Students learn about two, three and even higher dimensional objects in geometry. There are several different three dimensional shapes; such as - cube, cuboid cylinder, cone, sphere, prism, hemisphere, pyramid etc. There is a term lateral surface which is explained in context with three dimensional shapes or solids. **Lateral surfaces **are the surfaces on the sides of a solid object. It is also known as **lateral face** or **lateral side**. A cylinder has one lateral side which is curved around the surface. On the other hand, a prism has the same number of lateral sides as number of sides of its base. The cube and cuboid have four lateral sides in each.

The area of all the lateral sides of a three-dimensional object is termed as **lateral area** or **lateral surface area**. The lateral area of a solid is defined to be the sum of
the surface areas of all its faces excluding the area of the base. In other words,
lateral area is the area of the sides without the top and bottom.
In lateral surface area, the areas on top and the bottom is not included. Lateral area is measured in terms of square units. In this page, we are going to focus on the lateral area of solids. Go ahead with us learn about lateral areas of different 3D shapes and figures.

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Lateral area can be found for any solid object around its outer area. The lateral area of a solid is equal to the sum of the area of the faces.

### Lateral Area of a Cylinder Formula

The lateral area of a right cylinder with altitude, h, and circumference, 2$\pi$ r, of the base is given by:

Lateral surface area of the cylinder (LSA) = 2 $\pi$ r h square unit### Total Surface Area of a Right Cylinder

### Lateral Area of a Cone Formula

**Lateral Area of a Cone Formula (L.S.A) = $\pi$ r s square unit**

### Lateral Area of a Pyramid Formula

The lateral area of a pyramid can be calculated by multiplying half of the perimeter of the base by the slant height.

Lateral area of a regular pyramid = $\frac{1}{2}$ Ps sq.units

P - Perimeter of the base

s - Slant height (length of the altitude of a lateral face of the pyramid)

### Lateral Area of a Square Pyramid

A pyramid with square base is called as square pyramid.

If 'a' be the side of a square, then perimeter of the base = 4a and

Lateral area = $\frac{1}{2}$ * 4a * s = 2as.

Lateral area of a square pyramid = 2as square units

where, a - base side length

s - slant height of the square pyramid

### Lateral Area of a Triangular Pyramid

A pyramid with triangular base is called as triangular pyramid. If triangular pyramid is regular with base side length 'a', then perimeter of the base is 3a and

Lateral area = $\frac{1}{2}$ * 3a * s = $\frac{3}{2}$ * a * s

Lateral area of a regular triangular pyramid = $\frac{3}{2}$ a s sq.units

where, a - base side length and s - slant height.

### Lateral Area of a Pentagonal Pyramid

A pyramid with pentagonal bases is a pentagonal prism. If a, b, c, d, e are the sides of the pentagonal base then its perimeter is a + b + c + d + e.

Lateral area formula = $\frac{1}{2}$ Ps = $\frac{1}{2}$ (a + b + c + d + e)s

Lateral area of a pentagonal pyramid formula = $\frac{1}{2}$ (a + b + c + d + e)s square units

where, a, b, c, d, e are the sides of an pentagon and s - slant height of the pyramid. The lateral area of any prism is equal to the perimeter of the base times the height of the prism.

###

Lateral Area of a Prism Formula

Lateral area of a right prism = Ph

Where, P - Perimeter of the base and h - height of the prism.

### Lateral Area of a Rectangular Prism

A prism with rectangular bases is a rectangular prism. A rectangular prism has two bases and four sides.

Lateral Surface Area of Rectangular Prism = Ph = 2h(l + b) sq.units

Where, l - length, b - width and h - height of the prism.

### Lateral Area of a Triangular Prism

A prism with triangular bases is a triangular prism. If a, b, c are the sides of the triangular base then its perimeter is a + b + c.

Lateral area of the triangular prism = Ph = (a + b + c)h sq. units

Where, P - Perimeter of base of the prism and h - Height of the prism.

### Lateral Area of a Hexagonal Prism

A prism with hexagonal bases is a hexagonal prism. The hexagonal prism is a prism with 2 hexagonal bases and six rectangular sides.

The lateral area of the regular hexagonal prism with base edge of s and height h is 6sh.

Lateral area of a regular hexagonal prism = Ph = 6sh sq.units

Where, Perimeter (P) = 6s and h = height of the prism. A cube is a prism whose bases and lateral faces are all equal squares. The lateral area of a cube is equal to the sum of the area of the four faces.

### Lateral Area of a Cube Formula

Lateral area = Sum of the area of the 4 faces

Lateral area of a cube = 4a$^2$ sq.units

where, a is the side of a cube. A sphere is the set of all points that are the same distance from a centre. A radius of a sphere is a segment whose endpoints are the centre and a point on the sphere. Since sphere is a closed circular curve, so its lateral area does not exit. But below you could see the formula for the surface area of a sphere.

Surface area of a sphere = 4$\pi$ r$^2$ sq.units

where, r - radius of a sphere Given below are some of the examples to find the lateral area of a solid.### Solved Examples

**Question 1: **Find the lateral surface area of cone with radius 4 cm and slant height 15 cm.

** Solution: **

**Question 2: **Find lateral surface area of cylinder with radius 4.5 cm and height 14 cm.

** Solution: **

**Question 3: **Find lateral surface area of cone with radius 3 cm and slant height 12 cm.

** Solution: **

The lateral area of a right cylinder with altitude, h, and circumference, 2$\pi$ r, of the base is given by:

Lateral surface area of the cylinder (LSA) = 2 $\pi$ r h square unit

** ** r - Radius

h - Height of the cylinder

The surface area of a cylinder is the sum of the lateral area and area of the two bases.

Surface area = Lateral Area + 2(Base area)

= 2$\pi$r h + 2$\pi$ r$^2$

= 2$\pi$r(r + h)

Total surface area of a cylinder = 2$\pi$r(r + h) sq. units

Where, r - Radius

h - Height of the cylinder

** ** r - Radius

s - Slant height of a cone

A pyramid is a polyhedron in which one face is a polygon and the other faces are triangles with a common vertex. An altitude is the perpendicular segment drawn from the vertex to the base of the pyramid.

The lateral area of a pyramid can be calculated by multiplying half of the perimeter of the base by the slant height.

Lateral area of a regular pyramid = $\frac{1}{2}$ Ps sq.units

P - Perimeter of the base

s - Slant height (length of the altitude of a lateral face of the pyramid)

A pyramid with square base is called as square pyramid.

If 'a' be the side of a square, then perimeter of the base = 4a and

Lateral area = $\frac{1}{2}$ * 4a * s = 2as.

Lateral area of a square pyramid = 2as square units

where, a - base side length

s - slant height of the square pyramid

A pyramid with triangular base is called as triangular pyramid. If triangular pyramid is regular with base side length 'a', then perimeter of the base is 3a and

Lateral area = $\frac{1}{2}$ * 3a * s = $\frac{3}{2}$ * a * s

Lateral area of a regular triangular pyramid = $\frac{3}{2}$ a s sq.units

where, a - base side length and s - slant height.

A pyramid with pentagonal bases is a pentagonal prism. If a, b, c, d, e are the sides of the pentagonal base then its perimeter is a + b + c + d + e.

Lateral area formula = $\frac{1}{2}$ Ps = $\frac{1}{2}$ (a + b + c + d + e)s

Lateral area of a pentagonal pyramid formula = $\frac{1}{2}$ (a + b + c + d + e)s square units

where, a, b, c, d, e are the sides of an pentagon and s - slant height of the pyramid. The lateral area of any prism is equal to the perimeter of the base times the height of the prism.

Lateral Area of a Prism Formula

Lateral area of a right prism = Ph

Where, P - Perimeter of the base and h - height of the prism.

A prism with rectangular bases is a rectangular prism. A rectangular prism has two bases and four sides.

Lateral Surface Area of Rectangular Prism = Ph = 2h(l + b) sq.units

Where, l - length, b - width and h - height of the prism.

A prism with triangular bases is a triangular prism. If a, b, c are the sides of the triangular base then its perimeter is a + b + c.

Lateral area of the triangular prism = Ph = (a + b + c)h sq. units

Where, P - Perimeter of base of the prism and h - Height of the prism.

A prism with hexagonal bases is a hexagonal prism. The hexagonal prism is a prism with 2 hexagonal bases and six rectangular sides.

The lateral area of the regular hexagonal prism with base edge of s and height h is 6sh.

Lateral area of a regular hexagonal prism = Ph = 6sh sq.units

Where, Perimeter (P) = 6s and h = height of the prism. A cube is a prism whose bases and lateral faces are all equal squares. The lateral area of a cube is equal to the sum of the area of the four faces.

Lateral area = Sum of the area of the 4 faces

Lateral area of a cube = 4a$^2$ sq.units

where, a is the side of a cube. A sphere is the set of all points that are the same distance from a centre. A radius of a sphere is a segment whose endpoints are the centre and a point on the sphere. Since sphere is a closed circular curve, so its lateral area does not exit. But below you could see the formula for the surface area of a sphere.

Surface area of a sphere = 4$\pi$ r$^2$ sq.units

where, r - radius of a sphere Given below are some of the examples to find the lateral area of a solid.

Given, Radius (r) = 4 cm

Height (h) = 15 cm

**Lateral surface area (L.S.A) = **$\pi$** r s square unit.**

Substitute the values of r and s in formula and simplify,

LSA = $\pi$ x 4 x 15

= 3.14 x 4 x 15

= 188.4

Lateral surface area of a cone = 188.4 cm^{2}

Given, Radius (r) = 4.5 cm

Height (h) = 14 cm

**Lateral surface area of the cylinder (LSA) = 2 **$\pi$** r h square unit**

= 2 x 3.14 x 4.5 x 14

= 395.64

Lateral surface area of the cylinder (LSA) = 395.64 cm^{2 }

Given, Radius (r) = 3 cm

Height (h) =12 cm

**Lateral surface area (L.S.A) =** $\pi$** r s square unit.**

Substitute the values of r and s in formula and simplify,

= $\pi$ x 3 x 12

= 3.14 x 3 x 12

= 113.04

Lateral surface area (L.S.A) = 113.04 cm^{2}

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