In this article, we have compiled a list of geometry formulas which are quite helpful in solving the questions related to areas, volumes and perimeters of geometrical figures. So, let us get started.

Area and Perimeter Formulas

In this section, we have compiled a list of area and perimeter formulas of various geometrical figures.

Area of a geometrical figure refers to the amount of space in square length units occupied by the surface of the geometrical figure. On the other hand, a perimeter refers to the distance around the closed geometrical figure or shape.

Triangle

Triangle is one of the most fundamental shape in geometry that consists of three sides and three vertices. The sum of the interior angles of a triangle is equal to 180 degrees. The three types of triangle are equilateral, isosceles and scalene.

The formula for area of the triangle is:

Area = frac {base times height}{2}

The formulas for perimeter of equilateral, isosceles and scalene triangles are a bit different. Basically, they all involve adding the length of three sides of the triangle.

Perimeter of an equilateral triangle = 3 x length of its side

Perimeter of an isosceles triangle = 2 x length + base

Perimeter of the scalene triangle = P = a + b + c

Square

Square refers to a regular quadrilateral which has four equal sides and angles. It means that all sides of the square are of the same length and all the angles are equal in measure.

The formula for area of the square is:
Area = l x l
The formula for calculating the perimeter of a square is:
Perimeter = 4 x length of its side

Rectangle

A rectangle is a type of quadrilateral with four right angles. The opposite sides of the rectangle are of the same length and are parallel to each other.

A Rectangle
The formula for area of the rectangle is:
Area = l x w, where l is the length of the rectangle, and w is its width
The formula for perimeter of the rectangle is:
Perimeter = 2 (l + w)

Rhombus

A rhombus refers to a parallelogram with four equal sides and opposite equal angles. It means that all the four sides of the rhombus are of the same length and opposite angles are congruent.

A Rhombus
The formula to calculate the area of the rhombus is:
Area = frac {D cdot d}{2}, here D and d represent the diagonals of the rhombus
The formula to calculate the perimeter of the rhombus is:
Perimeter = 4 x length of the side of rhombus

Rhomboid

A rhomboid is a type of parallelogram in which the adjacent sides have different lengths and angles are not equal to 90 degrees.

A Rhomboid
The formula to calculate the area of the rhomboid is:
Area = base x height
The formula to calculate the perimeter of the rhomboid is:
Perimeter = 2 . (a + b), where a and b are the sides of the rhomboid

Area of Trapezoid

Trapezoid is a quadrilateral with one pair of parallel sides as shown in the figure below:

A Trapezoid
The formula to calculate the area of the trapezoid is:
Area = frac {(B + b) cdot h}{2}

Area of a Regular Polygon

In geometry, a regular polygon is a polygon that is equilateral and equiangular. It means that all sides of the regular polygon are of the same length and all of its angles are of the same measure.

A Regular Polygon
The formula to calculate the area of the regular polygon is:
Area = frac{perimeter cdot apothem}{2}
The formula to calculate the perimeter of the regular polygon is:
Perimeter = n x l, where n depicts the number of sides of the polygon

Polygon

If you have an irregular polygon which seems like the figure given below, then you can calculate the area by triangulating the polygon and adding the area of these triangles.

A = T_1 + T_2 + T_3 + T_4

A polygon

Circle

A circle is a geometrical figure in which all points are located at equal distance from its center.

A Circle
Instead of the perimeter, a circle has the circumference. The formula for calculating the circumference of the circle is:
C = 2 pi r, where pi has a fixed value and "r" is the radius of the circle
Since the radius of the circle is half of its diameter, hence we can also write the formula of the circumference of the circle as shown below:
C = pi cdot d
The formula to calculate the area of the circle is:
A = pi r^2, where r is the radius of the circle

Circular Sector

A circular sector, also referred to as a disk center or circle center is the portion of the circle that is enclosed by an arc and two radii of the circle.

Circular Sector
The formula for computing the area of the circular sector is given below:
A = frac {pi cdot r^2 cdot alpha}{360^0}
The formula for calculating the arc length of the circular sector is:
L = frac {2 cdot pi cdot r cdot alpha}{360^0}

 

Circular Segment

A circular segment refers to the region of the circle that is "cut off" from the remaining circle by a chord or secant.

A Circular Segment
The formula for calculating the area of the circular segment is:

The circular area of the segment AB = Area of the circular sector AOB - Area of triangle AOB

 

Lune of Hippocrates

It refers to the lune bounded by two arcs of the circle. You can read our article here to learn more about this concept.

Lune of Hippocrates

 

Area of the lune = area of the semicircle − area of circular segment.

Area of the lune = Area of the right triangle

 

Circular Trapezoid

In two given concentric circle, a circular trapezoid refers to the area that lies between two non-crossing chords of the circle.

A Circular Trapezoid
The formula for calculating the area of the circular trapezoid is given below:
A = frac {pi (R^2 - r^2) cdot alpha}{360^0}

 

Area Enclosed between Two Concentric Circles

Area Enclosed Between Two Concentric Circles
The formula for calculating the area between two concentric circle us given below:
A = pi cdot (R^2 - r^2)

 

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Surface Area and Volume Formulas

In this section of the article, we have compiled a list of surface area and volume formulas of various geometrical figures.

Tetrahedron

A tetrahedron refers to a solid that contains four plane triangular faces

A tetrahedron
The formula for calculating the area of the tetrahedron is:
A = sqrt{3} cdot a^2
The formula for calculating the volume of the tetrahedron is:
V = frac {sqrt{2}} {12} a^3

Octahedron

An octahedron refers to a three-dimensional shape which has eight plane faces.

An octahedron
The formula for calculating the area of the octahedron is:
A = 2 sqrt{3} cdot a^2
The formula for calculating the volume of the octahedron is:
V = frac {sqrt{2}} {3} a^3

Icosahedron

Icosahedron refers to a solid figure that contains 20 plane faces.

An icosahedron
The formula for calculating the area of an icosahedron is:
A = 5 cdot sqrt{3} cdot a^2

The formula for calculating the volume of the tetrahedron is:

V = frac {5}{12} (3 + sqrt{5})a^3

Dodecahedron

A dodecahedron refers to a three-dimensional figure with twelve plane faces.

A dodecahedron
The formula for calculating the area of the dodecahedron is:
A = 30 cdot a cdot ap

The formula for calculating the volume of the dodecahedron is:

V = frac {1}{4} (15 + 7 sqrt{5})a^3

Cube

A three-dimensional shape that contains six equal square is known as a cube.

A cube
The formula for calculating the volume of the cube is:
V = a^3
The formula for calculating the surface area of the cube is:
A = 6 cdot a^2

Cuboid

A three-dimensional geometrical figure that contains six rectangular faces is known as a cuboid

A cuboid

The formula for calculating the area of the cuboid is:

A = 2 (a cdot b + a cdot c + b cdot c)

The formula for calculating the volume of the cuboid is:

V = a cdot b cdot c

Prism

A prism refers to a solid geometrical figure in which two ends are equal, similar and parallel rectilinear figures, and sides are parallelograms as shown in the figure below:

 

A Prism
P_{B} = Perimeter of the base
A_{L} = P_B cdot h
A_{T} = A_{L} + 2 cdot A_{B}
V = A_{B} cdot h

Pyramid

In geometry, a pyramid refers to polyhedron that is formed by joining the polygonal based an point, known as apex

A Pyramid

P_{B} = Perimeter of the base

Ap = apothem of the pyramid

ap = apothem of the base

Ap^2 = h^2 + ap^2

A_{L} = frac {P_{b} cdot Ap} {2}

A_{T} = A_{L} + A_ {B}

V = frac {A_{B} cdot h}{3}

Truncated Pyramid

A truncated pyramid results from cutting a pyramid by a plane parallel to the base and segregating the portion that contains the apex.

A Truncated Pyramid

P = Perimeter of the larger base

P' = Perimeter of the smaller base

A = Area of the larger base

A' = Area of the smaller base

A_{L} = frac {P + P'} {2} cdot Ap

A_{T} = frac {P + P'}{2} cdot Ap + A + A'

V = frac {h}{3} cdot (A + A' + sqrt {A cdot A'})

Cylinder

A cylinder as shown below refers to a surface that contains all the points on all the lines that are parallel to a given line and which pass through a fixed plane curve in a plane that is non-parallel to the given line.

A cylinder
A_ {L} = 2 cdot pi cdot r cdot h
A_{T} = 2 cdot pi cdot r cdot (h + r)
V = pi r^2 h

Cone

A cone in geometry refers to a three-dimensional shape that narrows smoothly from a flat base to a point known as a  vertex or apex.

A cone
s^2 = h^2 + r^2

A_{L} = pi cdot r cdot s

A_{T} = pi cdot r cdot (s + r)

V = frac {pi cdot r^2 cdot h}{3}

Truncated Cone

a cone section or pyramid lacking an apex and terminating in a plane usually parallel to the base.

It refers to the cone section or a pyramid that lacks an apex and terminates in a plane usually parallel to the base.

A truncated cone
A_{L} = pi (R + r) cdot s
A_{T} = pi [s (R + r) + R^2 + r^2]

V = frac {1}{3} cdot pi cdot h(R^2 + r^2 + R cdot r)

Sphere

Just like circle, a sphere refers to the set of points that are all located at the equal distance "r" from a given point.

A sphere
A = 4 cdot pi cdot r^2

V = frac {4}{3} pi cdot r^3

Spherical Wedge

A spherical wedge, also known as ungula is a part of a ball bounded by two plane semi disks and a spherical lune.

A Spherical wedge
A = frac{4 cdot pi ]cdot r^2}{360} cdot n

V = frac {4}{3} cdot frac {pi cdot r^3}{360} cdot n

Spherical Cap

A spherical cap, also known as a spherical done is a part of sphere or a ball that is "cut off" by a plane.

A Spherical Cap
R = frac {r^2 + h^2}{2h}

A = 2 pi Rh

V = frac {1}{3} pi cdot h^2 cdot (3R - h)

Spherical Segment

A spherical segment refers to a solid defined by cutting a ball or a sphere with a pair of parallel lines.

A Spherical Segment
A = 2 pi Rh
V = frac {1}{6} pi h (h^2 + 3 cdot R^2 + 3 r^2)
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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.