Among several three-dimensional shapes studied in geometry, the pyramids are quite commonly used. You would have heard about great pyramids of Egypt. You can also observe many objects in your surrounding in the shape of a pyramid.

A pyramid is a special figure which is made up of polygons. It has one polygonal surface that is horizontal and is a known as the base of the pyramid. Other surfaces are known as lateral surfaces or side surfaces which are all triangles. They meet at one point called apex of the pyramid. A pyramid is named after the shape of its base. The triangular pyramid (triangle base), square pyramid (square base) and hexagonal pyramid (hexagonal base) are shown in the diagram below :

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The lateral surface area, as the name suggests, of any solid shape is the area of all the lateral faces or side faces. Note that the area of the base is not included in the lateral surface area. In a similar way, in case of a pyramid, the lateral surface area is the area of all the lateral triangles excluding the area of the base. On the other hand, the surface area that includes the area of base is termed as the total surface area which is expressed as the sum of the lateral surface area and area of the base. But we are going to focus only on the lateral surface area on this page.

A pyramid in which height (length of perpendicular drawn from apex to the base) touches the base exactly in the center, is called a right pyramid. While, the pyramid in which height does not meet base in the center is termed as an oblique pyramid whose lateral surface area can be calculated by just finding the area of all lateral surfaces individually and then adding them. On the other hand, one may use a formula to find the lateral surface area of a right pyramid. The formula for calculating lateral surface area of a pyramid is written by :

In short, we may write it as

Note that the slant height is the height of lateral triangular face in a right pyramid. The height and slant height are shown in the image below.

Once you are aware of the formula, finding the lateral surface area of a pyramid becomes an easy task. It can be done in the following simple steps.

**Step 1 :** Read the question carefully and write down the slant height of pyramid denoting by l. Mention its unit as that of length.

**Step 2 :** Calculate the perimeter of the base by adding all the sides together, denote it by P. Its unit is also the unit of length.

**Step 3 :** Substitute both values in the formula of lateral surface area LSA = $\frac{1}{2}$ P l and calculate the required value of the lateral surface area. Express the answer with appropriate unit of area, i.e. unit$^{2}$.
Read the following solved examples to have a better understanding.

**Example 1 :** In a pentagonal pyramid, each side of the base is 5 cm and slant height is 8.4 cm. Find its lateral surface area.

Solution : Each side of pentagon = 5 cm

Perimeter of base = Perimeter of pentagon

P = 5 $\times$ 5 = 25 cm

Slant height, l = 8.4 cm

Lateral surface area = $\frac{1}{2}$ P l

= $\frac{1}{2}$ $\times$ 25 $\times$ 8.4

= 25 $\times$ 4.2

= 105 cm$^{2}$**Example 2 :** 90 square feet canvas is used to make a tent in the shape of a square pyramid. If the slant height of tent 12 feet, then calculate the length of the side of the square base.

**Solution :** The area of canvas used in making tent is equal to the lateral surface area.

Therefore, LSA = 90 sq ft

Slant height, l = 12 ft

The formula is

180 = P $\times$ 12

P = $\frac{180}{12}$

P = 15 cm

A pyramid in which height (length of perpendicular drawn from apex to the base) touches the base exactly in the center, is called a right pyramid. While, the pyramid in which height does not meet base in the center is termed as an oblique pyramid whose lateral surface area can be calculated by just finding the area of all lateral surfaces individually and then adding them. On the other hand, one may use a formula to find the lateral surface area of a right pyramid. The formula for calculating lateral surface area of a pyramid is written by :

Lateral Surface Area = $\frac{1}{2}$ Perimeter of base $\times$ Slant height

In short, we may write it as

LSA = $\frac{1}{2}$ P l

Note that the slant height is the height of lateral triangular face in a right pyramid. The height and slant height are shown in the image below.

Once you are aware of the formula, finding the lateral surface area of a pyramid becomes an easy task. It can be done in the following simple steps.

Solution :

Perimeter of base = Perimeter of pentagon

P = 5 $\times$ 5 = 25 cm

Slant height, l = 8.4 cm

Lateral surface area = $\frac{1}{2}$ P l

= $\frac{1}{2}$ $\times$ 25 $\times$ 8.4

= 25 $\times$ 4.2

= 105 cm$^{2}$

Therefore, LSA = 90 sq ft

Slant height, l = 12 ft

The formula is

LSA = $\frac{1}{2}$ P l

90 = $\frac{1}{2}$ P $\times$ 12

180 = P $\times$ 12

P = $\frac{180}{12}$

P = 15 cm

Length of each side of square base = $\frac{15}{4}$ = 3.75 cm

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