How to Get the Lateral Area of a Pentagonal Pyramid

A pentagonal pyramid has a pentagon as its base.
••• Images

The lateral area of a solid is defined as the combined area of all of its lateral faces. The lateral faces are the sides of the solid excluding the base and top. For a pentagonal pyramid, the lateral area is the combined area of the five triangular sides of the pyramid. To calculate this, you must find the areas of the triangular sides and add them together.

Area of a Triangle

Each of the sides of a pentagonal pyramid is a triangle. Therefore, the area of one of the sides is equal to one-half the base of the triangle times its height. When you add up the area of each of the triangular sides of the pentagonal pyramid, you will get the total lateral area of the pyramid.

Set Up Your Equation

The height of each of the triangle sides of a pyramid is known as the slant height. The slant height of a side is the distance from the apex of the pyramid to the midpoint of one of the sides of the base. Therefore, the formula for the lateral area of the pentagonal pyramid is 1/2 x base one x slant height one + 1/2 x base two x slant height two + 1/2 x base three x slant height three + 1/2 x base four x slant height four + 1/2 x base five x slant height five. If all of the triangular faces of the pentagonal pyramid are identical, this formula can be simplified to 5/2 x base x slant height. Because all of the bases combine to equal the perimeter of the pentagon, you could represent the formula as 1/2 x perimeter of pentagon x slant height.

Finding the Slant Height

If you are not given the slant height of the pyramid, you must find it by considering the various triangles that exist within the solid. For example, in a right pentagonal pyramid, the apex of the pyramid is above the center of its base. This creates a right triangle with a base between the center of the pentagon and the midpoint of one of its sides, a height between the center of the pentagon and the apex of the pyramid and a hypotenuse equal to the slant height. Because of this arrangement, you can use the Pythagorean Theorem to determine the slant height.

Regular Vs. Irregular Pyramids

If the base of the pentagonal pyramid is a regular pentagon, this means that all of the sides of the base are identical, as are the angles between the sides. If the base of the pyramid is not a regular pentagon, each of its triangular faces may be different. Depending on the location of the apex of the pyramid, this may mean that each triangle's area is different. In this case, the formula may not simplify to 5/2 x base x slant height. Instead, you must add the area of each one of the sides.