Maths is a subject of concepts, logic and fundamentals. The basic fundamentals we learn are arithmetic, algebra, calculus, geometry, etc. Geometry is one of the important concepts where we learn about different shapes and sizes. But to define these shapes it is very necessary to understand their properties and dimensions. Mensuration is a topic which explains the properties and features of all the shapes.
The properties of all the shapes vary from each other. For two dimensional shapes, the two major properties are area and perimeter. Area defines the region covered by a shape whereas the perimeter of the shape defines how much distance its boundary covers. The area gives the space occupied and the perimeter gives the length of the boundary of the shape. Let us see the formulas of area and perimeter for different shapes.
A circle is a closed curve shape, whose outer boundary is equidistant from the center. For a circle, the perimeter is called the circumference. The formula for the circumference of a circle is 2πr and area of circle is πr2.
A triangle is a three-sided polygon, whose perimeter is equal to the sum of all three sides.
Perimeter of triangle = Sum of all its sides
Area = ½ (Product of base and height of the triangle)
A square is a four-sided enclosed shape whose all sides are equal and all the vertices have angles equal to 90 degrees. Perimeter is equal to the sum of all its four sides.
Perimeter = 4 x side
Area of a square = side2
A rectangle is also a four-sided polygon, where the opposite sides are parallel and equal. Also, all the angles equal to 90 degrees.
Perimeter of rectangle = 2(Length + Breadth)
Area of rectangle = Length x breadth
Apart from these important shapes, there are other shapes in Geometry such as Kite, Rhombus, Parallelogram and n sided polygons, which fall under 2D shapes.
When we speak of three-dimensional shapes, the parameters that define these shapes are surface area and volumes. The surface area is the total area occupied by the surface of any 3d shape whereas volume defines the amount of capacity.
In 3d geometry, we will come across different solid shapes such as Cube, Cuboid, Sphere, Cone, Cylinder. These are the basic shapes. Let us see their surface area and volume.
It has eight faces, eight vertices and twelve edges. All the faces are square in shape.
Surface area of cube = 6 (edge-length)2
Volume of cube = (Edge-length)3
It has a circular base, which narrows from bottom to the top at a single point, called the vertex.
Surface area of cone = π r(r+l)
Volume of cone = 1/3π r2h
Also, if the cone is cut by a plane, horizontally, then the upper part still remains the cone and lower is called frustum of a cone. Now, the formula for the volume of a frustum will be different from the volume of cone formula, since the shape has been changed and it has an additional circular surface at the top.
Like the cube, it also has eight faces, eight vertices and twelve edges, but face of the cuboid is rectangular.
Surface area = 2(Length x Breadth+Breadth x Height + Length x Height)
Volume = Length x Breadth x Height
So by using all these formulas we can easily find the areas and volumes of the shapes in real life as well which resembles the same shape. This shows the importance of mensuration in our day to day life.