- We have learnt that for a closed plane figure, the perimeter is the distance around its boundary and its area is the region covered by it.
- We found the area and perimeter of various plane figures such as triangles, rectangles, circles etc.
- We have also learnt to find the area of pathways or borders in rectangular shapes.
- We will try to solve problems related to perimeter and area of other plane closed figures like quadrilaterals.
- We will also learn about surface area and volume of solids such as cube, cuboid and cylinder.
- Mensuration is the branch of mathematics that studies the measurement of the geometric figures and their parameters like length, volume, shape, surface area, lateral surface area, etc.
- Here, the concepts of mensuration are explained and all the important mensuration formulas provided.
- Also, the properties of different geometric shapes and the corresponding figures are given for a better understanding of these concepts.
- A branch of mathematics that talks about the length, volume, or area of different geometric shapes is called Mensuration.
- These shapes exist in 2 dimensions or 3 dimensions.
- Let’s learn the difference between the two.
Differences Between 2D and 3D shapes in Mensuration
2d shapes in mensuration
- If a shape is surrounded by three or more straight lines in a plane, then it is a 2D shape.
- These shapes have no depth or height.
- These shapes have only two dimensions say length and breadth.
- We can measure their area and Perimeter.
3D SHAPES in mensuration
- If a shape is surrounded by a no. of surfaces or planes then it is a 3D shape.
- If a shape is surrounded by a no. of surfaces or planes then it is a 3D shape.
- These are called Three dimensional as they have depth (or height), breadth and length.
- We can measure their volume, CSA, LSA or TSA.
Volume in mensuration
It refers to the quantity of three dimensional space enclosed by a closed surface, for example, the space that a substance ( solid, liquid, gas, or plasma.) or shape occupies or contains.
volume is often quantified numerically using SI derived unit, the cubic meter.
Given below are some standard definitions for volumes of recognizable objects-:
- volume : volume is the amount of space enclosed by a shape or object, how much 3-dimensional space (length,width,and height) it occupies. this is measured in cubic unit like cm3, m2, etc..
- volume of cuboid : length X Breadth X height
- volume of cube : side X side X side
- volume of sphere : 4/3 X pi X radius X radius X radius
- volume of prism : area of base X height
- volume of cone : 1/3 X pi X radius X radius X height
PERIMETER IN MENSURATION
A Perimeter is a path surrounds a two-dimensional shape.
The term may be used either for the path or its length it can be thought of as the length of the outline of a shape.
the perimeter of a circle or ellipse is called its circumference. as mentioned before, since perimeter is nothing but a measure of length, its SI unit will be the meter itself. given below are some formulae for perimeter of two dimensional figures -:
- perimeter of circle : 2 X pi X radius
- perimeter of triangle : side A + side B + side C
- perimeter of square : 4 X side
- perimeter of rectangle : 2 X ( length X breadth )
AREA IN MENSURATION
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. this is measured in square unit like cm2, m2 etc..
- Area of rectangle : length X breadth
- Area of triangle : 0.5 X base X height
- Area of square : side X side ( or side^2 )
- Area of circle : PI X radius X radius
- Surface area of a cylinder = 2 X PI X radius X ( radius + height )
- Surface area of a sphere = 4 X PI X radius X radius
- Surface area of a cube = 6 X side X side
- Surface area of a cuboid = 2 X ( length X breadth + breadth X height + length X height )
AREA OF TRAPEZIUM
So to find the area of a trapezium we need to know the length of the parallel sides and the perpendicular distance between these two parallel sides. Half the product of the sum of the lengths of parallel sides and the perpendicular distance between them gives the area of trapezium
Area of a General Quadrilateral
A general quadrilateral can be split into two triangles by drawing one of its diagonals. This “triangulation” helps us to find a formula for any general quadrilateral. Area of quadrilateral ABCD
= 1/2 d ( h1+ h2) where d denotes the length of diagonal AC.
Area of special quadrilaterals
We can use the same method of splitting into triangles (which we called “triangulation”) to find a formula for the area of a rhombus. a rhombus. Therefore, its diagonals are perpendicular bisectors of each other.
= Area of rhombus ABCD = (area of Δ ACD) + (area of Δ ABC)
= 1/2 AC × BD = 1/2 d1× d2 where AC = d1 and BD = d2
Area of a Polygon
We split a quadrilateral into triangles and find its area. Similar methods can be used to find the area of a polygon. Observe the following for a pentagon By constructing two diagonals AC and AD the pentagon ABCDE is divided into three parts. So, area ABCDE = area of Δ ABC + area of Δ ACD + area of Δ AED.
By constructing one diagonal AD and two perpendiculars BF and CG on it, pentagon ABCDE is divided into four parts. So, area of ABCDE = area of right angled Δ AFB + area of trapezium BFGC + area of right angled Δ CGD + area of
Δ AED. (Identify the parallel sides of trapezium BFGC.)
In your earlier you have studied that two dimensional figures can be identified as the faces of three dimensional shapes.
Observe the solids which we have discussed so far
- cuboidal box = All six faces are rectangular, and opposites faces are
identical. So there are three pairs of identical faces.
- cubical box = All six faces are squares and identical.
- cylindrical box = One curved surface and two circular faces which are
identical. circular base and top are identical.
Now take one type of box at a time. Cut out all the faces it has. Observe the shape of each face and find the number of faces of the box that are identical by placing them on each other. Write down your observations.
The cylinder has congruent circular faces that are parallel to each other. Observe that the line segment joining the center of circular faces is perpendicular to the base. Such cylinders are known as right circular cylinders. We are only going to study this type of cylinders, though there are other types of cylinders as well.
Surface Area of Cube, Cuboid and Cylinder
Imran, Monica and Jaspal are painting a cuboidal, cubical and a cylindrical box respectively of same height.
They try to determine who has painted more area. Hari suggested that finding the surface area of each box would help them find it out.
To find the total surface area, find the area of each face and then add. The surface area of a solid is the sum of the areas of its faces. To clarify further, we take each shape one by one.
Volume of Cube, Cuboid and Cylinder
Amount of space occupied by a three dimensional object is called its volume. Try to compare the volume of objects surrounding you. For example, volume of a room is greater than the volume of an almirah kept inside it.
Similarly, volume of your pencil box is greater than the volume of the pen and the eraser kept inside it. Can you measure volume of either of these objects?
Remember, we use square units to find the area of a region. Here we will use cubic units to find the volume of a solid, as cube is the most convenient solid shape (just as square is the most convenient shape to measure area of a region).
For finding the area we divide the region into square units, similarly, to find the volume of a solid we need to divide it into cubical units.
Observe that the volume of each of the adjoining solids is 8 cubic units
We can say that the volume of a solid is measured by
counting the number of unit cubes it contains. Cubic units which we generally use to measure volume are
1 cubic cm = 1 cm × 1 cm × 1 cm = 1 cm3
= 10 mm × 10 mm × 10 mm = …………… mm3
1 cubic m = 1 m × 1 m × 1 m = 1 m3
= …………………………. cm3
1 cubic mm = 1 mm × 1 mm × 1 mm = 1 mm3
= 0.1 cm × 0.1 cm × 0.1 cm = …………………. cm3
We now find some expressions to find volume of a cuboid, cube and cylinder. Let us take each solid one by one.
The cube is a special case of a cuboid, where l = b = h. Hence, volume of cube = l × l × l = l
We know that volume of a cuboid can be found by finding the product of area of base and its height. Can we find the volume of a cylinder in the same way?
Just like cuboid, cylinder has got a top and a base which are congruent and parallel to each other. Its lateral surface is also perpendicular to the base, just like cuboid. So the Volume of a cuboid = area of base × height
= l × b × h = lbh
Volume of cylinder = area of base × height
= pi r2 × h = pi r2h
MENSURATION IN DAILY LIFE
Mensuration is the mathematical study of geometric polygons and scalar parameters related to them like area, volume, lateral area and divisions.
If you take a look at the arousing topic of mensuration, we will find many concepts and formulas which help us in day to day activities.
Let’s say your mom asked you to bring fish(I hope you like it) from supermarket which is 10 kms away. You have to watch your favorite TV show which starts in 25 mins then mensuration could be extremely helpful. You would chose the shortest path to the supermarket, like hypotenuse in case of triangles and diagonals in case of quadrilaterals.
If you have to water your garden and you know the amount of water utilized for 1 m² area then you could get a rough estimation of the total amount if water.
Mensuration is the weapon which can save you from excessive burden, it can help you in saving time and heights and distances give us the aptitude of skill, like engineers have to calculate the height, length, breadth, angle on inclination and many more stuff to erect a building.
APPLICATION OF MENSURATION IN DAILY LIFE
Mensuration is a branch of geometry which deals with the calculation of area,perimeter,volume etc.
it is applied in our daily lives for instance if we need to build a house we will calculate the volume of bricks as well as walls which are to be made to in order to consider the percentage and the number of bricks needed to build a house.
it minimizes the wastage of material, this method is beneficial if certain amount of products are to be made in order to increases its efficiency with reducing wastage.