To measure a length, area, or volume one of the following methods is used.

• Measurement by Superimposition

• Measurement by Calculation

• Measurement by Equivalence

We quickly reviewed "measurement by superimposition" and "measurement by calculation".

Concept of " Measurement by Equivalence" is easy to understand. Three methods are used to find equivalence. These three methods are little involved in explanation and are hard to understand. These are essential to solve area of shapes of curves or volume with curved surfaces.

The following three methods are explained later in the course.

• Equivalence by infinitesimal pieces

• Equivalence by Cavalieri's Principle (2D)

• Equivalence by Cavalieri's Principle (3D)

In this topic, let us quickly review the concept.

*click on the content to continue..*

Consider the shape given in the figure. It is a parallelogram of base `b cm` and height `h cm`. The grid of unit squares is superimposed and the number of unit squared are counted. Which of the following is true?

- accurate area is directly possible to find
- since unit-squares will not fit exactly at the vertices, only approximate area can be computed
- since unit-squares will not fit exactly at the vertices, only approximate area can be computed

The answer is "since unit-squares will not fit exactly at the vertices, only approximate area can be computed".

Consider finding area of a parallelogram of base `b cm` and height `h cm`. The figure shows the parallelogram `ABCD` in orange. A rectangle `ABRQ` of length `b cm` and width `h cm` is shown in blue. Consider the region `/_\ BRC` and the region `/_\ AQD`. The area of these two triangles are equal.

The area of the parallelogram is sum of areas of the region `ABRD` and `/_\ BRC`. This can be equivalently given as sum of areas of the region `ABRD` and `/_\ AQD`.

What does this imply?

- area of parallelogram cannot be computed
- area of the parallelogram equals the area of the rectangle
- area of the parallelogram equals the area of the rectangle

The answer is "area of the parallelogram equals the area of the rectangle".

**Measurement by Equivalence**: Length, Area, or Volume can be measured by constructing an equivalent shape or object involving geometrical principles.

This method suits best for

• area of plane figures of curved sides or curved surfaces

• volume of solid figures of curved edges or curved faces

*slide-show version coming soon*