The word Geometry is derived from the two ancient Greek words, “Geo” and “Metron” where Geo stands for Earth and Metron stands for measurements. 

Geometry is the branch of mathematics that deals with shapes (Circle, triangle, rectangle, oval, square, etc.), angles, dimensions, and sizes of a variety of things that we experience in our daily life.  This branch of mathematics lets us learn about different angles, transformations, and similarities in the figures. Some of the examples of topics of geometry are explained below:

1.     Calculate The Area of a Circle

To calculate the area of a circle the following formula is used:

A=πr²   Where,  A = Area of the circle , π = constant = 3.14,  r = radius

 Implementation: Calculate the area of a circle with a radius of 15cm.

 π = 3.14, r = 15 cm .  A = (3.14) (15)²   A = (3.14) (225  Thus  A = 706.5 cm²

 2.     Calculate The Circumference of Circles

 To calculate the circumference of a circle the following formula is used:

c = 2πr ,  Where, c = circumference, r = for radius  and π = constant = 3.14

 Implementation: Calculate the circumference of a circle with an 8 cm radius.  C = 2πr , C = 2(3.14) (8)  Thus C = 2 (25.12)  C = 50.24 cm

3.     Perimeter and Area of Equilateral Triangles  

A type of triangle whose all three sides are equal, as well as all the three angles, is equal to 60° each is known as an Equilateral Triangle.  To calculate the perimeter of an equilateral triangle the sum of its three sides is calculated hence the formula is,  P = 3a , Where, P = Perimeter , a = one side of the triangle.  Since an equilateral triangle has equal sides the sum is represented as 3a

 Implementation: Find the perimeter of an equilateral triangle with a side of 9 units. P = 3a, a = 9,  P = 3 (9), thus P= 27 units.

4.     The Area of Circles with Diameter Values.

The area of a circle can also be calculated using the diameter when the radius is unknown with the following formula: A = (π/4) × d2. 

WhereA = area , π = constant = 3.14 and d = diameter

 Implementation: How to Calculate the area of a circle with a diameter of 20 inches. A = (π/4) × d²  A = (3.14/4) × (20)²  A = 0.785 × 400

Thus A = 314 cm²

5.     The Area of Rectangles

The area of a rectangle is the easiest to find. The formula for calculating the area of a rectangle is as follows:

Area of rectangle = Length x Width

Implementation: How to Calculate the area of the following rectangle                        

l= 2 units   L= 6 units.   Since Area of rectangle = Length x Width  this means A = 6 x 2  thus A = 12 cm²

 6.    Find The Volume of Cubes

The total three-dimensional space occupied by a cube is known as its volume. To find the volume of a cube, the edge length is multiplied three times hence the following formula volume:  The volume of a cube = s3 or s x s x s  Where,  S = the side of the cube.

Implementation:

Find the volume of a cube with an edge length of 14. The volume of a cube = (14)³ Or 14 x 14 x 14. The volume of a cube = 2744

7.     How to find the  Volume and Area of Cylinders ? The volume of a cylinder can be calculated with the help of the radius and height of the cylinder.

The volume of a cylinder (V) = π r² h   Where, π = Constant = 3.14. r = radius. h = height

 Implementation: Finding the volume of a cylinder having a radius of 50 cm and a height of 100 cm.

V = π r2 h , V = (3.14) (50)² (100); V = (3.14) (2500) (100)  this means V = 785,000 cm.³   The Surface Area of the Cylinder is the sum of the curved surface and the area of two circular bases of the cylinder.

Formula:  Total Surface Area of a Cylinder = Curved Surface + Area of Circular bases.  Curved Surface Area = 2π × r × h

Area of Circular Base .= 2 πr²Total Surface Area of a Cylinder = (2 πrh) + (2 πr²).  Hence,   A = 2πr (h + r) sq. Unit     Where,  π = 3.14. r = radius

  Implementation:  Find the surface area of a cylinder having a 50 cm radius and 100 cm height.   A = 2πr (h + r)  A = 2(3.14)(50) (100+50).  A = (314) (150). A = 464 sq. unit

 8.     Geometry Measure Angles. There are six angles in geometry:   Acute Angle – 90 degree ;  Right Angle – 90 degree;  Obtuse Angle – Greater than 90 degrees and less than 180 degrees.  Straight Angle – 180 degreReflex Angle – Greater than 180 degrees.  Full Rotation – Exact 360 degree

 Perimeter and Area of an Isosceles Triangle

A type of triangle whose two sides are of an equal length and the two angles opposite to the equal side are the same is known as an isosceles triangle. To calculate the perimeter and area of a triangle, the lengths of the equal sides and base are measured.

 The Perimeter of an isosceles triangle (P) = 2a + b . Where, a = length of equal sides ; b = base. Hence, the The area of an isosceles triangle would be          A = ½ × b × h  Where, b = base and h = height.

Example, Find the perimeter and area of an isosceles triangle having two equal sides measuring 4 cm, height 5 cm, and a base of 5 cm.

P = 2a + b

P = 2 (4) + 5

P = 8+5

P = 13 cm

A = ½ × b × h

A = ½ × 5 × 5
A = ½ × 25

A = 12.5 cm²

10. Perimeter and Area of a Parallelogram

A type of rectangle that is slanted and has equal parallel/opposite sides is called a parallelogram.

Perimeter of a parallelogram = 2 (a + b)

Area of Parallelogram = b × h

Where,
a = adjacent side length

b = base length

h = height

Implementation:

Find the area of a parallelogram having a 10 cm base, 3.5 cm height, and 5 cm adjacent side length.

Perimeter = 2 (a + b)

P = 2 (5 + 3.5)
P = 2 (8.5)
P = 17 cm

Area = b x h

A = 10 x 3.5

A = 35 cm²

11. Perimeter and Area of L Shapes

To find the area of L shapes, first, we need to divide the shape into to rectangles. Secondly, find the area of both rectangles. The sum of the area of those two rectangles is the total area of an L shape.

Implementation:

The area of rectangle A is 14 × 5 = 70 cm²

The area of rectangle B is 2 × 3 = 6 cm²

The area of L shape = 70 + 6

A = 76 cm²

12. Perimeter and Area of Right Angled Triangles

The perimeter of a right-angled triangle is calculated by adding up all its sides. Hence,

The perimeter of right-angled triangle P = a + b + c

Where,

c = hypotenuse

a = height

b = base

When one of the sides is unknown, we can apply the Pythagoras theorem to find the missing side by (Hypotenuse)² = (Base)² + (Height)²

The area of a right-angled triangle is calculated using A = ½ × b × h

Where,

b = base

h = height

13. Perimeter and Area of Trapezoids

A quadrilateral having only two of its sides parallel is known as a trapezoid. These parallel sides are called the bases of a trapezoid. The area of a trapezoid is calculated using

The area of a trapezoid is A=½ h(b1+b2)

Where,

h = the perpendicular distance between the bases. 

b1 and b2 = bases or parallel sides.

Implementation:

Find the area of a trapezoid.

        21 cm

                                               41 cm

A = ½ h(b1+b2)

A = ½ (18) (21 + 41)

A = (9) (62)

A = 558 units²

14. Perimeter of Rectangles

The perimeter of a rectangle is calculated using its length and width.

The formula states:

P = (L + W) × 2

Where,

P = perimeter

L = length

W = width.

Implementation:

                                             18 feet

                                                                                    10 feet

P = (L +W) x 2

P = (18 + 10) x 2

P = 28 x 2

P = 56 feet

15. Pythagorean Theorem

The fundamental relation between the three sides of a right triangle is called Pythagoras or Pythagorean Theorem. This theorem states that the square of the hypotenuse (the side opposite to the right angle) is equal to the sum of the areas of the squares on the other two sides.

It is represented as:

                        (Hypotenuse)² = (Base)² + (Height)²

Implementation:

  3 units²                                                    5 units²

                               4 units²

Applying Pythagoras theorem:

(Hypotenuse)² = (Base)² + (Height)²

(5) ² = (4) ² + (3) ²

25 = 16 + 9

25 = 25 Proved.

16. Perimeter and Area of Scalene Triangles

A type of triangle whose all sides are of different measures, as well as all the angles are also of different measures is known as the Scalene triangle.

The area of the scalene triangle is equal to half of the product of its base length and height.

Area = (1/2) x b x h square units

Where,

b = base

h = height

Perimeter is equal to the sum of its three unequal sides.

Perimeter = a + b + c units

17. Volume and Area of a Cone

A three-dimensional geometric shape having a circular base that tapers from a flat base to a point called apex or vertex is known as a cone.

To calculate the volume of a cone following formula is used:

V = (1/3) πr²h

To calculate the area of a cone following the formula is used:

                                      A = 𝜋r (l + r) square units

Where,

r = radius of the circular base

l = height of the cone

Implementation:

Calculate the volume and area of a cone having a radius of 2 cm and height of 5 cm.

The volume of the cone:

V = (1/3) πr²h

V = (1/3) (3.14) (2) ² (5)

V = 20.93 cm³

The area of the cone:

A = 𝜋r (l + r)

A = (3.14) (2) (5 + 2)

A = (6.28) (7)

A = 43.96 sq. Units

18. Volume and Area of Rectangular Prisms

A solid shape that is bound on all its sides by plane faces is known as a prism.

To calculate the volume of a rectangular prism following formula is used:

V = base area × height of the prism

To calculate the area of a rectangular prism following formula is used:

                                    A = length × width

Implementation:

Calculate the volume of a rectangular prism whose height is 8 in and the base area is 90 square inches.

V = 90 x 8

V = 720 cubic inches

Calculate the area of a rectangular prism whose base length is 6 in and the width is 9 in

A = 6 x 9

A = 54 cubic inches