)

Lateral surface area of the hexagonal pyramid = 3(s × l)

Total surface area of the hexagonal pyramid = 3(s × l) + 3√3/2 (s)²

Where,

• s is side length of the base, and
• l is slant height of the pyramid.

Read More,

• Surface Area of a Square Pyramid
• Total Surface Area of Rectangular Pyramid Formula
• Area of a Pentagonal Pyramid
• Surface Area of a Triangular Pyramid
• Surface Area Formulas for Different Geometrical Figures

Pyramid is a three-dimensional geometric structure with a polygonal base and all its triangular faces meet at a common point called the apex or vertex. The real-life examples of pyramids are the pyramids of Egypt; rooftops; camping tents, etc. Depending upon the shape of the polygonal base, pyramids are classified into different types, such as triangular pyramids, square pyramids, rectangular pyramids, etc. The side faces, or lateral surfaces of a pyramid, are triangles that meet at a common point called the apex. The height, or altitude, of a pyramid, is the perpendicular distance between the apex and the center of the base, whereas a slant height is a perpendicular distance between the apex and the base of a lateral surface.