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