Measurement by superimposition

Overview

**Measurement by Superimposition** : Length, Area, or Volume can be measured by superimposition of corresponding unit-measures.

overlay a grid

Consider the shape given in the figure. It is a rectangle of length $7$ cm and width $5$ cm. The figure also provides the reference unit-square on the top-right. The area of the shape is "$35c{m}^{2}$". The area is computed by creating a grid of unit-squares. The number of unit squares contained within the shape is the area of the rectangle.

simplify

Area of the given shape is calculated by superimposing the reference unit-squares. Based on this, the area of a square or rectangle is simplified to the formulas

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

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

approximate

Consider the shape given in the figure. The measurements are not provided. The reference unit-square is shown on the top-right corner. Can we calculate the area of the given shape? Yes, "approximate area can be calculated by superimposing a grid of unit-squares".

The area is approximately $15c{m}^{2}$.

.

The area is approximated to the count of large squares within the figure.

suitable for some shapes

Measurement by superimposition provides accurate results for

• area of squares and rectangles : The unit-square has $90}^{\circ$ angles at the vertices. And so, the unit-square fits within Squares and Rectangles.

• volume of cubes and cuboids : The unit-cube has $90}^{\circ$ angles at the vertices. And so, the unit-cube fits within cubes and cuboids.

For shapes like triangles or prisms, measurement requires some geometrical calculation or the measurement will be approximate.

summary

**Measurement by Superimposition** : Length, Area, or Volume can be measured by superimposition of corresponding unit-measures.
This method suits best for

• area of squares and rectangles

• volume of cubes and cuboids

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__