Mensuration : Length, Area, and Volume

Welcome to the *refreshingly new views* to calculating perimeter, area, and volume of 2D and 3D shapes.

• Length is Distance-Span, measured in reference span of to $1$ unit long line

• Area is Surface-Span, measured in reference to span of $1\times 1$ square

• Volume is Space-Span, measured in reference to span of $1\times 1\times 1$ cube

In this basic course, the following are covered.

• Perimeter and area of simple 2D shapes

• surface area and volume of simple 3D shapes

Introduction to Standards

This topic introduces the following

• What are standard in measurement?

• What are Absolute Standards?

• What are Derived Standards?

**Absolute Standards** : An unit of measurement, widely accepted and used in reference to a standardized prototype.

**Derived Standards** : An unit of measurement that is defined using other absolute standards.

Introduction to Measuring Length

This topic introduced measuring length as "distance-span". It is an absolute standard and measured in reference to a standard $1$ unit.
**Length** : The distance-span between two points is the length. It is measured in meter or in one of its other forms. Length is specified as a number in reference to the reference-prototype-standard meter (or in one of other derived or similar forms).

Accurate measure and Approximate measure

In this topic accuracy of measurement is explained.

**Accuracy of Measurements** : Measurement can be performed to a desired accuracy level.

When the accuracy is specified to a lower level, the measurement is approximate measurement.

Introduction to Measuring Area

This topic introduced measuring area as "surface-span". It is a derived standard and measured in reference to a standard $1x1$ square.**Area of a plane figure** : The surface-span of a plane figure is the area of the surface. It is measured in square meter (or in one of other derived or similar forms).
Area is specified as a number in reference to the surface-span of a square of $1$ meter side.

Introduction to Measuring Volume

This topic introduced measuring volume as "space-span". It is a derived standard and measured in reference to a standard $1\times 1\times 1$ cube.

**Volume of a solid** : The space-span of a solid is the volume of the solid. It is measured in cubic meter (or in one of other derived or similar forms.)
Volume is specified as a number in reference to the space-span of a cube of $1$ meter side.

Conversion of Units of Measure

This topic provides a very brief overview of conversion of Units of Measure.

Perimeter: Square, Rectangle, Triangle, Polygons

Finding perimeter of simple figures (squares, rectangle, triangle, polygon) are revised.**Perimeter ** :

Perimeter of a figure is the length of the line or curve forming the boundary of the figure.

Perimeter of square $=4\times \phantom{\rule{1ex}{0ex}}\text{side}$

Perimeter of rectangle $=2\times \phantom{\rule{1ex}{0ex}}\text{length + width}$

Perimeter of triangle $=\phantom{\rule{1ex}{0ex}}\text{sum of three sides}$

Perimeter of polygon $=\phantom{\rule{1ex}{0ex}}\text{sum of sides}$

Area of Square and Rectangle

**Area of a Square and a Rectangle** :

Area of square $={\phantom{\rule{1ex}{0ex}}\text{side}}^{2}$

Area of rectangle $=\phantom{\rule{1ex}{0ex}}\text{length}\phantom{\rule{1ex}{0ex}}\times \phantom{\rule{1ex}{0ex}}\text{width}$

Area of a Triangle

The area of a triangle is calculated based on geometrical properties. In this chapter, the different configurations of triangles are illustrated and a common formula is derived.**Area of a Triangle**:

$\frac{1}{2}\times \phantom{\rule{1ex}{0ex}}\text{base}\phantom{\rule{1ex}{0ex}}\times \left(height\right)$

Area of Polygons

**Area of a Polygon** : Consider a polygon to be combination of known geometrical forms, mostly triangles.
The geometrical forms and the formula for area are:

Area of a triangle $=\frac{1}{2}\times \phantom{\rule{1ex}{0ex}}\text{base}\phantom{\rule{1ex}{0ex}}\times \phantom{\rule{1ex}{0ex}}\text{height}$

Area of a trapezium $=\frac{1}{2}\times \phantom{\rule{1ex}{0ex}}\text{sum of bases}\phantom{\rule{1ex}{0ex}}\times \phantom{\rule{1ex}{0ex}}\text{height}$

Area of a parallelogram $=\phantom{\rule{1ex}{0ex}}\text{base}\phantom{\rule{1ex}{0ex}}\times \phantom{\rule{1ex}{0ex}}\text{height}$

Area of a kite $=\frac{1}{2}\times \phantom{\rule{1ex}{0ex}}\text{major-diagonal}\phantom{\rule{1ex}{0ex}}\times \phantom{\rule{1ex}{0ex}}\text{d2}$

Circumference and Area of a Circle

In this page, the formula to find circumference of a circle is introduced.

**Circumference of a Circle:** :

Circumference $=2\pi r$ and $=\pi d$

$r$ is the radius of the circle

$d$ is the diameter of the circle
**Area of a Circle** :

$=\pi {r}^{2}$

$r$ is the radius of the circle

Perimeter and Area of Various Quadrilaterals

In this page, the formula to find area of various quadrilaterals is introduced.

**Area of Some Quadrilaterals** : Consider the polygon shapes as combination of triangles and find sum of area of the triangles.
$\phantom{\rule{1ex}{0ex}}\text{area of a parallelogram}\phantom{\rule{1ex}{0ex}}$$=\phantom{\rule{1ex}{0ex}}\text{base}\phantom{\rule{1ex}{0ex}}\times \phantom{\rule{1ex}{0ex}}\text{height}$
$\phantom{\rule{1ex}{0ex}}\text{area of a trapezium}\phantom{\rule{1ex}{0ex}}$$=\frac{1}{2}\times \phantom{\rule{1ex}{0ex}}\text{sum of bases}\phantom{\rule{1ex}{0ex}}\times \phantom{\rule{1ex}{0ex}}\text{height}$
$\phantom{\rule{1ex}{0ex}}\text{area of a kite}\phantom{\rule{1ex}{0ex}}$$=\frac{1}{2}\times \phantom{\rule{1ex}{0ex}}\text{product of the diagonals}$

Surface Area of Cube, Cuboid, Cylinder

In this page, Finding surface area of simple figures (cube, cuboid, and cylinder) are revised without much discussion.

**Surface Area of Some Shapes**:
Surface Area of cube $=6{q}^{2}$
Surface Area of cuboid $=2(lb+bh+hl)$
Curved Surface Area of Cylinder $=2\pi rh$

Surface Area of cylinder $=2\pi r(r+h)$

Volume : Cube, Cuboid, Cylinder

In this page, Finding volume of simple figures (cube, cuboid, and cylinder) are revised without much discussion.

**Volume of Some Solids** :
Volume of cube $={a}^{3}$
Volume of cuboid $=lbh$
Volume of cylinder $=\pi {r}^{2}h$