Geometrical Concept part -2

Dear Readers,
Today in this post, we are providing you remaining part of the necessary formula and concepts  related to Geometry and Mensuration  from the Quant section.

TRIANGLES : Triangles are closed figures containing three angles and three sides.

 

General Properties of Triangles:
The sum of the two sides is greater than the third side: a + b > c, a + c > b, b + c > a
The sum of the three angles of a triangle is equal to 180°: In the triangle  ∠A + ∠B + ∠C = 180°
 Area of a Triangle:
Area of a triangle = 1/2 x base x height=1/2 x a x h
Area of a triangle= 1/2 bcsinA = 1/2ab sinC=1/2 acsinB
Area of a triangle = abc/4R where R circumradius
Area of a triangle= r x s where r inradius and s = (a+b+c)/2

REGULAR POLYGONS : A regular polygon is a polygon with all its sides equal and all its interior angles equal. All vertices of a regular lie on a circle whose center is the center of the polygon.

Each interior angle of a regular polygon = 180(n-2)/n
Sum of all the angles of a regular polygon = n x180(n- 2)/n = 180(n-2).
Quadrilateral: A quadrilateral is any closed shape that has four sides. The sum of the measures of the angles is 360°. Some of the known quadrilaterals are square, rectangle, trapezium, parallelogram and rhombus.
Square: A square is regular quadrilateral that has four right angles and parallel sides. The sides of a
square meet at right angles. The diagonals also bisect each other perpendicularly.
If the side of the square is a, then its
Perimeter = 4a,
•Area = a^2 and the length of the diagonal = √2a
 
Rectangle: A rectangle is a parallelogram with all its angles equal to right angles.
Area of a rectangle = length × breadth
Perimeter = 2(sum of length and breath)
Parallelogram: A parallelogram is a quadrangle in which opposite sides are equal and parallel.
Any two opposite sides of a parallelogram are called bases, a distance between them is called a height.
Area of a parallelogram = base × height
Perimeter = 2(sum of two consecutive sides)
 
Rhombus: If all sides of parallelogram are equal, then this parallelogram is called a rhombus.
Area of a rhombus = 1/2 product of diagonals
Perimeter = 4a,

 

Trapezoid: Trapezoid is a quadrangle two opposite sides of which are parallel.

Area of a trapezoid = 1/2(Sum of parallel sides)height

CIRCLE: A circle is a set of all points in a plane that lie at a constant distance from a fixed point. The fixed point is called the center of the circle and the constant distance is known as the radius of the circle.

Arc: An arc is a curved line that is part of the circumference of a circle. A minor arc is an arc less than the semicircle and a major arc is an arc greater than the semicircle.

Chord: A chord is a line segment within a circle that touches 2 points on the circle.
Diameter: The longest distance from one end of a circle to the other is known as the diameter. It is
equal to twice the radius.
Circumference: The perimeter of the circle is called the circumference.
 circumference = 2πr, where r is the radius of the circle.
 Area of a circle: Area = π x (radius)^2 = πr^2.
Sector: A sector is like a slice of pie (a circular wedge).
Area of Circle Sector: (with central angle θ) Area = θ/360 xπ x r^2
Length of a Circular Arc: (with central angle θ) The length of the arc = θ/360 x2π x r
Tangent of circle: A line perpendicular to the radius that touches ONLY one point on the circle
 
Cuboid: A parallelepiped whose faces are rectangular is called a cuboid. The three dimensions
associated with a cuboid are its length, breadth and height (denoted as l, b and h here.)
• The total surface area of the cuboid = 2(lb + bh + hl)
• Volume of a cuboid = lbh

 

Cube: A cube is a parallelepiped all of whose faces are squares.

Total surface area of the cube = 6a^2

• Volume of the cube = a^3

 

Right Circular Cylinder: A right circular cylinder is a right prism whose base is a circle.  the cylinder has a base of radius r and a height of length h.

Curved surface area of the cylinder = 2πrh

• Total surface area of the cylinder = 2πrh + 2πr^2
• Volume of the cylinder = πr^2h

 

Right Circular Cone: a right circular cone is a pyramid whose base is a circle. In , the right circular cone has a base of radius r and a height of length h.
Curved surface area of the cone = πrl
• Total surface area of the cone = πrl + πr^2
• Volume of the cone =1/3πr^2h

 

Sphere: A sphere is a set of all points in space which are at a fixed distance from a given point. The fixed point is called the centre of the sphere, and the fixed distance is the radius of the sphere.
Surface area of a sphere = 4πr^2
• Volume of a sphere = 4/3πr^3

 

 
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