Mensuration Formulae for Triangles and Quadrilaterals

Mensuration Formulae

Perimeter and Area of Various Shapes

Triangle

• Perimeter: P = a + b + c where a, b, and c are side lengths
• Area: A = (b \* h) / 2 where b is base and h is height
• Heron's Formula: A = √(s(s-a)(s-b)(s-c)) where s is semi-perimeter (s = (a + b + c) / 2)

Quadrilateral

• Perimeter: P = a + b + c + d where a, b, c, and d are side lengths
• Area: A = (d \* h) / 2 where d is diagonal and h is height
• Trapezium: A = (1/2) \* (sum of parallel sides) \* height

Circle

• Circumference: C = 2 \* π \* r where r is radius
• Area: A = π \* r^2
• Sector: A = (θ / 360) \* π \* r^2 where θ is angle in degrees

Polygon

• Perimeter: P = sum of side lengths
• Area: A = (n \* s^2) / (4 \* tan(π/n)) where n is number of sides and s is side length

Volume and Surface Area of 3D Shapes

Cube

• Volume: V = s^3 where s is side length
• Surface Area: SA = 6 \* s^2

Cylinder

• Volume: V = π \* r^2 \* h where r is radius and h is height
• Surface Area: SA = 2 \* π \* r \* (r + h)

Sphere

• Volume: V = (4/3) \* π \* r^3 where r is radius
• Surface Area: SA = 4 \* π \* r^2

Mensuration Formulae

Perimeter and Area of Various Shapes

Triangle

• The perimeter of a triangle is the sum of its side lengths: P = a + b + c
• The area of a triangle can be calculated using base and height: A = (b \* h) / 2
• Heron's Formula provides an alternative method for finding the area of a triangle: A = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter

Quadrilateral

• The perimeter of a quadrilateral is the sum of its side lengths: P = a + b + c + d
• The area of a quadrilateral can be calculated using diagonal and height: A = (d \* h) / 2
• For a trapezium, the area can be found using the formula: A = (1/2) \* (sum of parallel sides) \* height

Circle

• The circumference of a circle is proportional to its radius: C = 2 \* π \* r
• The area of a circle can be calculated using its radius: A = π \* r^2
• The area of a sector can be found using the angle and radius: A = (θ / 360) \* π \* r^2

Polygon

• The perimeter of a polygon is the sum of its side lengths: P = sum of side lengths
• The area of a polygon can be calculated using its number of sides and side length: A = (n \* s^2) / (4 \* tan(π/n))

Volume and Surface Area of 3D Shapes

Cube

• The volume of a cube is proportional to the cube of its side length: V = s^3
• The surface area of a cube is proportional to the square of its side length: SA = 6 \* s^2

Cylinder

• The volume of a cylinder is proportional to the radius squared and height: V = π \* r^2 \* h
• The surface area of a cylinder is proportional to the radius and height: SA = 2 \* π \* r \* (r + h)

Sphere

• The volume of a sphere is proportional to the cube of its radius: V = (4/3) \* π \* r^3
• The surface area of a sphere is proportional to the square of its radius: SA = 4 \* π \* r^2

Tense

Definition and Types

• Tense refers to the form of a verb that indicates when an action takes place
• There are 12 basic tenses in English grammar

Present Tense

• Describes an action that is happening now
• Formed using the base form of the verb
• Examples:
  • I go to school every day (Simple present)
  • I am studying for my exam (Present continuous)
  • I have studied English for three years (Present perfect)
  • I have been studying English for three years (Present perfect continuous)

Past Tense

• Describes an action that happened in the past
• Formed using the past form of the verb
• Examples:
  • I went to the park yesterday (Simple past)
  • I was studying at 9 pm (Past continuous)
  • I had eaten dinner before I went to the park (Past perfect)
  • I had been studying for three hours before I took a break (Past perfect continuous)

Future Tense

• Describes an action that will happen in the future
• Formed using the future form of the verb
• Examples:
  • I will go to the party tonight (Simple future)
  • I will be studying at 10 pm (Future continuous)
  • I will have studied English for four years by next year (Future perfect)
  • I will have been studying English for four years by next year (Future perfect continuous)

Other Tenses

Conditional Tense

• Used to describe hypothetical or uncertain situations
• Examples:
  • If you heat ice, it melts (Zero conditional)
  • If it rains, I will take an umbrella (First conditional)
  • If I won the lottery, I would buy a house (Second conditional)
  • If I had studied harder, I would have passed the exam (Third conditional)

Subjunctive Mood

• Used to express doubt, uncertainty, or possibility
• Examples:
  • It is necessary that he be there
  • I suggest that he take a break