Important Formulas of Geometry
Geometry is a wing of mathematics. In geometry, we deal with objects of different shapes and sizes and try to calculate the required values from the given data. Geometry is a combination of two words Geo which means Earth and Metron mean measurement. It is not today’s practice as we measure length, Area and volume very often in our regular life.
Geometry can be divided into two types: Plane Geometry and Solid Geometry. Plane Geometry is the geometry of two-dimensional objects and deals with shapes such as circles, triangles, rectangles, squares, Trapeziums and many more flat shapes. Whereas, Solid Geometry is the geometry of three-dimensional objects such as cubes, cuboids, cylinders, spheres, cones and many related geometrical shapes and structures.
In competitive exams, the general concern of all most all students is the Aptitude and in Aptitude, it’s the problems of Geometry. Many candidates even tell that they skip these problems as all Geometric formulas are very complicated. However, there are some basic formulas that we use frequently and many problems require these basic formulas to solve.
Following are some basic but important Formulas of Geometry.
Plane Geometry
This is the Geometry related to flat two dimensional objects. The objects belonging to this category are Circle, Triangle, Rectangle, Square, Parallelogram, Rhombus, etc.
Circle
The circle is a shape where all points of the shape are at a given distance from a given point or centre. The basic formulas related to circles are given in the following table.
Parameter |
Formula |
|---|---|
| Circumference of a circle | 2πr |
| Area of a circle | πr2 |
| ARC (AB) | (2πrθ)/360o |
| Area of Sector AOB | (θ/360o) × πr2 |
Triangle
A triangle is a closed shape polygon made with three vertices. These are of the following types.
Parameter |
Formula |
|---|---|
| Area of Any Triangle | (1/2) × Base × Height |
| Perimeter of Any Triangle | a + b + c |
| s = Perimeter/2 | (a + b + c) / 2 |
| Area of Any Triangle | √{s(s-a)(s-b)(s-c)} |
| Area of Equilateral Triangle | (√3/4) × a2 |
| Perimeter of Equilateral Triangle | 3 × a |
| Area of Isosceles Triangle | (b/4) × √(4a2 – b2) |
| In Any Triangle | ∠α + ∠β + ∠θ = 180° |
| Area of the triangle if two sides and the angle between them is given (refer to Scalene Triangle of the above image) |
(1/2) × a × b sinθ (1/2) × b × c sinα (1/2) × c × a sinβ |
Rectangle
A rectangle is a quadrilateral with four right angles and Two of the sides are larger than the other two.
Parameter |
Formula |
|---|---|
| Perimeter of rectangle | (a+b) × 2 |
| Area of rectangle | a × b |
| Diagonal (d) | √(a2 + b2) |
Square
a square is a quadrilateral with four right angles and 4 equal sides.
Parameter |
Formula |
|---|---|
| Perimeter of square | 4 × a |
| Area of square | (1/2) × d2 × a2 |
| Diagonal (d) | √2 × a |
Parallelogram
A parallelogram is a simple quadrilateral with two pairs of parallel sides.
Parameter |
Formula |
|---|---|
| Area of Parallelogram | b × h |
Rhombus
A rhombus is a quadrilateral whose four sides have the same length.
Parameter |
Formula |
|---|---|
| Area of Rhombus | a × h (1/2) × d1 × d2 |
| Sides (a) of a Rhombus | (1/2) × √(d1+d2) |
Trapezium
A trapezium is a quadrilateral with at least one pair of parallel sides.
Parameter |
Formula |
|---|---|
| Area of Trapezium | (1/2) × (a + b) × h |
| s | (m+c+d)/2 ( if :- (a-b)=m ) |
| h | (2/m) × √{s(s-m)(s-c)(s-d)} |
Cuboid
A cuboid is a polyhedron having six quadrilateral faces.
Parameter |
Formula |
|---|---|
| Volume of a Cuboid | l × b × h √(A1 × A2 × A3) |
| Total Surface area of a cuboid | 2 (l×b + b×h + l×h) |
| Diagonal of cuboid | √(l2 + b2 + h2) |
Cube
A cube is a polyhedron having six square faces.
Parameter |
Formula |
|---|---|
| Volume of a cube | a3 |
| Total Surface Area | 6 × a2 |
| Diagonal of a Cube | √3 × a |
Cylinder
A cylinder is a three-dimensional solid that holds two parallel circular bases joined by a curved surface having same cross-section from one end to the other.
Parameter |
Formula |
|---|---|
| Volume of a Cylinder | πr2h |
| Curved Surface Area of a Cylinder | 2πrh |
| Total Surface Area of a Cylinder | 2πr2 + 2πrh |
Sphere
A sphere is a three-dimensional round-shaped object where every point on the surface is the same distance from the centre.
Parameter |
Formula |
|---|---|
| Volume of a Sphere | (4/3) × πr3 |
| Surface area of a Sphere | 4πr2 |
| Volume of a Hemisphere | (2/3) × πr3 |
| Curved surface area of a Hemisphere | 2πr2 |
| Total Surface area of Hemisphere | 3πr2 |
Right Circular Cone
A Right Circular Cone is a three-dimensional solid that holds a circular base and an equidistance point from the circumference of the circle.
Parameter |
Formula |
|---|---|
| Slant Height (s) of a Right Circular Cone | √(h2 + r2) |
| Volume of the Right Circular Cone | (1/3) × πr2h |
| Curved Surface area of Right Circular Cone | πrs |
Frustum of Right Circular Cone
A cutting section parallels to the base but not passing through the vertex of a cone portion forms the Frustum of a Right Circular Cone.
Parameter |
Formula |
|---|---|
| Volume of the Frustum of Right Circular Cone | (πh/3) × (r2 + R2 + Rr) |
| Slant Height (s) of Frustum of Right Circular Cone | √{h2 + (R – r)2} |
| Curved Surface Area of Frustum of Right Circular Cone | π(r+R)s |
| Total Surface Area of Frustum of Right Circular Cone | π{(r + R)s + r2 + R2} |