Clarity on Mensuration

The concept of Mensuration deals with the profound study of all the two-dimensional and three-dimensional objects that have a proper shape and size. And there can be different formulae to determine the diameter, area, circumference, perimeter, or total surface area of any given figure whether 2D or 3D. Mensuration of all formulae can be used to determine the same. The different figures and formulas of measurement have been critically analyzed in this writeup.

2D Shapes

Shapes that are known as 2D are two-dimensional objects that have mainly two segments, which are height and width but don’t have thickness. A rectangle, square, circle, or triangle, for reference. There are two axes in mathematical form (Y and X). These 2D objects, which have only two-axis don’t really exist in reality and can only be depicted using flat surfaces and are widely used in mensuration.

Rectangle: 

It is a two-dimensional form with four sides and four corners. Every angle that is present in the rectangle is 90 degrees because it is a quadrilateral and it has 4 right angles.

Consider Length to be “L” and Breadth to be “B”

Perimeter = 2 (L + B)

Area = L x B

Square:

A 2D-shaped plane having 4 equivalent edges with four 90-degree angles is known as a Square. The diagonals of the shape have a similar length It is generally used in mensuration under problems based on a closed square room.

Area = Side x Side

Perimeter = 4 x Side

Circle:

It is a fundamental two-dimensional shape that consists of a set of equidistant points along a plane. The length from the point of center to point that is present anywhere along the circumference is referred to as the radius of the circle.

Considering the Radius to be “r”

Diameter = 2 × r

Circumference = π × Diameter or 2 × r x π

Area = π × r2

Triangle:

Three edges and three included angles make up a triangle. All the 3 angles of the triangle have a sum of 180 degrees. In mensuration the problems on triangles of different types such as scalene, isosceles, or equilateral are common.

Triangle = ½ × base × height

3 Dimensional Shapes

Any three-dimensional object in the actual world has three segments (height, width, and depth). It has three axes in mathematical form (X, Y, and Z). 3D objects have a volume and a Total Surface Area (TSA) that employs all three dimensions of the object: length, breadth, and depth. 

Parallelogram:

It is a two-dimensional object with opposing sides that are parallel to one another. A parallelogram conventionally has 4 sides with equal lengths.

Consider one side to be “a”, the other side to be “b”

Perimeter = 2 (a + b)

Area = b × height

Cube:

A 3D object that consists of six square planes, 12 edges, and out of 12, three of them meet at a single vertex. The total number of vertices is 8. A crystal of Sugar is an instance of a cube.

Volume = Side3 

Lateral Surface Area = 4 × side2 

Total Surface Area = 6 × side2 

Cuboid:

A three-dimensional object with 3 sides that are not all equal is called a cuboid. It has six faces, eight vertices, and twelve edges in all of its facets.

Consider Length to be “L”, Width to be “W”, and Height to be “H”

Volume = (L + W + H)

Lateral Surface Area = 2 × H (L + W).

Total Surface Area = 2[(L × W)+ (L × H) + (H × W)]

Diagonal length = L2 + B2 + H2 

Sphere:

It is considered to be a figure in a 3D space that has a shape of a round geometric entity. The center has several points that are noted to be equidistant. The sphere has a hub of points that are all equally distant from its center The distance measured from the center (R) of the sphere to any point is called Radius. Spheres are an essential element of mensuration.

Consider the Radius to be “r”

Volume e = 4/3 x π x r³ 

Surface = 4 x π x r²

Cone:

A 3D geometric object is known as a cone. It is created by connecting the vertex with a group of vertical lines. Since the cone has a circular shape at the bottom, we can easily calculate its radius The distance is calculated from any point marked on the base’s perimeter to the topmost spot of the cone is known as Slant height

Considering the Radius to be “r”, the Height to be “h” and the Slant Height to be “l”

The volume = 1/3 × × π × r² × h 

Total Surface Area = π x r (l + r)

The above formula of mensuration is generally utilized in the solution of problem statements based on common figures like cube or square, rectangle or parallelogram or cuboid, or cone or a triangle.

Conclusion:

With the help of mensuration, all formulas can easily calculate whatever is asked in question whether the area, total surface area, diameter or perimeter, or even circumference by using the given information about the dimensions like length, height, width, radius, perpendicular distance, etc.

It will not be a tough task to estimate the answer to any problem based on mensuration as these formulas are concept-oriented. All that is required is an adequate understanding of the shapes. Then it will be easier to just draw the shapes, locate the dimensions, put the formula and solve the query.