Measurement by Equivalence

Overview

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

consider a problem

Consider the shape given in the figure. It is a parallelogram of base $b$ centimeter and height $h$ centimeter. To find the area of the parallelogram, the grid of unit squares is superimposed and the number of unit squared are counted. Since unit-squares will not fit exactly at the vertices, only approximate area can be computed.

equivalence

A rectangle $ABRQ$ of length $b$ cm and width $h$ cm is shown in blue. Consider the region $\u25b3BRC$ and the region $\u25b3AQD$. The area of these two triangles are equal.

The area of the parallelogram is sum of areas of the region $ABRD$ and $\u25b3BRC$. This can be equivalently given as sum of areas of the region $ABRD$ and $\u25b3AQD$.

This implies that "area of the parallelogram equals the area of the rectangle".

summary

**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

Outline

The outline of material to learn *Mensuration : Length, Area, and Volume* is as follows.

Note 1: * click here for the detailed overview of Mensuration High *

Note 2: * click here for basics of mensuration, which is essential to understand this. *

• ** Basics of measurement**

→ __Summary of Measurement Basics__

→ __Measurement by superimposition__

→ __Measurement by calculation__

→ __Measurement by equivalence__

→ __Measurement by infinitesimal pieces__

→ __Cavalieri's Principle (2D)__

→ __Cavalieri's Principle (3D)__

• **Perimeter & Area of 2D shapes**

→ __Circumference of Circles__

→ __Area of Circles__

• **Surface area & Volume of 3D shapes**

→ __Prisms : Surface Area & Volume__

→ __Pyramids : Surface Area & Volume__

→ __Cone : Surface Area & Volume__

→ __Sphere : Surface Area & Volume__

• **Part Shapes **

→ __Understanding part Shapes__

→ __Circle : Sector and Segment__

→ __Frustum of a Cone__