Mensuration Formulae - Part 2

Mensuration Formulas for 3D Figures

8. Mensuration Formulas for CUBOID

In the following formulae, l = length, b = breadth and h = height

Total surface area of cuboid = 2 (lb + bh + lh)

Length of diagonal of cuboid= √(l²+b²+h²)

Volume of cuboid = l × b × h

9. Mensuration Formulas for CUBE

In the following formulae, a = side of a cube

Volume of cube = a³

Total surface area of cube = 6a²

Length of Leading Diagonal of Cube = a√3

10. Mensuration Formulas for CONE

In the following formulae, r = radius of base, l = slant height of cone and h = height of the cone (perpendicular to base)

Slant height of a cone = l =√(h²+r² )

Curved surface area of a cone = C = π × r × l

Total surface area of a cone = π × r × (r + l)

Volume of right circular cone =1/3 πr²h

11. Mensuration Formulas for CYLINDER

In the following formulae, r = radius of base, h = height of cylinder

Curved surface area of a cylinder = 2πrh

Total surface area of a cylinder = 2πr(r + h)

Volume of a cylinder = πr²h

12. Mensuration Formulas for SPHERE

In the following formulae, r = radius of sphere, d = diameter of sphere

Surface area of a sphere = 4πr² = πd²

Volume of a sphere = (4/3) πr³ = (1/6)πd³

13. Mensuration Formulas for HEMISPHERE

In the following formulae, r = radius of sphere

Volume of a hemisphere =(2/3)πr³

Curved surface area of a hemisphere = 2πr²

Total surface area of a hemisphere = 3πr²

14. Mensuration Formulas for HOLLOW CYLINDER

Hollow cylinder made by cutting a smaller cylinder of same height and orientation out of a bigger cylinder.

Volume of hollow cylinder = πh(R²– r²)
(Where, R = radius of cylinder, r = radius of cavity, h = height of cylinder)

15. Mensuration Formulas for FRUSTUM OF A RIGHT CIRCULAR CONE

Frustum is created when a plane cuts a cone parallel to its base.
In the following formulae, R = radius of the base of the frustum, r = radius of the top of the frustum,
h = height of the frustum, l = slant height of the frustum

If a cone is cut by a plane parallel to the base of the cone, the lower part is called the frustum of the cone.

Slant height of the frustum =l=√(h²+(R-r)²)

Curved surface area of frustum = π(R + r)l

Total surface area of frustum = π(R + r)l + π(R² + r²)

Volume of the frustum=(1/3)πh(R²+r²+Rr)

16. Mensuration Formulas for PRISM

Prism consists of two polygonal bases which are parallel to each other.

These bases are joined by lateral faces, which are perpendicular to the polygonal bases.

The number of lateral faces is equal to the number of sides in the polygonal base. Thus, the base of a prism could be of various shapes, namely, triangular, quadrangular, pentagonal etc.

Volume of prism = Base area × height

Lateral surface area of prism = perimeter of base × height

Total surface area of prism = Lateral surface area + (2 × base area)

17. Mensuration Formula for PYRAMID

Pyramid consists of a polygonal base and triangles at its sides. These triangles are called faces.

The base could be of any shape, whereas the faces are generally isosceles triangles.

All these triangular faces meet in a single point called the apex.

Total surface area of pyramid = base area + (number of sides × ½ × slant height × base length)

Volume of pyramid = (1/3) × base area × height

Mensuration Formulas Part 1

1. Mensuration Formulas for RECTANGLE

Area of Rectangle = Length × Breadth.                               

Perimeter of a Rectangle = 2 × (Length + Breadth)

Length of the Diagonal = √(Length² + Breadth²)

2. Mensuration Formulas for SQUARE

 

Area of a Square = Length × Length = (Length)² 

Perimeter of a square = 4 × Length

Length of the Diagonal = √2 × Length

3. Mensuration Formulas for PARALLELOGRAM

Area of a Parallelogram = Length × Height

Perimeter of a Parallelogram = 2 × (Length + Breadth)

4. Mensuration Formulas for TRIANGLE

Area of a triangle=(1/2)(Base × Height)=(1/2)(BC×AD)

For a triangle with sides measuring a, b and c, respectively:

Perimeter = a + b + c

s = semi perimeter = perimeter/2 =  (a+b+c)/2

Area of Triangle, A=

(This is also known as “Heron’s formula”)

Area of isosceles triangle = 

(Where a = length of two equal side, b = length of base of isosceles triangle.)

Area of an equilateral triangle = 

(Where, a is the side of an equilateral triangle)

5. Mensuration Formulas for TRAPEZIUM

Area of a trapezium = (1/2) × (sum of parallel sides) × (distance between parallel sides)
= (1/2) × (AB+DC) × AE

Perimeter of a Trapezium = Sum of All Sides

6. Mensuration Formulas for RHOMBUS

 

Area of a rhombus=(1/2)×Product of diagonals

 

Perimeter of a rhombus = 4 × l
(where l = length of a side)

7. Mensuration Formulas for CIRCLE and SEMICIRCLE

In the following formulae, r = radius and d = diameter of the circle

Area of a circle = πr²= (πd²)/4

Circumference of a circle = 2πr = πd

Circumference of a semicircle = πr

Perimeter of the semicircle = (πr+d)

Area of semicircle =(πr²)/2 

Length of an arc = (2πrθ)/360, where θ is the central angle in degrees.

Area of a sector = (1/2) × (length of arc) × r = (πr²θ)/360

Part 1 deals largely with 2D formulas. You can also Read Part 2  of Mensuration Formula

Tips & Tricks For Spotting Errors in English

Even those of you who are well-versed with English end up making the silliest of errors in grammar. It is a very natural tendency but error-spotting is not an art that one can’t master. It is just a matter of swearing by some guidelines and practice! Here are 26 tips for error spotting in English:

1. Certain nouns possess a singular form but still represent plurality and thus, take a plural verb when used in a sentence.
E.g. Cattle, peasantry, people, clergy, police.
Thus,

The Police has come (Incorrect)

The Police have come (Correct)

2. Certain nouns always take the plural verb because their form is always plural.
E.g. Scissors, trousers, spectacles, thanks, premises.
Thus,

The scissors is kept on the table. (Incorrect)

The scissors are kept on the table. (Correct)

3. When a number is followed by a noun denoting measure, length, money, number or weight, the form of the nouns does not change so long as they are followed by another noun or pronoun.
E.g. Million, pair, metre, year, dozen, foot, head.
Thus,

This is a nine-metres cloth. (Incorrect)

This is a nine-metre cloth. (Correct)

4. When a number is followed by a noun denoting measure, length, money, number or weight, but these are not followed by another noun or pronoun, then they take the plural form.
E.g. Million, pair, metre, year, dozen, foot, head.
Thus,

This sari is nine yard long. (Incorrect)

This sari is nine yards long. (Correct)

5. Certain nouns, especially of the collective category, are used as singular when they specify a unit.
E.g. Public, team, committee, government, audience, orchestra, company, jury.
Thus,

The public were unanimous in their opinion. (Incorrect)

The public was unanimous in its opinion. (Correct)

6. Certain nouns, especially of the collective category, are used in plural when they specify a difference of opinion or class.
E.g. Public, team, committee, government, audience, orchestra, company, jury.
Thus,

The jury was divided in its opinion. (Incorrect)

The jury were divided in their opinion. (Correct)

7. If the pronoun ‘one’ is used, it must be maintained throughout the sentence.
Thus,

One must respect his elders.  (Incorrect)

One must respect one’s elders. (Correct)

8. The word ‘whose’is used for living people and ‘which’ is used for non-living things or ideas.
Thus,

Which box is kept on the table? (Incorrect)

Whose box is kept on the table?  (Correct)

9. ‘Less’ is used to denote quantity while ‘fewer’ is used to denote number.
Thus,

No less than twenty people were (Incorrect)

No fewer than twenty people were (Correct)

10. ‘One of’is always followed by noun in the plural form.
Thus,

She is one of the least important person in the office. (Incorrect)

She is one of the least important people in the office. (Correct)

11. Only ‘than’ should be used after ‘no other’
Thus,

I like no other movie but Titanic. (Incorrect)

I like no other movie than Titanic. (Correct)

12. After the word ‘Know’, ‘how, ‘what’ or ‘when’ should be used before using the infinitive.
Thus,

I know to speak English. (Incorrect)

I know how to speak English. (Correct)

13. If the verb indicates a purpose, an infinitive must be used and if the verb indicates a cause, a gerund must be used.
Thus,

He went to the mall for watching a movie. (Incorrect)

He went to the mall to watch a movie. (Correct)

He was suspended to show indiscipline. (Incorrect)

He was suspended for showing indiscipline.(Correct)

14. ‘As’ is not used with verbs like ‘appointed’, ‘elected’ , ‘considered’, ‘called’ but it is used with the word ‘regard’.
Thus,

He was elected as Secretary of the organisation. (Incorrect)

He was elected Secretary of the organisation. (Correct)

I regard Sahil my best friend. (Incorrect)

I regard Sahil as my best friend. (Correct)

15. Adverbs should not be confused with adjectives. An adjective describes the characteristic of the subject while an adverb describes the action of the verb.
Thus,

The horse looked beautifully. (Incorrect)

The horse looked beautiful. (Correct)

16. Question tags are always the opposite of the sentence which means that if the sentence is positive, the question tag is negative and vice versa.
Thus,

You were dancing, were you? (Incorrect)

You were dancing, weren’t you? (Correct)

17. An infinitive verb should never be split.
Thus,

I request you to kindly tell me. (Incorrect)

I request you kindly to tell me. (Correct)

18. A pronoun after ‘like’, ‘unlike’ and ‘let’ takes an objective case.
Thus,

You will never find a woman like she. (Incorrect)

You will never find a woman like her. (Correct)

Let I do it. (Incorrect)

Let me do it. (Correct)

19. The relative pronoun ‘that’is used instead of ‘who’ or ‘which’ after adjectives in the superlative degree.
Thus,

This is the best which she could do. (Incorrect)

This is the best that she could do. (Correct)

20. To show equality ‘as’ is used both before and after the adjective.
Thus,

I can run as fast, if not faster than you. (Incorrect)

I can run as fast as, if not faster than you. (Correct)

21. Even though more than one’indicates a plural sense, it agrees with a singular noun and takes a singular verb.
Thus,

More than one students completed their project. (Incorrect)

More than one student completed his project. (Correct)

22. ‘Scarcely’and ‘hardly’ are followed by ‘when’ and not by ‘than’.
Thus,

Hardly had the teacher left the room than the pupils started enjoying. (Incorrect)

Hardly had the teacher left the room when the pupils started enjoying. (Correct)

23. ‘Though’ is followed by ‘yet’and not by ‘but’.
Thus,

Though he is poor but he is honest. (Incorrect)

Though he is poor, yet he is honest. (Correct)

24. ‘Lest’must be followed by ‘should’ or by nothing at all and ‘Such’ must be followed by ‘as’.
Thus,

Work hard lest you will fail. (Incorrect)

Work hard lest you should fail. (Correct)

Work hard lest you fail. (Correct)

He is such a writer that everybody should read his books. (Incorrect)

He is such a writer as everybody should read his books. (Correct)

25. ‘Unless’ expresses a condition and is always used in the negative sense. Thus ‘not’ is never used with ‘unless’.
Thus,

Unless you do not work hard, you will not excel in the examination. (Incorrect)

Unless you work hard, you will not excel in the examination. (Correct)

26. ‘When’denotes a general sense and ‘while’ implies a time duration of doing something.
Thus,

When learning how to sing, technique is of utmost importance. (Incorrect)

While learning how to sing, technique is of utmost importance. (Correct)

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