Geometry Formulas with solved examples
Geometry, one of the oldest branches of mathematics, is related to the study of properties and dimensions of different types of shapes, figures, surfaces and solids etc. and Geometry formulas help to find the same.
The Geometry Formulas assist in calculation of the perimeters, areas, volumes and surface areas etc. of different types of 2-dimensional and 3-dimensional figures.
The list of all the important Geometry Formulas is as given below
Name of the Shape | Formulas |
Triangle | Area = ½ x base x perpendicular Perimeter = Sum of all sides |
Rectangle | Area = Length x Breadth Perimeter = 2 (Length + Breadth) Diagonal (d) = √ ((length)^{2} + (breadth)^{2}) |
Square | Area = side x side Perimeter = 4 X Side Diagonal (d) = side x √2 |
Circle | Area = π x r x r Circumference = 2 x π x r Diameter = 2r where, r is the radius of the circle |
Trapezium | Perimeter = Sum of all sides Area = ½ x sum of parallel sides x distance between the parallel sides |
Parallelogram | Perimeter = Sum of all sides Area = base x height |
Cube | Surface Area = 6 x (side)^{2} Volume = (side)^{3} |
Cuboid | Surface Area = 2(lb + bh + hl) Volume = l x b h where, l = length of cuboid b = breadth of cuboid h = height of cuboid |
Sphere | Surface Area = 4 x π x r^{2} Volume = 4/3 x π x r^{3} Diameter = 2r where, r is the radius of the sphere |
Cone | Base Area = πr^{2} Total Surface Area = πr (r + l) Volume = 1/3πr^{2}h Slant height (l) = √ (h^{2} + r^{2}) where, r = radius of cone h = height of the cone l = slant height of the cone |
Cylinder | Total Surface Area = 2πr (r + h) Curved Surface Area = 2πrh Volume = πr^{2}h Base Area = πr^{2} where, r = radius of the cylinder h = height of the cylinder |
Solved Examples
Question1. Find the area and perimeter of the following:
Rectangle of length 5 cm and breadth 8 cm
Circle of radius 7 cm
Right-angled triangle whose length of the perpendicular, base and the hypotenuse are given to be 5cm, 6cm and 9 cm respectively.
Solution:
a) Area of rectangle = Length x Breadth
Putting the values, Area = 5cm x 8cm = 40 cm^{2}
Perimeter = 2 (Length + Breadth)
Putting the values, Perimeter = 2 (5 + 8) = 26 cm
b) Area of Circle = π x r x r, where r is the radius of the circle
Putting the values, Area = 22/7 x 7 x 7 = 154 cm^{2}
Circumference = 2 x π x r
Putting the values,
Circumference = 2 x 22/7 x 7 = 44 cm
c) Area of triangle = ½ x base x perpendicular
Putting the values,
Area = ½ x 6 x 5
Or Area = 15 cm^{2}
Perimeter of triangle= sum of all sides
Or Perimeter = Perpendicular + Base + Hypotenuse
Putting the values,
Perimeter = 5 + 6 + 9
Or Perimeter = 20 cm
Question2. Find the surface area of a cubical water tank of side 1.5 m.
Solution: Total Surface area of the cube = 6 (side)^{2}
Therefore, total surface area = 6 x 1.5 x 1.5
Or total surface area = 6 x 2.25
Or total surface area = 13.50 sq. m
Question 3. Find the curved surface area of a cylinder whose height is 14 cm and the radius is 10 cm.
Solution: Radius (r) = 10 cm
Height (h) = 14 cm
Lateral/Curved Surface area of cylinder = 2πrh
= 2 x 22/7 x 10 x 14
= 880 sq. cm
Question 4. Calculate the volume of the cone if its slant height and radius of the base are given to be 9 cm and 4 cm respectively.
Solution: Now, Slant height (l) = √ (h^{2} + r^{2})
Putting values in the above formula to find h
9 = √ (h^{2} +(4)^{2})
Squaring both sides
81 = h^{2} + 16
Therefore, h^{2} = 81 – 16
Or h = √65
Or h = 8.06 cm
Putting values in the formula, Volume = 1/3πr^{2}h to find the volume of the given cone
Volume = 1/3 x 22/7 x 4 x 4 x 8.06 = 128.96 x 0.33 x 3.14 = 133.62 cm^{3}