MathWithNaziaa : Surface area and volume

Surface area and volume

Surface area and volume are calculated for any three-dimensional geometrical shape. The surface area of any given object is the area or region occupied by the surface of the object. Whereas volume is the amount of space available in an object.

In geometry, there are different shapes and sizes such as sphere, cube, cuboid, cone, cylinder, etc. Each shape has its surface area as well as volume. But in the case of two-dimensional figures like square, circle, rectangle, triangle, etc., we can measure only the area covered by these figures and there is no volume available. Now, let us see the formulas of surface areas and volumes for different 3d-shapes.

 

What is the Surface Area?

The space occupied by a two-dimensional flat surface is called the area. It is measured in square units. The area occupied by a three-dimensional object by its outer surface is called the surface area. It is also measured in square units.

Generally, Area can be of two types:

(i) Total Surface Area

(ii) Curved Surface Area/Lateral Surface Area

Total Surface Area

Total surface area refers to the area including the base(s) and the curved part. It is the total of the area covered by the surface of the object. If the shape has a curved surface and base, then the total area will be the sum of the two areas.

Curved Surface Area/Lateral Surface Area

Curved surface area refers to the area of only the curved part of the shape excluding its base(s). It is also referred to as lateral surface area for shapes such as a cylinder.

What is Volume?

The amount of space, measured in cubic units, that an object or substance occupies is called volume. Two-dimensional doesn’t have volume but has area only. For example, the Volume of the Circle cannot be found, though the Volume of the sphere can be. It is so because a sphere is a three-dimensional shape.

 

CONE:

This is a 3D geometric shape created by two line segments connecting a common point. This point is known as vertex or apex. This shape tapers from a smooth and flat base to the apex.

Here are the formulas of volume and surface area of cone- 

If the vertical height of the cone = h Radius of the base = r Slant height = l 

Then, 

volume = 1/3 π r2h

Total surface area = π r2+ πrl

Lateral area = πrl

Sphere:

It is a geometrical shape in 3D space that is similar to the outer surface of a planet or ball. Mathematically, it can be defined as a set of points that are situated at an equal distance from any given point. 

To solve mathematical problems related to the sphere, you need to understand the formulas of surface area and volume of spheres.

If the radius = r, 

Then,

 volume = 4/3 πr3

Surface area = 4π r2

Cylinder:

This is a solid curvilinear shape with the surface formed by equal and fixed-distant points from a given line. 

Now, the formula of volume and surface area of cylinder-

If the radius of a cylinder = r Height = h

The lateral surface area without the top and bottom surface = 2πrh

The total surface area including top and bottom surface = 2πr (r + h)

Volume = π r2h

 

Cube and Cuboid

A cube is formed with six square faces or sides, and each vertex is the meeting point of three sides. However, a cuboid also has six faces, but the cuboid faces can be any quadrilateral. Mostly, cuboid faces are rectangular

Following are the formulas of volume and surface area of a cube -

If the length of one side of a cube = a, Then, 

surface area = 6a3

Volume = x3

Now, let’s move to the surface area and volume formula of a cuboid-

If the height of a cuboid = h Length = l  Width = w

Then, 

surface area = 2 (hl + lw + wh)

Volume = hlw

With the help of these above formulas, you can easily solve volume and surface area problems of cone, cube, cuboid, sphere, cylinder.