# Lateral Area

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### Lateral Area

In geometry, a 3-dimensional figure is the object which has 3 dimensional measurements of length, width and height. Using these 3 measurements, various calculations of volume, surface areas are analyzed. Shapes such as polyhedrons, cylinders, cones and spheres are 3-dimensional figures. Polyhedrons are the shapes which have flat surfaces known as faces, and these faces are made of polygons. Examples of polyhedrons are pyramids and prisms. Cylinders, cones, spheres are 3-dimensional but are not polyhedrons as they do not have flat surfaces. They have curved surfaces. Cylinders have 2 congruent base circles connected by a curved surface. A cone is a figure which has a base circle connected to the vertex on top by a curved surface. A sphere is also one such space figure which has all its points equidistant from the center point.

### What is Lateral Area?

Lateral Area is the sum area of all the surfaces of the figure except the base and the top area. That means, lateral area is the sum of area of all the faces or lateral surfaces only. Based on the shape of the figure, the lateral area can be calculated accordingly. Lateral area is measured in square units. For instance, if the dimensions are in ‘meters’, then the unit of lateral area would be square meters.

### Lateral Area of Geometric Shapes:

Lateral area of various geometric shapes can be calculated using the dimensions of that particular shape. For calculating the lateral area, we do not add the areas of the top surface and the bottom surface of the figure. While calculating the Total Surface Area of a figure, we add up the areas of all the surfaces (including the top and the bottom), but for Lateral Area only the areas of the lateral faces need to be added up.

Now let us calculate the Lateral Area of various geometric shapes with different sizes:

### 1) Lateral Area of a Prism:

A prism is a very popular 3-dimensional figure which consists of flat faces and identical bases. The bases are congruent and parallel to each other. All along the length, the prisms have the same cross-section. The prism is a polyhedron, so it does not have any curved sides. Its faces are flat and it has edges (or sides) as straight lines. We can classify different types of prisms based on the cross-section or the base of the prism. If the base or the cross-section of a prism is a square, then it is known as a Square Prism. If the cross-section along the length is a triangle, then it is known as a Triangular Prism.

Lateral area of any prism can be calculated by using the formula as shown below:

Lateral Area of a Prism = (Perimeter of the Base) * (Height of the prism) ==> L = P * h

a) Lateral Area of a Rectangular Prism: A rectangular prism has 6 rectangular faces including the top and the bottom surface. Since the base of the cross-section of the prism is a rectangle, hence it is known as the Rectangular Prism. To calculate the Lateral area of a rectangular prism, we consider only the area of the 4 lateral faces and do not calculate the area of the 2 bases of the prism. The perimeter of the base of a rectangular prism is nothing but the perimeter of the base rectangle. The perimeter of a rectangle is the sum of all its side lengths. This implies Perimeter of a rectangle, P = 2l +2w (where l = length and w = width of the rectangle).

Hence the Lateral Area of a Rectangular prism can now also be written as:
Lateral Area of a Rectangular Prism = Perimeter of the Base * Height of the Prism
L = P * h
L = (2l + 2w) * h
Or L = 2lh + 2wh (where l = length, w = width, h = height)

Example: Calculate the lateral area of a rectangular prism if given that the length is 6m, width is 5m, and height is 8m.

Given that length l = 6m, width w = 5m and height, h = 8m.
Lateral area of the rectangular prism, L = Perimeter of the base * Height
L = 2lh + 2wh
==> L = (2* 6* 8) + (2* 5 * 8)
==> L = 96 + 80 = 176m
Hence, the Lateral Area, L = 176 square meters.

b) Lateral area of a Triangular Prism:A triangular prism is a prism whose base of the prism (or the cross-section along the length) is a triangle. If the sides of the base triangle are ‘a’, ‘b’ and ‘c’, then the Perimeter of a triangle is the sum of all its sides = (a + b + c).

Lateral Area of a Triangular Prism = (Perimeter of the base triangle) * (Height of the Prism) L = (a + b + c) * h

Example: What is the lateral area of a triangular prism whose height is 12cm and which has a base triangle of side length 6cm, 4cm and 5cm?

Given height of the prism, h = 12cm
The side lengths of the base triangle are a = 6cm, b =4cm and c = 5cm.
Lateral Area of a Triangular Prism = (Perimeter of the base triangle) * (Height of the Prism)
L = (a + b + c) * h
Hence, L = (6cm + 4cm + 5cm) * 12cm ==> L = 180 square centimeters.

c) Lateral area of a Regular Hexagonal Prism:A hexagonal prism is a prism whose base of the prism (or the cross-section along the length) is a hexagon. A hexagon is a polygon with 6 sides. A hexagonal prism consists of 2 identical hexagonal bases and 4 rectangular faces. A regular hexagon is a polygon which has 6 equal sides.

If the side length of the base regular hexagon is ‘s’, then the perimeter of the base hexagon is the sum of all its sides = s + s + s + s + s + s = 6s.

Lateral area of a Hexagonal Prism, L = (Perimeter of the base regular hexagon) * (Height of the Prism) L = (6* s) * h

Example: If the height of the prism is 10cm and the base is regular hexagon of side length 4cm, then what is the perimeter of this hexagonal prism?
Given height of the prism, h = 10cm
The side length of the base regular hexagon, s = 4cm
Lateral area of a Hexagonal Prism, L = (Perimeter of the base hexagon) * (Height of the Prism)
L = (6* s) * h
Hence, L = (6* 4cm) * 10cm ==> L = 240 square centimeters.

2)      Lateral area of a Pyramid:
A pyramid is a 3-dimensional figure whose base is a polygon and has triangular faces meeting at the top vertex (also known as the ‘apex’). Lateral area of a pyramid is the sum of the areas of the lateral faces of the pyramid structure, without including the area of the base. Just like a prism, there are different types of pyramids based on the shape of its base. If the base of the pyramid is a triangle, then it is known as a Triangular Pyramid. If the base of the pyramid is a rectangle, then it is known as the Rectangular Pyramid.

If the base polygon is a regular polygon, then we get a regular pyramid. If the base polygon is an irregular polygon, then the pyramid formed is an irregular pyramid.

Lateral Area of a Regular Pyramid = 1/2 * (Perimeter of the base) * (Slant height of the pyramid)
(Note: Slant height is the perpendicular altitude drawn from the apex (vertex) to the base of the lateral triangle as shown in the above figure).

Lateral Area of an Irregular Pyramid = Sum of the areas of each lateral triangular faces

a)      Lateral Area of a Square Pyramid:
A square pyramid is a pyramid which has a square base.

If the side length of the square base is ‘a, then the perimeter of the square base = 4 * a
Let the slant height which is the perpendicular distance drawn from the apex to the base of the lateral triangle be = s
Lateral Area of a Square Pyramid = 1/2 * (Perimeter of the Square base) * (Slant height of the pyramid) = 1/2 * 4a * s = 2 * a * s
Therefore, Lateral Area of the Square Pyramid = 2 * a * s

Example: Find the lateral area of a square pyramid whose square base has a side length of 5m and its slant height is 9m.
Given side length of the square base of the pyramid, a = 5m
Slant height of the pyramid, s = 9m
Lateral area of the Square Pyramid = 2* a* s = 2* 5m * 9m = 90m2

b)      Lateral Area of a Triangular Pyramid:
A pyramid consisting of a triangular base is known as the Triangular Pyramid. In general cases, the base triangle is an equilateral triangle and therefore it is an equilateral triangular pyramid, also known as the regular triangular pyramid. But in case the base triangle does not have equal sides, then the pyramid is known as the irregular pyramid.

If the side lengths of the base triangle are ‘a’, ‘b’, and ‘c’, then the perimeter of the triangle = (a+ b+ c)
Let the slant height of the pyramid = s
Then, Lateral Area of the Triangular pyramid = 1/2 * (a+ b+ c) * s

Example: Calculate the lateral area of an equilateral triangular pyramid of base side of 6m and slant height of 10m.
Given the side of the base equilateral triangle, a = 6m (Equilateral triangles have equal sides)
Hence, a = b = c = 6m
Slant height of the pyramid, s = 10m
Lateral area of the triangular pyramid = 1/2 * (a+ b+ c) * s ==> L = 1/2 * (6+ 6+ 6) * 10 = 90m2

c)       Lateral area of a Pentagonal Pyramid:
A pyramid consisting of a pentagonal base is known as the pentagonal pyramid. A pentagon is a polygon consisting of 5 sides.

If the base pentagon has side lengths of ‘a’, ‘b’, ‘c’, ‘d’ and ‘e’, then perimeter of the pentagon = sum of all its sides =  (a + b + c + d + e)
Let the slant height of the pyramid = s
Lateral Area of a Pentagonal Pyramid = 1/2 * (Perimeter of the base pentagon) * (Slant height)
So, Lateral area of a Pentagonal Pyramid = 1/2 * (a+ b+ c+ d+ e) * s

Example: Given the side lengths of a regular pentagonal pyramid as 5cm and the slant height of the pyramid as 12cm. What is the lateral area of this regular pentagonal pyramid?
A regular pentagon has 5 equal sides.
Given the side lengths of the base pentagon as a = b = c = d = e = 5cm
Slant height of the pentagonal pyramid, s = 12cm
Lateral area of a Pentagonal Pyramid = 1/2 * (5+ 5+ 5+ 5+ 5) * 12 = 150cm2

2)      Lateral Area of a Cylinder:
Cylinders are commonly observed in our daily life. A cylinder is a 3-dimensional solid closed figure and it consists of 2 congruent circular bases that are connected by a curved surface. A cylinder has 2 congruent circular bases and they are parallel to each other. The perpendicular length between the 2 circular bases is known as the ‘height of the cylinder’ or the ‘altitude’’.

For a given cylinder, let the radius of the circular base = ‘r’
Let the height (or altitude) which is the perpendicular distance between the 2 circular bases = ‘h’
Then the lateral area of the cylinder is given by the equation below:

Lateral Area of a Cylinder = (Circumference of the circular base) * (Height of the cylinder)
Therefore, Lateral Area of a Cylinder = 2 * ???? * r * h

Example: Calculate the lateral area of a cylinder whose radius of the circular base is 6m and the height of the cylinder is 8m.

Given the radius of the circular base, r = 6m
Height of the cylinder, h = 8m
Lateral area of the cylinder = 2 * ???? * r * h ==> L = 2 * 3.14 * 6m * 8m = 301.44 m2

3)      Lateral Area of a Cone:
A cone is a 3-dimensional figure which has a circular base connected with the single vertex on top (also known as the ‘apex’) by a curved surface. The height of the cone is the perpendicular distance from the vertex to the center of the circular base. The slant height is the distance along the surface of the cone from the vertex to the circle, as shown in the figure on the right.

For a given cone, let the radius of the circular base = r
Let the height of the cone = h and the slant height of the cone = s
From the figure we can see that slant height, s = √(h2 + r2) (Using the Pythagorean Theorem)
Then the lateral area of the cone can be calculated as follows:
Lateral area of the cone = ???? * r * s
Therefore, Lateral Area of the Cone = ???? * r * √(h2 + r2)
Example: Calculate the slant height and the lateral area of the cone if the radius of the cone is 6cm and the height of the cone is 8cm.

Given radius of the cone, r = 6cm
Height of the cone, h = 8cm
Slant height, s = √(h2 + r2) = √(62 + 82) = 10cm
Lateral Area of the Cone = ???? * r * √(h2 + r2) = 3.14 * 6 * √(62 + 82) = 188.4 cm2

4)      Lateral Area of a Sphere and Hemisphere:
A sphere is a 3-dimensional geometric figure perfectly symmetrical in shape. It is a closed figure formed by points which are equidistant from the center. A sphere has no edges (sides) or vertices (corners).

If the radius of a sphere is ‘r’, then we can calculate the lateral area as shown below:
Lateral Area of a Sphere = 4* ????* r2

When a sphere is cut into equal halves, then we get a Hemisphere. Therefore, the lateral area of a hemisphere is half of the lateral area of the sphere.
Lateral Area of a Hemisphere = 2* ????* r2

Example: If the radius of a sphere is 5cm, then what is the lateral area of the sphere and the hemisphere?
Given the radius of the sphere, r = 5cm
Lateral Area of the Sphere = 4* ????* r2 ==> L = 4* 3.14* (5cm)2 = 314 cm2
Lateral Area of the Hemisphere = 2* ????* r2 ==> L = 2* 3.14* (5cm)2 = 157 cm2