Page 1 HERONS FORMULA Measuring the length, area and volume of plane figures is called â€˜Mensurationâ€™. Perimeter Perimeter of a figure is the length of the boundary of that figure. A B C D In quadrilateral ABCD, Perimeter = Total Length Of Boundary Of ABCD = AB + BC + CD + DA Area Area of a figure is the total area enclosed by that figure. In quadrilateral ABCD, Area = Area Enclosed By ABCD Triangle L B C Perimeter = a + b + c Where a, b, c Sides Of Triangle Area 1. Area = Altitude Base 2 1 × × 2. Area = c) (s b) (s a) (s s - - - Where a, b, c Sides Of Triangle s 2 c b a + + Special Triangle Cases There are few formulas, which can be applied only in case of a particular type of triangle, such as right angled triangle, isosceles triangle, equilateral triangle etc. Note These are not general formulas and can not be applied to every type of triangle. 1. Right Angled Triangle B C Perimeter = a + b + c Area = 2 1 × Base × Altitude = 2 1 × a × b Pythagoras Theorem Hypotenuse 2 = Perpendicular 2 + Base 2 2. Right Angled Isosceles Triangle B C Perimeter = 2a + b Where a Equal Side b Unequal Side Proof Perimeter = Equal Side1 + Equal Side2 + Unequal Side = (2 × Equal Side) + Unequal Side Area = 2 a 2 1 Proof Area = 2 1 × Base × Altitude = a a 2 1 × × = 2 a 2 1 Hypotenuse = a 2 Proof Hypotenuse 2 = Perpendicular 2 + Base 2 Page 2 HERONS FORMULA Measuring the length, area and volume of plane figures is called â€˜Mensurationâ€™. Perimeter Perimeter of a figure is the length of the boundary of that figure. A B C D In quadrilateral ABCD, Perimeter = Total Length Of Boundary Of ABCD = AB + BC + CD + DA Area Area of a figure is the total area enclosed by that figure. In quadrilateral ABCD, Area = Area Enclosed By ABCD Triangle L B C Perimeter = a + b + c Where a, b, c Sides Of Triangle Area 1. Area = Altitude Base 2 1 × × 2. Area = c) (s b) (s a) (s s - - - Where a, b, c Sides Of Triangle s 2 c b a + + Special Triangle Cases There are few formulas, which can be applied only in case of a particular type of triangle, such as right angled triangle, isosceles triangle, equilateral triangle etc. Note These are not general formulas and can not be applied to every type of triangle. 1. Right Angled Triangle B C Perimeter = a + b + c Area = 2 1 × Base × Altitude = 2 1 × a × b Pythagoras Theorem Hypotenuse 2 = Perpendicular 2 + Base 2 2. Right Angled Isosceles Triangle B C Perimeter = 2a + b Where a Equal Side b Unequal Side Proof Perimeter = Equal Side1 + Equal Side2 + Unequal Side = (2 × Equal Side) + Unequal Side Area = 2 a 2 1 Proof Area = 2 1 × Base × Altitude = a a 2 1 × × = 2 a 2 1 Hypotenuse = a 2 Proof Hypotenuse 2 = Perpendicular 2 + Base 2 = 2 2 a a + [Perpendicular = Base] = 2 2 a a + = 2 2a = a 2 3. Equilateral Triangle B C Perimeter = 3a Where a Equal Sides Proof Perimeter = Side1 + Side2 + Side3 = a + a + a = 3a Area = 2 a 4 3 Proof Area = Altitude Base 2 1 × × = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? - × × 2 2 2 a a a 2 1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? - = 2 2 2 a a Altitude Theorem, Pythagorus By = 4 a a a 2 1 2 2 - × × = 4 3a a 2 1 2 × × = 2 3 a a 2 1 × × = 2 a 4 3 Altitude = a 2 3 Proof In right angled ? ABD, Altitude = 2 2 2 a a ? ? ? ? ? ? ? ? - = 4 a a 2 2 - = 4 3a 2 = a 2 3 Quadrilateral A B C D Perimeter = a + b + c + d Where a, b, c, d Sides Of Trapezium Area = ( ) ( ) ?B CD Area ? ABD Area + = ( ) ( ) ( ) 1 1 1 1 1 1 1 c s b s a s s - - - + ( ) ( ) ( ) 2 2 2 2 2 2 2 c s b s a s s - - - Special Quadrilateral Cases There are few formulas, which can be applied only in case of a particular type of quadrilateral, such as rectangle, square, parallelogram etc. 1. Trapezium A B C D Perimeter = a + b + c + d Where a, b, c, d Sides Of Trapezium Area = Altitude ) S (S 2 1 2 1 × + × Where 1 S , 2 S Parallel Sides Of Trapezium Proof Area ( ) ( ) ( ) ?B CD Area ? ABD Area ABCD + = = ? ? ? ? ? ? ? ? + × + ? ? ? ? ? ? ? ? × × h S 2 1 h S 2 1 2 1 = ( ) 2 1 S S h 2 1 + × × 2. Parallelogram A B C D Perimeter = ( ) b a 2 + Where Page 3 HERONS FORMULA Measuring the length, area and volume of plane figures is called â€˜Mensurationâ€™. Perimeter Perimeter of a figure is the length of the boundary of that figure. A B C D In quadrilateral ABCD, Perimeter = Total Length Of Boundary Of ABCD = AB + BC + CD + DA Area Area of a figure is the total area enclosed by that figure. In quadrilateral ABCD, Area = Area Enclosed By ABCD Triangle L B C Perimeter = a + b + c Where a, b, c Sides Of Triangle Area 1. Area = Altitude Base 2 1 × × 2. Area = c) (s b) (s a) (s s - - - Where a, b, c Sides Of Triangle s 2 c b a + + Special Triangle Cases There are few formulas, which can be applied only in case of a particular type of triangle, such as right angled triangle, isosceles triangle, equilateral triangle etc. Note These are not general formulas and can not be applied to every type of triangle. 1. Right Angled Triangle B C Perimeter = a + b + c Area = 2 1 × Base × Altitude = 2 1 × a × b Pythagoras Theorem Hypotenuse 2 = Perpendicular 2 + Base 2 2. Right Angled Isosceles Triangle B C Perimeter = 2a + b Where a Equal Side b Unequal Side Proof Perimeter = Equal Side1 + Equal Side2 + Unequal Side = (2 × Equal Side) + Unequal Side Area = 2 a 2 1 Proof Area = 2 1 × Base × Altitude = a a 2 1 × × = 2 a 2 1 Hypotenuse = a 2 Proof Hypotenuse 2 = Perpendicular 2 + Base 2 = 2 2 a a + [Perpendicular = Base] = 2 2 a a + = 2 2a = a 2 3. Equilateral Triangle B C Perimeter = 3a Where a Equal Sides Proof Perimeter = Side1 + Side2 + Side3 = a + a + a = 3a Area = 2 a 4 3 Proof Area = Altitude Base 2 1 × × = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? - × × 2 2 2 a a a 2 1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? - = 2 2 2 a a Altitude Theorem, Pythagorus By = 4 a a a 2 1 2 2 - × × = 4 3a a 2 1 2 × × = 2 3 a a 2 1 × × = 2 a 4 3 Altitude = a 2 3 Proof In right angled ? ABD, Altitude = 2 2 2 a a ? ? ? ? ? ? ? ? - = 4 a a 2 2 - = 4 3a 2 = a 2 3 Quadrilateral A B C D Perimeter = a + b + c + d Where a, b, c, d Sides Of Trapezium Area = ( ) ( ) ?B CD Area ? ABD Area + = ( ) ( ) ( ) 1 1 1 1 1 1 1 c s b s a s s - - - + ( ) ( ) ( ) 2 2 2 2 2 2 2 c s b s a s s - - - Special Quadrilateral Cases There are few formulas, which can be applied only in case of a particular type of quadrilateral, such as rectangle, square, parallelogram etc. 1. Trapezium A B C D Perimeter = a + b + c + d Where a, b, c, d Sides Of Trapezium Area = Altitude ) S (S 2 1 2 1 × + × Where 1 S , 2 S Parallel Sides Of Trapezium Proof Area ( ) ( ) ( ) ?B CD Area ? ABD Area ABCD + = = ? ? ? ? ? ? ? ? + × + ? ? ? ? ? ? ? ? × × h S 2 1 h S 2 1 2 1 = ( ) 2 1 S S h 2 1 + × × 2. Parallelogram A B C D Perimeter = ( ) b a 2 + Where a Side1 b Side2 Proof Perimeter = a + b + a + b = 2a + 2b = ( ) b a 2 + Area = Base × Altitude 3. Rhombus A B C D Perimeter = 4a Proof Perimeter = a + a + a + a [Sides of rhombus are equal] = 4a Area = 2 1 d d 2 1 × × Where d1 Diagonal1 d2 Diagonal2 4. Rectangle A B C D Perimeter = ( ) b a 2 + Where a Side1 b Side2 Proof Perimeter = a + b + a + b = 2a + 2b = ( ) b a 2 + Area = Length × Breadth Diagonal = 2 2 b a + [By Pythagorus Theorm] 5. Square A B C D Perimeter = 4a Where a Side Proof Perimeter = a + a + a + a [Sides of square are equal] = 4a Area = 2 a Proof Area = Breadth Lenglth × = a a × [Sides of square are equal] = 2 a Diagonal = a 2 Proof In right angled ? ABD, d = 2 2 a a + [Pythagoras Theorem] = 2 2a = 2 aRead More