Geometry is an important branch of mathematics that involves studying various shapes, figures, dimensions, angles, etc. It provides formulas to calculate the area, perimeter, circumference, and volume of various figures. Geometrical figures are divided into two types based on dimensions:
Triangles are three-sided 2D figures that satisfy the following conditions:
Area of triangle = ½ × (b × h), where b is base and h is the height of the triangle
The perimeter of a triangle = Sum of all sides
Types of Triangles
Triangles are differentiated based on their angles and sides.
Based on the angles:
Based on the sides:
Two triangles are said to be congruent if both are of the same shape and size. For example, two equilateral triangles of side 6 cm are congruent to each other.
Two triangles are said to be similar if both are of the same shape but different sizes. For example, two right-angled triangles are similar to each other but might not be congruent.
Also known as Thales’s theorem, it states that if a line DE is drawn parallel to the base BC of triangle ABC, then AD/DB = AE/EC.
This theorem applies only to right-angled triangles where one angle is equal to 90°. The longest side of the right-angled triangle is opposite the 90° angle and is known as hypotenuse.
According to this theorem, the square of the hypotenuse of a right-angled triangle is always equal to the sum of the squares of the other two sides of the triangle.
Example: In a right-angled triangle ABC, where AB is the hypotenuse, AB^2 = BC^2 + AC^2.
Closed geometrical figures made of straight lines are known as rectilinear figures. In layman’s terms, closed figures made of at least three line segments are known as polygons or rectilinear figures.
Polygons are classified based on the sides:
A regular polygon is a figure where all sides, exterior angles, and interior angles are equal.
The following are the formulas for polygons with side ‘n’:
Circles are 2D figures measured on the basis of the radius. As it is a 2D shape, it has both area and perimeter.
Area = πr^2
Circumference (Perimeter) = 2πr
Properties of Circle
This chapter is an important topic for class 9th mathematics and an essential concept for advanced and higher-level mathematics. Sound knowledge of this chapter will prepare you for advanced lessons.