Area of a Closed Figure

We will discuss here about the area of a closed figure, measurement of area, area axiom for rectangle, area axiom for congruent figures and addition axiom for area.

Area of a closed figure

The measure of the reason bounded by a closed figure in a plane is called its area. In the following the areas of the figures are shaded.

Measurement of area

The area of a square of sides of length 1 unit is called an area of 1 unit². The area of a closed figure is measured by the number of unit squares contained in the region.

Area axiom for rectangle

The area of a rectangle is the product of its length and breadth. PQRS is a rectangle region. Its area = PQ × QR.

Area axiom for congruent figures

Any two congruent figures have equal area.

Let ∆PQR ≅ ∆XYZ. Then the area of the ∆PQR is equal to the area of the ∆XYZ.

We write ar(∆PQR) for the area of the ∆PQR.

Therefore, ∆PQR ≅ ∆XYZ     ⟹ ar(∆PQR) = ar (∆XYZ).

In the same way, if two polygons are congruent then their areas will be equal.

Note: Two triangles (or closed figures) may have equal areas but they may not be congruent.

Addition axiom for area

If a closed reason R is divided into two regions R\(_{1}\) and R\(_{2}\) that enclose no common region then

ar (region R) = ar(region R\(_{1}\)) + ar(region R\(_{2}\)).

Here, ar(quadrilateral PQRS) = ar(∆PQS) + ar(∆QRS).

9th Grade Math

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