Theorems on Area

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What are Area Theorems?

The area of the theorem has basically few properties which are as follows:

  • A parallelogram is divided into two triangles of equal areas by the diagonal.

  • The ratio of the areas of 2 triangles with a similar height is equivalent to the ratio of their bases.

  • The ratio of the areas of 2 triangles on the same base is equivalent to the ratio of their heights.

  • The area of triangles that are congruent is equal.

Area Theorems

The theorems state some link between the areas of these geometric objects under the condition when they lie between the same parallel lines and on the same base (or equal bases). Following are the area theorem axioms:

  • Theorem 1

Diagonal of a parallelogram cut it half into 2 triangles of the same area. Parallelograms between the same parallels and on the same base are equal in area. A diagonal of a parallelogram divides it into two triangles of the same area In this case area of (△ABC) = area of (△ADC). Also area of (△ABD) = area of (△BCD)

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  • Theorem 2

The area of a parallelogram is equivalent to the area of the rectangle of the same altitude and on the same base, i.e., between the same parallels. That is to say, the area of (||gm ABCD) = Area of (rectangle ABFE) since they lie between the same parallels AB and DE and on the same base.

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  • Theorem 3

The area of a triangle is half of the area of a parallelogram lying between the same parallels and on the same base. From the figure below, the area of (∆ APB) = ½ × Area of (||gm ABCD) seeing that they lie between the same parallels AB and PC and on the same base. Area of a parallelogram is the product of its base and the corresponding height.

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  • Theorem 4

Triangles between the same parallels as well on the same base are equivalent in area. Area of (∆ ABD) = Area of (∆ ABC) since they remain between the same parallels AB and DC and on the base AB. The area of a triangle is half the product of its corresponding height and any of its sides. This theorem is also called Heron's Theorem.

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  • Theorem 5

If a parallelogram and a parallelogram lies between the same parallels and on the same base, thus the area of the triangle will be equivalent to the half of the parallelogram.

Area of (△ABCD) = area of (△BCD)

Area of (||gmABCD) = AB × h

Area of (△ABE) = ½ AB × h

Area of (△ABE) = Area of (||gmABCD)

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  • Theorem 6

This is a Trapezium Area Axiom. According to this theorem, the area of a trapezium is half the product of the sum of its parallel sides and the altitude. A trapezium is a type of a quadrilateral that has two of its sides parallel to each other. There is also a type of trapezium which we call an isosceles trapezium whose non-parallel sides are equal.  Having said that, suppose that we have ‘a’ and ‘b’ the parallel sides and ‘h’, the distance between the parallel sides of a parallelogram ABCD. Then Area = (½ A+B) × h

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  • Theorem 7

Triangles between the same parallels and on the same base share the same area.

Area of (△ABD) = ½ × AB × h

Area of (△ABC) = ½ × AB × h

Hence, Area of (△ABD) = Area of (△ABC)

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  • Theorem 8

Triangles with equal areas and whose one side of one of the triangles equivalent to one side of the other triangle, with their corresponding heights the same.    

  • Theorem 9

Two triangles whose bases are the same (or equal bases) and equal area remain between the same parallels.

FAQ (Frequently Asked Questions)

Q1. What is the Area of a Plane?

Answer: Area of a plane is basically the region bounded by a plane figure. Simply know that equal figures mean the figures equivalent in areas. Understanding the different theorems related to the areas helps in knowing the properties and surface areas of different polygons.

Q2. What is a Planar Region?

Answer: Geometry originated from the necessity of measuring land or remodelling/refixing its boundaries in the mechanism of distribution of a particular field or land among different people. The part of the plane circumscribed by a simple closed curve is known as a planar region relative to that figure and the magnitude of this plane region is known as the area of that figure.

Q3. What are the Examples of a Closed Plane Figure?

Answer: Some of the different simple closed plane figures are as follows:

  • Rectangle

  • Triangle

  • Square

  • Parallelogram

  • Trapezium etc.

Q4. What are the Examples of a Closed Plane Figure?

Answer: This magnitude of measure is always demonstrated using a number (in some unit) like 15 cm², 32 m², 3·5 hectares etc. So, we can assert that the area of a simple closed plane figure is a number (in some unit) linked with the part of the plane circumscribed by the figure.