Geometry Formula: Area, Perimeter, Solved Examples

Geometry is one of the oldest branches of mathematics which deals with properties of space in relation to points, lines, surface areas, solids, the position of figures and angles. Euclid is called the father of geometry and the great mathematician introduced the concept of the fundamental concepts. 

Keyterms: Lines, surface areas, Solid, Position, Two dimensional objects, Three-dimensional objects, Area, Perimeter, Volumes

Also Read: Statistics Class 10 Revision Notes

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What is Geometry?

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Geometry is the study of mathematics that deals with the properties of space in relation to points, lines, surfaces, solids, the position of figures and angles. With the help of geometry, we can study and find the various characteristics of two dimensional and three-dimensional objects.

Geometry

The study comes with certain geometrical formulas which have made calculations easy and time-saving. These formulas help us find the area, perimeter, volumes of different kinds of geometrical figures. 

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Classification of Geometry

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Geometry has been broadly classified into two major groups. These are,

Plane geometry: It is concerned with two-dimensional geometrical shapes like circles, squares, triangles, and rectangles.

Plane Geometry

Solid geometry: It basically deals with finding the length, perimeter, area, volume of three-dimensional shapes like cubes, cuboids, and cones. 

Solid Geometry

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Geometric formulas

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The various geometric formulas for different shapes are given below.

Square

In geometry, a square is a figure consisting of four equal sides and four right angles. It means every angle is 90°. 

Square

• Perimeter of a square: P = 4a
• Area of a Square: A = a²
• Length of the Diagonal of a Square: d =  a\({ \sqrt{}}\)2

Where,

a → Side of a Square

Rectangle 

A rectangle is called an equilateral quadrilateral because all of its angles are equal, that is 90°.

Rectangle

• Perimeter of a Rectangle: P = 2 (l+b)
• Area of a Rectangle: A = l x b
• Length of the Diagonal of a Rectangle: d =  \({ \sqrt{}}\)(l² + b²)

Where,

l → Length of Rectangle 

b = Breadth/ Width of rectangle

Circle 

A circle is a perfectly round shaped figure. Any point on the line is at the same distance from the centre of the circle called the radius of the circle. 

Circle

• Circumference of a Circle: C = 2πr
• Area of a Circle: A = π × r²
• Diameter of the Circle: d = 2r

Where, 

r → radius of the circle

Triangle 

Triangle is a geometric shape with three lines and three angles. The place where two lines meet and a constituting angle is formed is called the vertex.

Triangle 

• Perimeter of a Triangle: P = a + b + c
• Area of a Triangle: A = ½ bh
• Height of a Triangle: h = 2 (a/b)

Where, 

a,b,c → Sides of the triangle

h → Height of a triangle

Right angle triangle

A triangle where one of the angles is 90° is known as a right-angle triangle. It is based on the Pythagoras Theorem. 

Right Angled Triangle

• Pythagoras Theorem: a² + b² = c²
• Area of a Right-angled Triangle: A = ½ ab
• Perimeter of a Right Angled Triangle: P = a + b + c

Where, 

a → Altitude of the Right-angled Triangle

b → Base of the Right-angled Triangle

c → Hypotenuse of the Right-angled Triangle

h → Height of a triangle

Also Read: Nature of Roots of Quadratic Equation

Cube 

A cube is a three-dimensional figure with six sides similar to that of a square. It is even called a hexahedron.

Cube

• Area of a Cube: A = 6a²
• Volume of a Cube: V = a³
• Space Diagonal: D = a√3

Where, 

a → Side of the cube

Also Read: Surface Areas and Volumes

Cuboid 

A cuboid is a three-dimensional solid geometric figure which has six rectangular faces at right angles to each other. 

Cuboid

• Surface area of a cuboid: A = 2(lb + bh + hl)
• Volume of a Cuboid: V = lbh
• Space Diagonal: D = \({ \sqrt{}}\)( l² + b² +h²)

Where, 

l → Length

b → Breadth

h → Height

Cylinder

A cylinder is a three-dimensional solid figure consisting of two parallel circular bases. These bases are joined by a curved surface and are situated at a particular distance from the centre.

Cylinder

• Total Surface Area of a Cylinder: A = 2πrh + 2πr²
• Curved Surface Area of a Cylinder: Ac = 2πrh
• Volume of a Cylinder: V = πr²h
• Base Area of a Cylinder: Ab = πr²
• Radius of the Cylinder: r = \({ \sqrt{}}\)(V/πh)

Where, 

r → Radius of the cylinder

h → Height of the cylinder

Cone 

It is a three-dimensional figure that has a flat surface called the base. The curved body of the cone moves upwards to a point which is called the vertex. It also contains width, height and radius. 

Cone

• Total Surface Area of a Cone: A = πr(r+l) = πr[r+\({ \sqrt{}}\)(h²+r²)]
• Curved Surface Area of a Cone: Ac = πrl
• Volume of a Cone: V = ? πr²h
• Slant Height of a Cone: l = \({ \sqrt{}}\)(h²+r²)
• Base Area of a Cone: Ab = πr²

Where, 

r = Radius of a cone

h = Height of a cone

l = Slant height

Sphere 

A Sphere refers to the set of all points lying the same distance from a given point. In other words, we can say that every point on its surface is equidistant from the centre itself.

Sphere

• Surface Area of a Sphere: A = 4πr²
• Volume of a Sphere: V = ?⁄?πr³
• Diameter of a Sphere: D = 2r

Where, 

r → Radius of a sphere

Parallelogram 

It is a quadrilateral where the opposite sides are parallel as well as equal. 

Parallelogram

• Perimeter of a Parallelogram: P = 2(a + b)
• Area of a Parallelogram: A = bh
• Height of a Parallelogram: h = A/b
• Base of a Parallelogram: b = A/h

Where,

a and b → Sides of a parallelogram

h → Height of the parallelogram

Also Read: Triangles Important Questions Class 10

Rhombus

A Rhombus is a quadrilateral whose opposite sides are parallel and all the sides are equal in length. 

Rhombus

• Perimeter of a Rhombus: P = 4a
• Area of a Rhombus: A =\(\frac{d_1 \times d_2}{2}\)
• Length of the diagonal of a Rhombus: d = \({{\sqrt {4a^2 - d^2}}}\)

Where,

a → Side of a Rhombus

d → Diagonal of the Rhombus

Trapezium 

A trapezium is a quadrilateral that has one pair of parallel opposite sides is known as a trapezium. The word trapezium took its origin from the Greek word "trapeza" which means a table.

Trapezium

• Area of a Trapezium: A = ½ (a + b)h
• Height of a Trapezium: h = 2A/(a + b)
• Base of a Trapezium: b = 2(A/h) - a

Where, 

a and b → Parallel sides of the Trapezium.

h → Distance between two parallel sides of the Trapezium.

Also Read: Trigonometry Identities

Arc 

In general, an arc is defined as any regular curve which ends up joining two points. The length of an arc is known as arc length. We may also mention that an arc is any portion apart from the entire curve of the circumference of a circle.

Arc

• Arc Length: L = rθ
• Area Enclosed by the Arc: A = ½ r²θ

Where, 

θ → Central angle of the radians

r → Radius of the arc

Also Read: Real Numbers Important Questions

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Things to Remember

• Geometry is a branch of mathematics that deals with different shapes and sizes. 
• It is divided into two types, plane geometry and solid geometry. 
• The basic formulas of geometry make our everyday calculations much easier by helping us find the length, breadth, width, perimeter, area, height, and volume of different figures. 
• Geometry has applications in almost all fields. Be it science, art, architecture, and other activities that are related to graphics, it has made sure to touch every aspect. 
• Geometry also has necessary applications in areas of mathematics. In the olden days, the concept of geometry was also used to measure orbits and study planetary movements. 
• Geometry is heavily used in the construction of buildings, dams, roads, etc. It has great usage in computer graphics, art, and interior design as well. Earlier it was also used so as to calculate the height of pyramids and the distance of ships from the shore.

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