What is Geometry?
Geometry is the branch of mathematics that deals with shapes, angles, dimensions and sizes of a variety of things we see in everyday life. In other words, Geometry is the study of different types of shapes, figures and sizes in Maths or real life. We get to learn about a lot many things in geometry such as lines, angles, transformations, symmetries and similarities. Due to its vast coverage, there are so many terms in geometry that often we need to refer to various books for the same. How about organising all the important terms in geometry in one place? Let us list down some important terms and definitions in geometry
Below is the list containing some important terms and definitions in geometry along with their graphical representations –
Point and Lines
A point is an exact location in space. It has no dimensions.
A line is a collection of points along a straight path that extends endlessly in both directions.
A line segment is a part of a line having two endpoints.
The length of this line segment will be denoted as AB.
A ray is a part of the line segment that has only one endpoint.
The above ray will be read as ray CD. It is important to note here that the endpoint of the ray is always the first letter.
Parallel lines are the lines that do not intersect or meet each other at any point in a plane. They are always parallel and are equidistant from each other. Parallel lines are non-intersecting lines. Symbolically, two parallel lines l and m are written as l || m.
Perpendicular lines are formed when two lines meet each other at the right angle or 90 degrees. Below if we have an example of perpendicular lines, where AB ⊥ XY
Two or more lines that share exactly one common point are called intersecting lines. This common point exists on all these lines and is called the point of intersection.
A transversal is defined as a line intersecting two or more given lines in a plane at different points.
When two rays combine with a common endpoint and the angle is formed.
Parts of an Angle
Arms – Arms are the two straight line segments from a vertex.
Angle – If a ray is rotated about its endpoint, the measure of its rotation is called the angle between its initial and final position.
An angle whose measure is ninety degrees (90°) is known as a right angle and it is larger than an acute angle. In other words, when the arms of the angle are perpendicular to each other they form a right angle.
An angle whose measure is more than zero degrees 0° and less than ninety degrees 90° is known as an acute angle.
An angle whose measure is more than ninety degrees (90°) and less than one hundred and eighty degrees (180°) is called the obtuse angle. An obtuse angle measures between ninety degrees (90°) to one hundred and eighty degrees (180°).
The angle if the arms of the angle are in an opposite direction to each other is known as the straight angle. In other words, the type of angle that measures 180 degrees (180°) is called a straight angle.
An angle whose measure is more than one hundred and eighty degrees (180°) and less than three hundred and sixty degrees (360°) is called the reflex angle.
If both the arms of the angle overlap each other then they form an angle that measures three hundred and sixty degrees is known as a complete angle. In other words, the type of angle that measures or equals to three hundred and sixty degrees (360°) is known as a complete angle.
When the sum of two angles is 90°, then the angles are known as complementary angles. In other words, if two angles add up to form a right angle, then these angles are referred to as complementary angles.
When the sum of two angles is 180°, then the angles are known as supplementary angles. In other words, if two angles add up, to form a straight angle, then those angles are referred to as supplementary angles.
The word triangle is made from two words – “tri” which means three and “angle”. Hence, a triangle can be defined as a closed figure that has three vertices, three sides, and three angles. The following figure illustrates a triangle ABC –
Triangles Based on Sides
A triangle is said to be a scalene triangle if none of its sides is equal. If none of the sides is equal, then the angles are not equal to each other.
A triangle is said to be an Isosceles triangle if its two sides are equal. If two sides are equal, then the angles opposite to these sides are also equal.
For example, in the following triangle, AB = AC. Therefore ∆ABC is an Isosceles triangle.
∠B = ∠C
A triangle is said to be an equilateral triangle if all its sides are equal. Also, if all the three sides are equal in a triangle, the three angles are equal.
Triangles Based on Angles
Acute Angled Triangle
An acute triangle is a triangle whose all three interior angles are acute. In other words, if all interior angles are less than 90 degrees, then it is an acute-angled triangle.
Right Angled Triangle
A triangle is said to be a right angled triangle if one of the angles of the triangle is a right angle, i.e. 90o. Suppose, we have a triangle, ABC where △ABC = 90o. Then such a triangle is called a right angled triangle which would be of a shape similar to the below figure.
Obtuse Angled Triangle
Obtuse triangles are those in which one of the three interior angles has a measure greater than 90 degrees. In other words, if one of the angles in a triangle is an obtuse angle, then the triangle is called an obtuse-angled triangle.
Circular Region – The part of the circle that consists of the circle and its interior is called the circular region.
Chord of a Circle – A line segment joining any two points on a circle is called a chord of the circle.
Circumference of a Circle – The perimeter of a circle is called the circumference of the circle. The ratio of the circumference of a circle and its diameter is always constant.
Concentric Circles – Circles having the same centre but with different radii are said to be concentric circles. Following is an example of concentric circles –
Arc of a Circle: An arc of a circle is referred to as a curve that is a part or portion of its circumference. Acute central angles will always produce minor arcs and small sectors. When the central angle formed by the two radii is 90o, the sector is called a quadrant because the total circle comprises four quadrants or fourths. When the two radii form a 180o or half the circle, the sector is called a semicircle and has a major arc.
Segment in a Circle: The area enclosed by the chord and the corresponding arc in a circle is called a segment. There are two types of segments – minor segment, and major segment.
Sector of a Circle: The sector of a circle is defined as the area enclosed by two radii and the corresponding arc in a circle. There are two types of sectors, minor sector, and major sector.
2 – Dimensional Shapes
Vertex – The meeting point of a pair of sides of a polygon is called its vertex. For example, the shapes such as cube and cuboid are 3-dimensional shapes. For example, in the below figure, ABCD, the vertices are A, B, C and D.
Side – The line joining two vertices is called a side. For example, in the above polygon, ABCD, AB is one of the sides of the polygon.
Adjacent Sides – Any two sides of a polygon having a common endpoint are called its adjacent sides. For example, in the given polygon ABCD, the four adjacent pairs of sides are ( AB, BC ), ( BC, CD ), ( CD, DA ) and ( DA, AB ).
A square is a quadrilateral that has four equal sides and four right angles.
A rectangle is a type of quadrilateral that has equal opposite sides and four right angles.
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
A trapezium is a quadrilateral in which one pair of opposite sides is parallel.
A rhombus is a quadrilateral with four equal sides.
3 – Dimensional Shapes
3 Dimensional shapes or 3D shapes are the shapes that have all three dimensions, i.e. length, breadth and height. The room of a house is a common example of a 3 d shape. Let us understand some of these shapes in detail. Some common terms used to define the 3D shapes are –
Faces – A face refers to any single flat surface of a 3D shape.
Edges – An edge is a line segment on the boundary joining one vertex (corner point) to another. It is similar to the sides we have in 2D shapes.
Vertices – The meeting point of a pair of sides of a polygon is called its vertex.
Let us now understand some of the common 3D shapes –
A 3D shape having six rectangular faces is called a cuboid. Ex a matchbox, a brick, a book etc. In other words, it is an extension of a rectangle in a 3D plane.
Below we have a general diagram of a cuboid
A cuboid has 6 rectangular faces, out of which the opposite sides are identical.
A cuboid has 12 eddges
A cuboid has 8 vertices
A cuboid whose length, breadth and height are equal is called a cube. Examples of a cube are sugar cubes, cheese cubes and ice cubes. In other words, it is an extension of a square in a 3D plane.
Below we have a general diagram of a cube
A cube has 6 rectangular faces, out of which all are identical.
A cube has 12 edges
A cube has 8 vertices
A cylinder is a solid with two congruent circles joined by a curved surface.
Below we have a general diagram of a Cylinder
A cylinder has one curved surface and two flat faces.
A cylinder has two curved edges.
A cylinder has no vertices.
A circular cone has a circular base that is connected by a curved surface to its vertex. A cone is called a right circular cone if the line from its vertex to the centre of the base is perpendicular to the base. An ice-cream cone is an example of a cone
Below we have a general diagram of a Cone
A cone has one flat face and one curved surface.
A cone has one curved edge.
A cone has one vertex.
A sphere is a solid formed by all those points in space that are at the same distance from a fixed point called the centre. In other words, it is an extension of a circle in a 3D plane.
Below we have a general diagram of a Sphere
A cone has one curved surface.
A cone has no edge.
A cone has no vertex.
A prism is a solid whose side faces are parallelograms and whose ends (bases) are congruent parallel rectilinear figures. A prism is a polyhedron that has two congruent and parallel polygons as bases. The rest of the faces are rectangles.
Base of a Prism – The end on which a prism may be supposed to stand is called the base of the prism.
Height of a Prism – The perpendicular distance between the ends of a prism is called the height of a prism.
Principal axis of a Prism – The straight line joining the centres of the ends of a prism is called the axis of the prism.
Length of a Prism – The length of a Prism is a portion of the axis that lies between the parallel ends.
Lateral faces – All faces other than the bases of a prism are called its lateral faces
Lateral edges – The lines of intersection of the lateral faces of a prism are called the lateral edges of a prism.
A solid shape bounded by polygons is called a polyhedron.
A rectangular prism is a polyhedron with two congruent and parallel bases. Some of the real-life examples of a rectangular prism are rooms, notebooks, geometry boxes etc. Following is the general representation of a rectangular prism.
Oblique Rectangular Prism
An oblique rectangular prism is a prism in which all the angles are not right angles. This means that a rectangular prism is a prism in which bases are not perpendicular to each other which is why it is called the oblique rectangular prism. In simple words, in an oblique rectangular prism, bases are not aligned one directly above the other. Following is the general representation of an oblique rectangular prism.
Right Rectangular Prism
A prism with rectangular bases is called a rectangular prism. In other words, a rectangular prism in which bases are perpendicular to each other is called the right rectangular prism. Following is the general representation of a right rectangular prism.
Right Triangular Prism – A right prism is called a right triangular prism if its ends are triangles. In other words, a triangular prism is called a right triangular prism if its lateral edges are perpendicular to its ends.
If the number of sides in the rectilinear figure forming the ends or the bases is 4, it is called a quadrilateral prism.
If the number of sides in the rectilinear figure forming the ends or the bases is 5, it is called a pentagonal prism.
If the number of sides in the rectilinear figure forming the ends or the bases is 6, it is called a hexagonal prism.
A pyramid is a polyhedron whose base is a polygon of any number of sides and other faces are triangles with the common vertex if all corners of a polygon are joined to a point not lying in its plane we get a pyramid. In other words, a pyramid is a solid whose base is a plane rectilinear figure and whose side faces are triangles having a common vertex, called the vertex of the pyramid.
Vertex – The common vertex of the triangular faces of a pyramid is called the vertex of the pyramid.
Height – The height of a pyramid is the length of the perpendicular from the vertex to the base. In other words, The length of the perpendicular drawn from the vertex of a pyramid to its base is called the height of the pyramid.
Axis – The axis of a pyramid is a straight line joining the vertex to the central point of the base.
Lateral edges – The edges through the vertex of a pyramid are known as its lateral edges.
Lateral faces – The side faces of a pyramid are known as its lateral faces.
A platonic solid is a polyhedron. It is interesting as well as surprising to know that there are exactly five platonic solids. These five platonic solids are tetrahedron, cube, octahedron, icosahedron, and dodecahedron.
Tetrahedron – Polyhedron or metallic solid whose faces are congruent equilateral Triangles is called the tetrahedron.
Octahedron – The platonic solid which has four equilateral triangles meeting at each vertex is known as the octahedron.
Dodecahedron – A platonic solid can have every face as a pentagon is known as a dodecahedron. In a Dodecahedron, three pentagons meet at every vertex.