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# Surface Area of a Pyramid(Definition, Formula & Examples)

September 24, 2022

You must have heard about ‘The Great Pyramid of Giza’, which is structured in the same concept. Every corner of this structure is linked to a single apex which makes it appear as a distinct shape. A pyramid is a 3D shape whose base is a polygon and whose side faces (that are triangles) meet at a point which is called the apex (or) vertex.

Let’s learn how to find the surface area of a pyramid and its methods and formulas.

## Pyramid – A 3D Solid Shape

A pyramid is a three-dimensional shape. A pyramid has a polygonal base and flat triangular faces, which join at a common point called the apex. A pyramid is formed by connecting the bases to an apex. Each edge of the base is connected to the apex and forms the triangular face, called the lateral face. If a pyramid has an $n$-sided base, then it has $n+1$ faces, $n+1$ vertices, and $2n$ edges.

## Types of Pyramids

The pyramids are classified depending on the following factors

• type of polygon, of the base
• alignment of the base
• shape of the base

A pyramid based on the type of the polygon of the base can be of the following two types.

• Regular Pyramid: If the base of the pyramid is in the shape of a regular polygon, the pyramid is a regular pyramid.
• Irregular Pyramid: If the base of the pyramid is in the shape of an irregular polygon, the pyramid is an irregular pyramid.

The location of the apex or the top of a pyramid decides whether a pyramid is a right pyramid or an oblique pyramid.

• Right Pyramid: A pyramid is named a right pyramid when the location of the apex is exactly over the middle of the base of the pyramid. In other words, when a perpendicular line from the apex intersects the centre of the base, it is a right pyramid.
• Oblique Pyramid: When the location of the apex is not exactly over the middle but slightly away, then that pyramid is called an oblique pyramid. When it does not intersect the center of the base, it is an oblique pyramid

There are different types of pyramids based on the shape of their base.

• Triangular Pyramid: If the base of a pyramid is in the shape of a triangle, it is said to be a triangular pyramid. A triangular pyramid has $6$ edges, $4$ vertices, and $4$ faces. This kind of pyramid can also be called a tetrahedron.
• Square Pyramid: A square pyramid is formed when the base of the pyramid is in the shape of a square. A square pyramid consists of one square base and three triangular faces. In other words, it has $8$ edges, $5$ vertices, and $5$ faces.
• Rectangular Pyramid: A rectangular pyramid is formed when the base of the pyramid is in the shape of a rectangle. A rectangular pyramid consists of one rectangular base and three triangular faces. In other words, it has $8$ edges, $5$ vertices, and $5$ faces.
• Pentagonal Pyramid: A pentagonal pyramid is one that has its base shaped like a pentagon, with the rest of the faces as triangles. This pyramid has $6$ vertices, $10$ edges, and $6$ faces.

## What is the Surface Area of a Pyramid?

The area occupied by the surface/boundary of a pyramid is known as the surface area of a pyramid. It is always measured in square units such as ($cm^{2}$, $m^{2}$, $in^{2}$, $ft^{2}$, etc.).

Since a pyramid has one base and $n$ number of triangular sides depending on the type of pyramid, there are two types of surface areas of a pyramid

• Lateral Surface Area
• Total Surface Area

### Lateral Surface Area of Pyramid

The lateral surface area of a pyramid is the area of its vertical faces(or lateral faces, which are triangles). Thus, the lateral surface area of a pyramid = $\frac {1}{2}pl$, where $p$ is the perimeter of a triangular face and $l$ is the slant height of the pyramid.

Consider the pyramid $ABCD$ as shown above.

The lateral faces of the pyramid are $\triangle ABC$, $\triangle ACD$, and $\triangle ABD$.

Lateral surface area of the pyramid = $\left( \text{Area of} \triangle ABC \right) + \left( \text{Area of} \triangle ACD \right) + \left( \text{Area of} \triangle ABD \right)$

The area of any triangle in the above figure is $\frac {1}{2}lh$, where $l$ is the length of the edge(or side) of a triangle and $h$ is the height of the triangle.

Therefore, sum of area of all the three triangles = $3 \times \frac {1}{2}lh = 3l \times \frac {1}{2}h = p \times \frac {1}{2}h = \frac {1}{2}ph$, where $p = 3l$ is the perimeter of a triangle.

### Total Surface Area of Pyramid

The total surface area of a pyramid is the sum of all the faces of a pyramid. There are two types of faces in a pyramid.

• Base (triangular in shape)
• Lateral faces (triangular in shape)

Area of Base (Polygon) = $\frac {nl^{2}}{4 \tan \frac {180}{n}}$,

where $l$ is the length of the edge(side) of a polygon

$n$ is the number of edges(sides) of a polygon

Total surface area of the pyramid = LSA + Area of Base = $\frac {1}{2}pl + \frac {nl^{2}}{4 \tan \frac {180}{n}} = \frac {1}{2}l \left(p + \frac {nl}{2 \tan \frac {180}{n}} \right)$.

Ex 1: What is the lateral surface area of a pyramid with a square base length of $15 m$ and a slant height (the height from the midpoint of one of the side lengths to the top of the pyramid) of $12 m? Length of the edge(side)$l =15 m$Height of the pyramid$h = 12 m$Number of sides of the base$n = 4$(Square base) Perimeter of a triangular face =$p = 3 \times 15 = 45 m$Lateral surface area =$\frac {1}{2}ph = \frac {1}{2} \times 45 \times 12 = 270 m^{2}$. Ex 2: Find the total surface area of the following pyramid. It’s a square pyramid with length of base$s = 10 m$Area of base =$10^{2} = 100 m^{2}$. Let the height of each triangle be$h$. Therefore,$h = \sqrt{13^{2} – 5^{2}} = 12 m$Perimeter$p = 10 \times 4 = 40 m$Lateral surface area =$\frac {1}{2}ph = \frac {1}{2} \times 40 \times 12 = 240 m^{2}$Therefore, total surface area of the pyramid is$100 + 240 = 340 m^{2}$. Ex 3: Find the total surface area of the following pyramid. It’s a square pyramid with the length of base$s = 37 cm$Area of base =$37^{2} = 1369 cm^{2}$. Height of pyramid$h = 44 cm$Perimeter$p = 37 \times 4 = 148 cm$Lateral surface area =$\frac {1}{2}ph = \frac {1}{2} \times 148 \times 44 = 3256 cm^{2}$Therefore, total surface area of the pyramid is$1369 + 3256 = 4625 cm^{2}$. ## Conclusion The surface area of a 3D shape (solid object) is a measure of the total area of the object’s surface. There are two types of surface areas in a pyramid, viz., lateral surface area calculated using the formula$\frac {1}{2}ph$and total surface area calculated by finding the sum of the lateral surface area and area of the base of the pyramid. ## Practice Problems 1. Determine the total surface area of a pyramid with a slant height of$15 ft$and a square base of the side of$10 ft$. 2. Find the total surface area of the given regular pyramid. 3. A square pyramid has a lateral surface area of$390 cm^{2}$and a slant height of$13 cm$. Determine the length of each side of its base. 4. In a square pyramid with a base side of$30 cm$, the distance between the apex and any base vertex is$56 cm$. Determine the total surface area of the pyramid. ## Recommended Reading ## FAQs ### How do you find the surface area of pyramids? A pyramid has a base of$n$-sided polygon and$n$triangular lateral faces. There are two types of surface areas of a pyramid. a) Lateral Surface Area: It is the sum of areas of all the lateral triangular faces. The formula to calculate LSA is$\frac {1}{2}ph$, where$p$is the perimeter of the triangular face and$h$is the height of the pyramid. b) Total Surface Area: It is the sum of LSA and the area of the polygonal base of the pyramid. ### What is the total surface area of a pyramid? The total surface area of a pyramid is obtained by adding the area of all its faces (both the base and the side faces). The total surface area of a pyramid whose base perimeter is$p$, the base area is$b$, and slant height is$l$is calculated using the formula TSA =$\frac {1}{2} pl + b$. ### What is the lateral surface area of a pyramid? The lateral surface area of a pyramid is the sum of the areas of all its side faces (which are triangles). The lateral surface area of a pyramid is calculated using the formula LSA =$\frac {1}{2} pl$, where$p$is the perimeter of the base and$l\$ is the slant height.

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