This Module describes various standard geometrical structures that exist and methods/formulas or calculating its characteristic values that can be used for various applications/calculations. The structures include circle, triangle, cylinder etc., that exists within or as a shape, in all the practical entities we see in our daily life. So, calculating its area, Volume and perimeter gives the mathematical insight about the structure.

For example, in order to determine the number of tiles of specific shape to be placed over the floor of specific dimension, area of both tile and the floor become the basis.

Prerequisites (Related Formulas) Area Calculations

Area is a quantity that expresses the extent of a two – dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat.
1 square kilometer = 1,000,000
square meters, 1 square meter = 10,000 square centimetres = 1,000,000 square millimetres
1 square centimetre = 100 square millimeters
1 square yard = 9 square feet
1 square mile = 3,097,600 square yards = 27,878,400 square feet

Volume is the quantity of three – dimensional space enclosed by some closed boundary, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is often quantified numerically using the SI derived unit , the cubic metre.

1 litre = (10 cm)3 = 1000 cubic centimeters = 0.001 cubic metres,
1 cubic metre = 1000 liters.
Small amounts of liquid are often measured in millilitres,
Where 1millilitre = 0.001 litres = 1 cubic centimetre

Surface Area Calculations

Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve.

a. Surface Area of a Cube = 6a² (a is the length of the side of each edge of the cube).

In words, the surface area of a cube is the area of the six squares that cover it. The area of one of them is a*a, or a2 . Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.

b. Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac (a, b, and c are the lengths of the 3 sides).

In words, the surface area of a rectangular prism is the area of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same.
The area of the top and bottom (side lengths a and c) = a*c. Since there are two of them, you get 2ac. The front and back have side lengths of b and c. The area of one of them is b*c, and there are two of them, so the surface area of those two is 2bc. The left and right side have side lengths of a and b, so the surface area of one of them is a*b. Again, there are two of them, so their combined surface area is 2ab.

c. Surface Area of Any Prism (b is the shape of the ends)

Surface Area = Lateral area + Area of two ends
(Lateral area) = (perimeter of shape b) * L
Surface Area = (perimeter of shape b) * L+ 2*(Area of shape b)

d. Surface Area of a Sphere = 4 pi r²(r is radius of circle)

e. Surface Area of a Cylinder = 2 pi r²+ 2 pi r h (h is the height of the cylinder, r is the radius of the top)

Surface Area = Areas of top and bottom +Area of the side
Surface Area = 2(Area of top) + (perimeter of top)* height
Surface Area = 2(pir2) + (2pir)* h

In words, the easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label that wraps around the middle. You can find the area of the top (or the bottom). That's the formula for area of a circle (pir2).

Since there is both a top and a bottom, that gets multiplied by two. The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides. One side is the height of the can, the other side is the perimeter of the circle, since the label wraps once around the can.
So the area of the rectangle is (2pir)* h. Add those two parts together and you have the formula for the surface area of a cylinder.

Perimeter Calculations

A perimeter is a path that surrounds an area. The word comes from the Greek peri (around) and meter (measure). The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape.

circle = pi d (where d is the diameter). The perimeter of a circle is more commonly known as the circumference.

Properties of various Geometrical structures

Triangle

In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.
In an isosceles triangle , two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the Isosceles triangle theorem.

In a scalene triangle , all sides are unequal. The three angles are also all different in measure. Some (but not all) scalene triangles are also right triangles.

A right triangle (or right - angled triangle , formerly called a rectangled triangle) has one of its interior angles measuring 90° (a right angle). The side opposite to the right angle is the hypotenuse ; it is the longest side of the right triangle. The other two sides are called the legs of the triangle.

Right triangles obey the Pythagorean theorem : the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse: a2 +b2=c2, where a and b are the lengths of the legs and c is the length of the hypotenuse.

Triangles that do not have an angle that measures 90° are called oblique triangles.

A triangle that has all interior angles measuring less than 90° is an acute triangle or acute - angled triangle.

A triangle that has one angle that measures more than 90° is an obtuse triangle or obtuse - angled triangle.

A "triangle" with an interior angle of 180° (and collinear vertices) is degenerate.

Quadrilateral

In Euclidean plane geometry, a quadrilateral is a polygon with four sides (or 'edges') and four vertices or corners . Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5- sided), hexagon (6- sided) and so on. The Word quadrilateral is made of the words quad (meaning "four") and lateral (meaning "of sides").

The diagonals of parallelogram bisect each other
Each diagonal of a parallelogram divides it into two triangles of the same area
The diagonals of rectangle are equal and bisect each other
The diagonals of square are equal and bisect each other at right angles
The diagonals of rhombus are unequal and bisect each other at right angles

Circle

The circle is the shape with the largest area for a given length of perimeter.
A circle's circumference and radius are proportional.
The area enclosed and the square of its radius are proportional.
The constants of proportionality are 2 π and π, respectively.
The circle which is centered at the origin with radius 1 is called the unit circle.

Some Important Metrics
1. 10,000 sq meters = 1 hectare
2. 100 hectares = 1 sq kilo meter
3. 1000 millimeters = 1 meter
4. 100 centimeters = 1 meter
5. 1000 metres = 1 kilometer
6. 1000 kilograms = 1 mega gram or 1 tonne
7. 3.6 kilometers per hour = 1 meter per second
8. 3600 kilometers per hour = 1 kilometer per second

Volume Surface Area and Perimeter Questions from Previous Year Exams

Volume Surface Area and Perimeter Video Lecture