Area Proposition

A planar region is the part of the plane enclosed by a simple closed figure. And the magnitude of this region is the area. By now, you already have an idea about the calculation of the area of parallelogram, square, rectangle, triangle, etc. in this article, we will use these formulas to understand the relationship between the areas of these geometric figures when they lie on the same base and/or between the same parallels.

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Area Proposition

Before we begin, it is important to note that two congruent figures have the same areas (Fig.1). However, if two figures have the same areas, then they are NOT necessarily congruent (Fig2.).

Figures on the Same Base and Between the Same Parallels

Take a look at these figures:

What do you observe? In each figure, the base is the same across two polygons and between different parallels.

• Trapezium ABCD and parallelogram EFCD have a common base DC. [Fig.3 – (i)]
• Parallelograms PQRS and MNRS are on the same base SR. [Fig.3 – (ii)]
• Triangles ABC and DBC are on the same base BC. [Fig.3 – (iii)]
• Parallelogram ABCD and triangle PDC are on the same base DC. [Fig.3 – (iv)]

Now, take a look at these figures:

What do you observe in these figures? In each figure, the base is the same across two polygons and between same parallels.

• Trapezium ABCD and parallelogram EFCD are on the same base DC and between the same parallels AF and DC. [Fig.4 – (i)]
• Parallelograms PQRS and MNRS are on the same base SR and between the same parallels PN and SR. [Fig.4 – (ii)]
• Triangles ABC and DBC lie on the same base BC and between the same parallels AD and BC. [Fig.4 – (iii)]
• Parallelogram ABCD and triangle PCD lie on the same base DC and between the same parallels AP and DC. [Fig.4 – (iv)]

Note: Two figures are said to be on the same base and between the same parallels, if they have a common base (side) and the vertices (or the vertex) opposite to the common base of each figure lie on a line parallel to the base.

Area of Parallelogram on the Same Base and between the Same Parallels

Take a graph sheet and draw two parallelograms as shown below:

Find the area of parallelogram ABCD and PQCD by counting the squares. What do you observe? Are the areas of the two parallelograms different or equal? In fact, they are equal. Let’s try to prove this relation:

Theorem 1

Parallelograms on the same base and between the same parallels are equal in area.

Proof:

In the above figure, two parallelograms ABCD and EFCD are given on the same base DC and between the same parallels. We need to prove that,

Area (ABCD) = Area (EFCD)

In ∆ ADE and ∆ BCF, ∠ DAE = ∠ CBF … (1)

These are corresponding angles from AD parallel to BC and transversal to AF.
∠ AED = ∠ BFC … (2)

These are corresponding angles from ED parallel to FC and transversal to AF. Therefore, using the angle sum property of triangles,
∠ ADE = ∠ BCF … (3)

Also, being the opposite sides of the parallelogram ABCD, AD = BC … (4)

So, by using the Angle-Side-Angle (ASA) rule of congruence and (1), (3) and (4), we have
∆ ADE ≅ ∆ BCF

We know that congruent figures have equal areas. Hence,
Area (ADE) = Area (BCF) … (5)

Now, Area (ABCD) = Area (ADE) + Area (EDCB)

From the equation (5), we can deduce that
Area (ABCD) = Area (BCF) + Area (EDCB) … [From (5)]
= Area (EFCD)

So, the area of parallelogram ABCD and EFCD is equal.

Triangles on the Same Base and between the Same Parallels

Now, draw two triangles as shown below:

What do you observe? Are the areas of the two triangles (ABC and PBC) different or equal? In fact, they are equal. Let’s try to prove this relation:

Theorem 2

Two triangles on the same base (or equal bases) and between the same parallels are equal in area.

In the above figure, ABCD is a parallelogram and AC is one of its diagonals. A line ‘AN’ is drawn which is perpendicular to the line ‘DC’. We know that a diagonal divides a parallelogram into two congruent triangles. Hence,

∆ ADC ≅ ∆ CBA

Also, we know that congruent figures have equal areas. So,

Area (ADC) = Area (CBA)

In simple words, the two triangles of the same (or equal) base and lying between the same parallels are equal in area. Taking the equation further, we can write

Area (ADC) = 1/2 [Area (ABCD)]

Now, the area of parallelogram is the length of the base multiplied by height. Hence, we have

Area (ADC) = 1/2 (DC x AN)

Or, Area of ∆ ADC = 1/2 (base DC × corresponding altitude AN)

Area of a triangle is half the product of its base (or any side) and the corresponding altitude (or height). This is the standard formula for calculation of the area of a triangle. Let’s look at the converse of Theorem 2.

Theorem 3

Two triangles having the same base (or equal bases) and equal areas lie between the same parallels.

In Theorem 2, we have proved that two triangles on the same base (or equal bases) and between the same parallels are equal in area. As a logical derivation of the same, it is easy to conclude that two triangles having the same base (or equal bases) and equal areas lie between the same parallels.

Solved Examples for You

Question: In the figure given below, E is any point on the median AD of a ∆ ABC. Show that Area (ABE) = Area (ACE).

Solution: Given – ABC is a triangle and AD is the median. Hence, BD = DC. Also, E is a point on AD.

To prove: Area (ABE) = Area (ACE)

Proof: Join points E and B and point E and C. Now, in triangle ABC, AD is the median. Since a median divides the triangle into two triangles having equal areas, we have
Area (ABD) = Area (ADC) … (a)

In triangle EBC, ED is the median (since point D divides BC in half). Hence,
Area (EBD) = Area (EDC) … (b)

Next, we subtract (b) from (a) and get,
Area (ABD) – Area (EBD) = Area (ADC) – Area (EDC)

If you look at the figure, ABD – EBD = ABE. Also, ADC – EDC = AEC. Hence,
Area (ABE) = Area (AEC). Proved.

Question- What is the formula of finding an area of the parallelogram?

Answer- As we know the opposite sides are equal in length and opposite angles are also equal in measure. Thus, in order to find out the area of a parallelogram, we will multiply the base by the height. The Formula is A = B * H. Over here, B is the base, H is the height, and * is multiplication.

Question- Is the area of a rectangle and parallelogram the same?

Answer- As you know already that parallelogram and rectangle compose of the same parts, they automatically have the same area. Moreover, we also notice how they have the exact same base length and height.

Question- How do you find the area of a parallelogram without height?

Answer- If the slant height and angle between the slant height and the base are known, we can make use of trigonometry, sin, to find out the height. Other than that, if the small segment forming a right triangle and slant height are known, we can make use of Pythagoras theorem.

Question- What is an example of a parallelogram?

Answer- A parallelogram comprises four sides where the sides opposite each other are parallel. Thus, examples of parallelograms are square, rhombus, and rectangle.