The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:
The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:
Introduction
Imagine that you want to purchase a carpet for your room, how would you know a perfect size? Simple, you would need to measure the area which you wish to cover. Similarly, if you want to fence your garden, you need to measure the entire length or distance around the boundary.
In both these examples, we are talking about the measurement of Perimeter, length, and area of different geometrical shapes. The field of mathematics dealing with all these parameters is called Mensuration.
Previously you have studied Perimeter and area of known and familiar shapes and before we continue, let us revise what we already know.
A quadrilateral with two parallel sides is known as trapezium. If the non-parallel sides are equal, it becomes isosceles trapezium.
You can find the area of the trapezium by dividing the figure into two or three plane figures; for example, we have divided the trapezium into two triangles and one rectangle.
You know how to find the area of triangle and rectangle which makes it easy for you to find the area of trapezium.
A = Area of triangle 1 + Area of triangle 2 + Area of rectangle
= ½ hx + ½ hy + ah
= ½ h (x + y + 2a), we know that total breadth, b = (x + y + a)
Therefore, the area of trapezium = 1/2h (a + b) or 1/2h (sum of both parallel sides)
First, draw a diagonal in the quadrilateral to divide it into two triangles, as shown below. After that draw perpendiculars h1 and h2 to AC.
Now, the area of quadrilateral ABCD will be the sum of areas of both the triangles.
A = (area of ∆ ADC) + (area of ∆ ABC)
= (1/2 h1 * AC) + (1/2 h2 * AC)
= ½ AC (h1 + h2), AC = d where d denotes the length of diagonal AC.
Therefore, Area of the quadrilateral = ½ d (h1 + h2)
There is no exact formula to calculate the area of polygons. However, you can figure it by splitting the given shape in possible known figures and then adding those areas.
For example, you can calculate the area of the octagon by dividing it into two trapeziums and one rectangle and adding their areas.
Area of octagon = Area of trapezium 1 + Area of trapezium 2 + Area of rectangle
Three-dimensional objects occupying some space and having length, breadth and depth or height are called solid shapes.
Some shapes have two or more identical or congruent faces, for instance, cube.
You can find the surface area of any solid by adding the areas of its faces.
You can use the rectangular solid to find the formula for the volume of the cylinder. For rectangle, V = Base * Height, here height = h and base = l * wHence, V = l * w * hMensuration is a significant topic, as these concepts and formulas also have real-life application. For example, if you know the area of the walls, you can very easily determine how much paint is needed to paint the entire wall.
In this chapter, you have explored relationships between different geometrical shapes and also learned to apply mathematical formulas. For instance, If you want to wrap a rectangular pencil box, then you can calculate how much paper is required by finding the surface area of that box.
