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Mensuration is a division of mathematics that studies geometric figure calculation and its parameters such as area, length, volume, lateral surface area, surface area, etc. It outlines the principles of calculation and discusses all the essential equations and properties of various geometric shapes and figures.
What is Mensuration?
Mensuration is a subject of geometry. Mensuration deals with the size, region, and density of different forms both 2D and 3D. Now, in the introduction to Mensuration, let’s think about 2D and 3D forms and the distinction between them.
What is a 2D Shape?
A 2D diagram is a shape laid down on a plane by three or more straight lines or a closed segment. Such forms do not have width or height; they have two dimensions-length and breadth and are therefore called 2D shapes or figures. Of 2D forms, area (A) and perimeter (P) is to be determined.
What is a 3D Shape?
A 3D shape is a structure surrounded by a variety of surfaces or planes. These are also considered robust types. Unlike 2D shapes, these shapes have height or depth; they have three-dimensional length, breadth and height/depth and are thus called 3D figures. 3D shapes are made up of several 2D shapes. Often known as strong forms, volume (V), curved surface area (CSA), lateral surface area (LSA) and complete surface area (TSA) are measured for 3D shapes.
Difference between 2D and 3D shapes
2D Shape |
3D Shape |
There is no depth or height to these shapes. |
These are also referred to as solid shapes, as they have height and depth, unlike 2D shapes. |
A 2D shape is bordered by three or more straight lines in a plane. |
A 3D shape is surrounded by several surfaces or planes. |
There are only two dimensions to these shapes: length and width. |
Because these shapes have depth (or height), breadth, and length, they are referred to as three-dimensional. |
We can take measurements of 2D shapes, area and perimeter. |
We can calculate their volume, CSA, LSA, and TSA. |
In the drawings, these shapes can be seen clearly. For example, in a square, all the edges may be seen. |
Due to overlapping, it is not visible completely. Like, in the case of the cube, it is impossible to show all its edges from a particular angle. |
Introduction to Menstruation: Important Terms
Until we switch to the list of important formulas for measurement, we need to clarify certain important terms that make these measurement formulas:
Area (A):
The area is called the surface occupied by a defined closed region. It is defined by the letter A and expressed in a square unit.
Perimeter (P):
The total length of the boundary of a figure is called its perimeter. Perimeter is determined by only two-dimensional shapes or figures. It is the continuous line along the edge of the closed vessel. It is represented by P and measures are taken in a square unit.
Volume (V):
The width of the space contained in a three-diMensional closed shape or surface, such that, the area by a room or cylinder. Volume is denoted by the alphabet V and the SI unit of volume is the cubic meter.
Curved Surface Area (CSA):
The curved surface area is the area of the only curved surface, ignoring the base and the top such as a sphere or a circle. The abbreviation for the curved surface area is CSA.
Lateral Surface Area (LSA):
The total area of all of a given figure’s lateral surfaces is called the Lateral Surface Area. Lateral surfaces are the layers covering the artefact. The acronym for the lateral surface area is LSA.
Total Surface Area (TSA):
The calculation of the total area of all surfaces is called the Cumulative Surface Region in a closed shape. For example, we get its Total Surface Area in a cuboid by adding the area of all six surfaces. The acronym for the total surface area is TSA.
Square Unit (/):
One square unit is simply the one-unit square area. When we quantify some surface area, we relate to the sides of one block square to know how many of these units will fit in the figure given.
Cube Unit (/):
One cubic unit is the one-unit volume filled by a side cube. When we calculate the volume of any number, we refer to this cube of one unit and how many these component cubes will fit in the defined closed form.
Tools Require for Mensuration
Calliper – A tool for measuring the diameter.
Try Square – A tool for determining the squareness and flatness of a surface.
Meter Stick – A measuring device with a one-meter length.
Compass – An arc and circle drawing tool.
List of Mensuration Formulas for 2D shapes:
As our introduction to Mensuration and the relevant words are over, let’s switch to the equations for Mensuration, as this is a discussion focused on an equation. The 2D figure has a list of formulas of measurement that define a relationship between the various parameters. Let’s look into detail about the estimation equations of some kinds.
Square:
Area = [(side)^{2}] sq. units.
Perimeter = [(4 times sides)] units.
Diagonal = [sqrt{2 times side}]units.
Rectangle:
Area = [(length times breadth) ]sq. units.
Perimeter = 2(length + breadth) units.
Diagonal, D = [sqrt{length^{2} + breath^{2}}]units.
Scalene Triangle:
Area, [A = frac{height times base}{2} ]sq. units
Perimeter = (side a + side b + side c) units.
Equilateral Triangle:
Area = [ frac{sqrt{3}}{4} times side^{2} ] sq. units.
Perimeter = [(3 times side)] units.
Isosceles Triangle:
Area = [A = frac{height times base}{2} ]sq. units
Perimeter = [(2 times side ] + base) units.
Right Angled Triangle:
Area = [A = frac{leg_{a}times leg_{b}}{2} ]sq. units
Perimeter = [leg_{a} + leg_{b} + sqrt {leg_{a}^{2} + leg_{b}^{2}}] units
Hypotenuse = [sqrt {leg_{a}^{2} + leg_{b}^{2}}] units
Circle:
Area = [ π times radius^{2} ] sq. units.
Circumference =[ 2π times radius] units.
Diameter, D = [2 times radius ] units.
List of Mensuration Formulas for 3D shapes:
The 3D figure has a list of formulas for measurement that define a relationship between the various parameters. Let’s look into the details about the estimation equations of some kinds.
Cube:
Volume = [ side^{3} ] cubic units.
Lateral Surface Area = [4 times side^{2}] sq. units.
Total Surface Area = [6 times side^{2}] sq. units.
Diagonal Length d = [sqrt {length^{2} + width^{2} + height^{2}}] units.
Cuboid:
Volume = (length + width + height) cubic units.
Lateral Surface Area =[ 2 times height (length + width) ]sq. units.
Total Surface Area = [2(length times width + length times height + height times width)] sq. units.
Diagonal Length = [ length^{2} + breadth^{2} + height^{2}] units.
Sphere:
Volume = [ frac{4}{3} prod times side^{2} ] cubic units.
Surface Area = [4prod times radius^{2} ]sq. Units.
Hemisphere:
Volume = [ frac{2}{3} prod times side^{2} ] cubic units.
Total Surface Area =[3prod times radius^{2} ]sq. Units.
Cylinder:
Volume = [prod times radius^{2} times height ] cubic units.
Curved Surface Area (excluding the areas of the top and bottom circular regions) = [(2 times prod times R times h)] sq. units.
Where, R = radius
Total Surface Area = [(2 times prod times R times h) + (2 times prod times R^{2}) ] sq. units
Cone:
Volume = [ frac{1}{3} prod times radius^{2} times height ] cubic units.
Curved Surface Area =[ ? times radius times height] sq. units.
Total Surface Area = ? x radius(length + height) sq. units
Slant Height of Cone =[ sqrt{height^{2} + base radius^{2}}] sq. units.
Using these above formulas for the Mensuration, most of the Mensuration problems can be solved.