Geometry: Branches, Formula, Plane and Solid Geometry

Geometry is the branch of mathematics that deals with the shape, size, angles, and dimensions of objects. We are able to understand all the shapes we see in our life better through various shapes in geometry. It helps us in measuring and calculating the volume, area, and perimeter of various shapes. In Geometry, we learn about different angles, lines, transformations, congruences and similarities in various figures. 

Read Also: Section Formula

Key Terms: Geometry, Two dimentional figure, Three dimentional figure, Polygons, Circles, Angles, Volume, Area, Geometric figure, Lines, Angles, Planes, Point

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Geometry

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Geometry can be defined as the branch of mathematics that deals with studying different types of sizes, shapes, figures, angles, and dimensions of various objects. Lines, angles, planes, and points are the base of geometry. All geometric figures are entirely based on these basic concepts. 

Geometry – Lines and Angles

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Branches in Geometry

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Algebraic Geometry

Algebraic geometry includes solving sets of zeros by using polynomial and linear algebraic equations. This branch of geometry involves studying zeros of polynomials. The applications of algebraic geometry include string theory, cryptography etc.

Differential Geometry

Differential geometry employs techniques from calculus and algebra for solving applications. Its applications include general relativity etc. It is used to differentiate the natural properties of surfaces, shapes, and curves.

Convex Geometry

Convex geometry is used for problem-solving in number theory in functional analysis and optimization. It comprises convex shapes. The techniques used are of real analysis.

Discrete Geometry

It is concerned with simple geometric objects like triangles, circles, lines, points, squares, etc. and their relative position.

Euclidean Geometry

Euclidean geometry includes the study of two-dimensional (Plane figures) and three-dimensional (solid figures) shapes. It includes theorems and axioms relating to angles, planes, similarities, congruence, points, and solid figures. This geometry is applied in crystallography, mathematical problem solving, computer science, etc.

Topology

Topology is employed and applied in consideration of completeness, uniform spaces, continuity, hyperspace topologies, metric spaces, nets, proximal continuity, grills, filters, compactness, separation axioms, initial and final structures, function spaces, clusters and bunches, and proximity spaces. This geometry is concerned with properties of space under continued mapping.

Also Check: Circles

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Formulas of Geometry

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Formulas of Geometry

Also Check: Triangles

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Two-Dimensional Geometry

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Two-dimensional geometry is also known as plane geometry. It deals with flat shapes. This includes two-dimensional circles, triangles, and lines. Two-dimensional shapes only have length and breadth. Some two-dimensional shapes are square, circle, triangle, rectangle, etc.

Important terms in two-dimensional geometry:

Point

Point refers to a specific location on the plane. Point only has a position and no dimension. A point is represented by a dot.

Point

Read Further: Area Related to Circle

Line

Line is a combination of infinite points. It has no curves. The line is straight and has no breadth. It stretches in both directions infinitely. In reference to geometry, a horizontal line is an x-axis and a vertical line is a y-axis.

• Line Segment – A line with a starting point and an ending point is known as a line segment.
• Ray - A line with a starting point but no ending point is known as a Ray.

Line

Angles

When two rays meet, an angle is formed. The rays share a common endpoint, known as the vertex of the angle. There are four types of angles:–

• Acute Angle – An angle that is smaller than the right angle is called an Acute angle. It ranges from anywhere between 0 to 90 degrees.
• Obtuse Angle – An angle that is less than 180 degrees but more than 90 degrees is called an Obtuse angle.
• Right Angle – A Right angle is an angle forming 90 degrees.
• Straight Angle – A straight angle is an angle formed by a straight line. The angle is 180 degrees.

Angles

Read Also: Perimeter and Area of a Circle

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Three-Dimensional Geometry

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Three-Dimensional geometry is also called solid geometry. It involves three-dimensional objects like cylinders, cubes, prisms, and spheres. Three-dimensional shapes have breadth, height, and length. Some three-dimensional shapes have faces and some such as spheres don’t.

Three-Dimensional Geometry

Attributes of three-dimensional shapes:

Edges

The line segment on the boundary that joins two vertices to each other is called an edge. Edges are the faces that meet in a straight line. This forms the skeleton in three-dimensional shapes. 

Also Check: Similarities of Triangle

Here are some three-dimensional shapes and the number of edges they have:

1. Triangular Pyramid - 6 edges
2. Square Pyramid - 8 edges 
3. Triangular Prism - 9 edges 
4. Pentagonal Pyramid - 10 edges 
5. Cube - 12 edges
6. Hexagonal Pyramid - 12 edges 
7. Rectangular prism - 12 edges
8. Pentagonal Prism - 15 edges 
9. Hexagonal Prism - 18 edges

Faces

Faces are flat surfaces that make up geometrical shapes. The face is always a two-dimensional figure for all three-dimensional shapes. The faces are enclosed together by the edges. 

Here are some three-dimensional shapes and the number of faces they have: 

1. Triangular Pyramid - 4 faces
2. Triangular Prism - 5 faces
3. Square Pyramid - 5 faces
4. Rectangular prism - 6 faces 
5. Cube - 6 faces
6. Pentagonal Pyramid - 6 faces
7. Pentagonal Prism - 7 faces
8. Hexagonal Pyramid - 7 faces
9. Hexagonal Prism - 8 faces

Read Also: Trigonometric Identities

Vertices

The point where the edges of the solid figure meet is called a vertex. Here are some three-dimensional shapes and the number of vertices they have: 

1. Triangular Pyramid - 4 vertices 
2. Square Pyramid - 5 vertices 
3. Triangular Prism - 6 vertices 
4. Pentagonal Pyramid - 6 vertices
5. Hexagonal Pyramid - 7 vertices
6. Rectangular prism - 8 vertices
7. Cube - 8 vertices 
8. Pentagonal Prism - 10 vertices
9. Hexagonal Prism - 12 vertices

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Polygon

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Polygon is a two-dimensional figure that is enclosed by a chain of straight line segments. The segments form a closed polygonal chain or circuit. There are 8 types of polygons:

• Triangle: A triangle has three sides and the sum of all the internal angles is always 180 degrees. A triangle can be an Equilateral, Isosceles, or Scalene triangle.
• Quadrilaterals: A quadrilateral has four sides and the sum of all the internal angles is always 360 degrees. It includes Square, Rectangle, Parallelogram, Rhombus, and Trapezium.
• Pentagon: A pentagon has five sides.
• Hexagon: A hexagon has six sides.
• Heptagon: A heptagon has seven sides.
• Octagon: An octagon has eight sides.
• Nonagon: A nonagon has nine sides.
• Decagon: A decagon has ten sides.

Types of Polygons

Read Also: Circumference of a Circle 

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Circle in Geometry

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A circle is a shape that has no edges. The distance from the center to any point on the circle is the same and is called the radius of the circle. The length of a line that passes through the center from one point on the circle to another point on the circle is the diameter of the circle.

Circle in Geometry

Also Read:

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Things to Remember

• Geometry can be defined as the branch of mathematics that deals with studying different types of sizes, shapes, figures, position angles, and dimensions of things. 
• There are six branches of geometry: Algebraic geometry, differential geometry, convex geometry, discrete geometry, euclidean geometry, and topology.
• Two-dimensional geometry deals with flat shapes. 
• The point, line, and angles are important attributes of two-dimensional geometry.
• Angles are of four types: acute, obtuse, right, and straight angle.
• Three-Dimensional geometry involves three-dimensional objects like cylinders, cubes, prisms, and spheres. 
• Edges, faces and vertices are important attributes of three-dimensional geometry.
• A circle is a shape that has no edges. 
• The distance from the center to any point on the circle is the same and is called the radius of the circle. 

Also Check: Construction Formula