Area Axioms

Area Axiom: Every polygonal region has an area and is measured in square units.

Congruent Area Axiom: If \triangle ABC and \triangle DEF are two congruent triangles, then ar(\triangle ABC) = ar(\triangle DEF) . ie. two congruent regions have equal area.

Rectangle Area Axiom: If ABCD is a rectangular  region such that AB = a units and BC = b units, then ar(ABCD) = ab square units.

 

Parallelograms on the same base and between the same parallels

2019-02-09_22-32-31Theorem 1: A diagonal of a parallelogram divides it into two triangles of same area

In this case ar (\triangle ABC) = ar (\triangle ADC)

Also ar (\triangle ABD) = ar (\triangle BCD)

Theorem 2: Parallelograms on the same base and between the same parallels are equal  in area.2019-02-10_9-54-47

ar (ABCD) = AB \times h

ar (ABFE) = AB \times h

\therefore ar (ABCD) = ar (ABFE)

Theorem 3: Area of a parallelogram is the product of its base and the corresponding altitude.2019-01-12_12-45-40

Let the two adjacent sides of the parallelogram be a and b .  Area = Base \times Height

 

Triangle Area Axiom

Theorem 4: The area of a triangle is half the product of any  of its sides and the corresponding altitude.

2019-01-12_12-49-14a, b, c denotes the sides of the Triangle. Then:

Area = \frac{1}{2} \times Base \times Height = \frac{1}{2} bh

Area = \sqrt{s(s-a)(s-b)(s-c)} . This is known as Heron’s Theorem.

Theorem 5: If a triangle and parallelogram are on the same base and between the same parallels, the area of the triangle is equal to the half of the parallelogram. 2019-02-10_10-00-24

ar (ABCD) = AB \times h

ar ( \triangle ABE) = \frac{1}{2} \times AB \times h

ar ( \triangle ABE) = ar (ABCD)

 

Trapezium Area Axiom

Theorem 6: The area of a trapezium is half the product of its heights and the sum of parallel sides.

2019-01-12_12-45-06A trapezium is a quadrilateral two of whose sides are parallel. A trapezium whose non-parallel sides are equal is known as an isosceles trapezium.

Let a and b be the parallel sides and h be the distance between the parallel sides. Then Area = \frac{1}{2} (a+b) \times h

 

Triangles on the same base and between the same parallels

Theorem 7: Triangles on the same base and the same parallels have the same area.2019-02-10_10-05-29

ar ( \triangle ABD) = \frac{1}{2} \times AB \times h

ar ( \triangle ABC) = \frac{1}{2} \times AB \times h

\therefore ar ( \triangle ABD) = ar ( \triangle ABC)

Theorem 8: Triangles having equal areas and having one side of one of the triangles equal to one side of the other triangle, have their corresponding altitudes the same.

Theorem 9: Two triangles having the same bases (or equal bases) and equal area lie between the same parallels.

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