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Pyramid Surface Area Calculator

Introduction

Welcome to our Pyramid Surface Area Calculator. A Pyramid is a three-dimensional shape with a polygonal base, and each side of the base is connected to the apex by triangular faces.

The triangular faces are called lateral faces, and the number of lateral faces depends on the number of edges of the base polygon.

Some real-life examples of pyramids include The Great Pyramids of Giza in Egypt, Tents built for camping, etc.

The lateral surface area of the pyramid is the sum of the areas of all the lateral faces. Subsequently, the total surface area of the pyramid is the sum of the lateral surface area and the areas of the base.

How to use the Pyramid Surface Area Calculator?

Using the Pyramid Surface Area calculator, you can calculate the pyramid’s lateral surface area and total surface area by inputting the values of the required variables for the specific type of Pyramid.

Triangular Pyramid Surface Area Calculator

You can calculate the lateral surface area and the total surface area of the triangular pyramid by inputting the required value for the calculation.

The variables in the pyramid surface area calculator include

Side of Base Triangle (a) The length of the base triangle of the pyramid

Height of Base Triangle (b) The length of the altitude or height of the base triangle

Slant Height (s) The length of height of the lateral face of the pyramid

Height (h) The length of the height of the Pyramid

Lateral Surface Area (LSA) The lateral surface area is the area of the lateral faces of the pyramid

Total Surface Area (TSA) The total surface area of the pyramid is the sum of the lateral surface area and the area of the base.

Square Pyramid Surface Area Calculator

Side (a) The length of the side of the square base

Slant Height (s) The length of height of the lateral face of the pyramid

Height (h) The length of the height of the Pyramid

Lateral Surface Area (LSA) The lateral surface area is the area of the lateral faces of the pyramid

Total Surface Area (TSA) The total surface area of the pyramid is the sum of the lateral surface area and the area of the base.

Pentagonal Pyramid Surface Area Calculator

Side (a) The length of the side of the pentagonal base

Height (h) The length of the height of the Pyramid

Apothem (b)

The distance from the center to the midpoint of a side of the base

Lateral Surface Area (LSA) The lateral surface area is the area of the lateral faces of the pyramid

Total Surface Area (TSA) The total surface area of the pyramid is the sum of the lateral surface area and the area of the base.

What is a Pyramid?

A pyramid is a polyhedron where the base is a Polygon, and the faces on the sides, the lateral faces, are all triangles. The lateral face connects each side of the base to a common point called the apex.

A pyramid can be classified into multiple types based on various parameters. Let’s look at some of the ways we can classify pyramids.

Classification of Pyramids

Firstly, Based on the Congruency of Lateral Sides

  1. Right Pyramid: In a right pyramid, the lateral faces are all congruent, meaning they are all the same shape and size. The point at which the lateral faces meet, called the apex, is directly above the center of the base.

  2. Oblique Pyramid: In an oblique pyramid, the lateral faces are not congruent, and the apex does not align with the center of the base.

Secondly, based on the Shape of the Base

  1. Triangular Pyramid: In a triangular pyramid, the pyramid’s base will be in the shape of a triangle. The triangular pyramid, also called a tetrahedron, has four faces, six edges, and four vertices.

  2. Square Pyramid: In a square pyramid, the base will be a square. A square pyramid comprises five faces, eight edges, and five vertices.

  3. Pentagonal Pyramid: in a pentagonal pyramid, the base will be a Pentagon, and the pentagonal pyramid has five faces, ten edges, and six vertices.

Thirdly, based on the Type of Polygon of the Base

  1. Regular Pyramid: If the pyramid’s base is a regular polygon, then the pyramid will be a regular pyramid.

  2. Irregular Pyramid: If the pyramid’s base is an irregular polygon, then the pyramid will also be irregular.

How is the Surface Area of the Pyramid Calculated?

Lateral Surface Area (LSA)

The lateral surface area is the surface area of the pyramid without including the area of the base. The generic formula for the lateral surface area of the pyramid is given by

LSA=12×Perimeter of the Base×Slant Height\text{LSA} = \normalsize \dfrac{1}{2} \times \text{Perimeter of the Base} \times \text{Slant Height}

Total Surface Area (TSA)

The total surface area of the pyramid is the sum of the lateral surface area and the area of the base. The generic formula for the total surface area of the pyramid is given by

TSA=(12×Perimeter of the Base×Slant Height)+Base Area\text{TSA} = \normalsize \bigg(\dfrac{1}{2} \times \text{Perimeter of the Base} \times \text{Slant Height} \bigg) + \text{Base Area}

Triangular Pyramid Lateral and Total Surface Area

The lateral surface area of the triangular pyramid is calculated using the following formula

LSA=32×b×s\text{LSA} = \normalsize \dfrac{3}{2} \times \text{b} \times \text{s}

Where,

b = Length of the side of the base

s = Slant height of the pyramid

The total surface area of the triangular pyramid is calculated using the following formula.

TSA=12(a×b)+32×a×s\text{TSA} = \normalsize \dfrac{1}{2}(a \times b) + \dfrac{3}{2} \times \text{a} \times \text{s}

a = length of the side of the base triangle

b = height of the base triangle

s = Slant Height of the pyramid

Square Pyramid Lateral and Total Surface Area

The lateral and total surface area of the square pyramid is calculated in two ways one using the height of the pyramid and another using the slant height of the pyramid

Using Height of Pyramid

Lateral Surface Area

LSA=aa2+4h2\text{LSA} = a\sqrt{a^2 + 4h^2}

Where,

a = length of the side of the base square

h = height/altitude of the pyramid

Total Surface Area

TSA=a2+aa2+4h2\text{TSA} = a^2 + a\sqrt{a^2 + 4h^2}

Where,

a = length of the side of the base square

h = height/altitude of the pyramid

Using Slant Height of the Pyramid

Lateral Surface Area

LSA=2as\text{LSA} = 2as

Where,

a = length of the side of the base square

s = Slant height of the pyramid

Total Surface Area

TSA=a2+2as\text{TSA} = a^2 + 2as

Where,

a = length of the side of the base square

s = Slant height of the pyramid

Pentagonal Pyramid Lateral and Total Surface Area

The lateral surface area of the pentagonal pyramid can be calculated in two ways, one using the apothem and slant height of the pyramid and another using the height of the pyramid

Using Apothem and Slant Height

Lateral Surface Area

LSA=52as\text{LSA} = \normalsize \dfrac{5}{2}as

Where,

a = length of the side of the base pentagon

s = Slant height of the pyramid

Total Surface Area

TSA=52×a×(b+s)\text{TSA} = \normalsize \dfrac{5}{2}\times a\times (b + s)

Where,

a = length of the side of the base pentagon

b = Apothem of the base pentagon

s = Slant height of the pyramid

Using Height of the Pyramid

Lateral Surface Area

LSA=5ah2+(atan(54°)2)22\text{LSA} = \normalsize 5a \dfrac{\sqrt{h^2 + \bigg(a\dfrac{tan(54\degree)}{2}\bigg)^2}}{2}

Where,

a = length of the side of the base pentagon

h = height of the pyramid

Total Surface Area

TSA=54tan(54°)a2+5ah2+(atan(54°)2)22\text{TSA} = \normalsize \dfrac{5}{4}tan(54\degree) a^2 + 5a \dfrac{\sqrt{h^2 + \bigg(a\dfrac{tan(54\degree)}{2}\bigg)^2}}{2}

Where,

a = length of the side of the base pentagon

h = height of the pyramid

Examples

Given a right pentagonal pyramid with a base length of 10 cm and height of 15 cm. What are the pyramid’s lateral surface area and total surface area?

The lateral surface area of the pyramid can be calculated using the following formula

LSA=5ah2+(atan(54°)2)22=412.58  cm2\begin{aligned} \text{LSA} &= \normalsize 5a \dfrac{\sqrt{h^2 + \bigg(a\dfrac{tan(54\degree)}{2}\bigg)^2}}{2} \\[10pt] &= 412.58 \; cm^2 \end{aligned}

The total surface area can be calculated using the following formula

TSA=54tan(54°)102+5⋅10152+(10tan(54°)2)22=584.63  cm2\begin{aligned} \text{TSA} &= \normalsize \dfrac{5}{4}tan(54\degree) 10^2 + 5 \cdot 10 \dfrac{\sqrt{15^2 + \bigg(10\dfrac{tan(54\degree)}{2}\bigg)^2}}{2} \\[10pt] &= 584.63 \; cm^2 \end{aligned}
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