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

• Measurement by Superimposition

• Measurement by Calculation

• Measurement by Equivalence

We studied about measurement by superimposition earlier. In this page, let us quickly review "measurement by calculation".

*click on the content to continue..*

Consider the shape given in the figure. It is a right-triangle of base `6cm` and height `3cm`. Let us superimpose a grid of unit-squares and count the area. 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 right-triangle using grid of unit-squares. The figure illustrates the problem. To find the area, large unit-squares are counted. What is the area?

- approximately `9cm^2`
- approximately `10cm^2`
- either of the above, as the area is approximate
- either of the above, as the area is approximate

The answer is "either of the above, as the area is approximate". Though we can further refine the approximation, using unit-squares of millimeter size, it will still have small errors. We are interested in a mathematically rigorous calculation.

Consider finding area of a right triangle of `6cm` base and `3cm` height. The figure shows the triangle in orange. A rectangle of length `6cm` and width `3cm` is visualized. The rectangle is shown in blue. Geometrically, the hypotenuse of the right triangle splits the rectangle into equal halves. One half is the given right triangle.

We know that the area of the rectangle is length multiplied width. And the right-triangle occupies exact half of the rectangle. What does this imply?

- it does not imply anything
- the area of the triangle is half of the area of the rectangle
- the area of the triangle is half of the area of the rectangle

The answer is "the area of the triangle is half of the area of the rectangle".

It is geometrically understood that the area of a right-triangle is half of that of the rectangle. And so, the formula for the area is derived as `1/2 text( base ) xx text( height )`.*Note: The proof is given for right-triangles. The same formula is applicable for triangles in general. The proof for all triangles will be given later in the course.* The solution to finding area of a triangle is an example of "measurement by calculation".

**Measurement by Calculation**: Length, Area, or Volume can be measured by calculations involving geometrical principles.

This method suits best for

• area of plane figures of straight lines

• volume of solid figures of straight edges and flat-faces

*slide-show version coming soon*