Areas of Parallelograms and Triangles — Study Notes

1. Introduction

We all know how to find the area of simple figures: a rectangle is length times breadth, and a triangle is half of base times height. But why do these formulas hold, and how can we compare the areas of figures that look completely different? This chapter answers such questions in a systematic way. We will learn that figures lying on the same base and between the same parallels have equal areas, which is one of the most powerful results in geometry.

The ideas in this chapter are important both for practical measurement and for theoretical geometry. When we know that two parallelograms on the same base and between the same parallels have equal areas, we can relate the areas of triangles to the areas of parallelograms and solve many problems without computing any lengths at all. These concepts prepare the way for Heron's formula and for the mensuration of the later chapters, and they appear again in the study of similarity of figures in higher classes.

2. Basic Concepts of Area

The area of a closed figure is the amount of region enclosed by it. Two figures can have the same area even if their shapes are completely different. The key ideas used in this chapter are:

• A parallelogram is a quadrilateral whose opposite sides are parallel and equal.
• A triangle is a figure with three sides.
• Two figures are said to be on the same base if they share a common side.
• Two figures are between the same parallels if they lie between two parallel lines, one of which contains their common base.

Basic Area Formulas

• Area of a parallelogram = base x height = b x h, where h is the perpendicular height.
• Area of a triangle = (1/2) x base x height = (1/2) b h.
• Area of a rectangle = length x breadth.
• Area of a square = side x side.

Note that the height in these formulas is always the perpendicular distance, not the length of a slanted side.

3. Parallelograms on the Same Base and Between the Same Parallels

Theorem: Equal Areas of Parallelograms

Parallelograms on the same base and between the same parallels have equal areas.

If two parallelograms ABCD and ABEF stand on the same base AB and lie between the parallels AB and DE, then their areas are equal. To prove this, we use the congruence of triangles. The area of each parallelogram equals the area of the rectangle with the same base and the same perpendicular height. Since the perpendicular distance between the two parallels is the same for both, both parallelograms have equal heights and equal bases, hence equal areas.

An immediate consequence is: the area of a parallelogram equals the area of a rectangle of the same base and height, because a rectangle is itself a parallelogram.

4. Triangles on the Same Base and Between the Same Parallels

Theorem: Equal Areas of Triangles

Triangles on the same base and between the same parallels have equal areas.

If two triangles ABC and ABD stand on the same base AB and lie between the parallels AB and CD, then their areas are equal. This follows because each triangle has the same base AB and the same perpendicular height (the distance between the parallel lines), so area = (1/2) x base x height gives equal areas.

Corollary 1: A Triangle and a Parallelogram

A triangle on the same base and between the same parallels as a parallelogram has half the area of the parallelogram. In other words: area of triangle = (1/2) x area of parallelogram, when they are on the same base and between the same parallels.

Corollary 2: Median Divides a Triangle into Two Equal Areas

The median of a triangle divides it into two triangles of equal area. Since the median joins the vertex to the midpoint of the opposite side, both smaller triangles have the same base (half the side) and the same height. Hence their areas are equal. In fact, the area of each is half the area of the original triangle.

Corollary 3: Midpoint Connection

If a triangle and a parallelogram are on the same base and between the same parallels, the area of the triangle is half that of the parallelogram. This is the same result as Corollary 1, expressed in another way.

5. Problem-Solving Strategies

Many problems in this chapter do not require direct calculation of lengths; they ask us to compare areas. The standard strategy is:

1. Identify the common base.
2. Check whether the two figures lie between the same parallels.
3. Apply the appropriate theorem: same base and same parallels implies equal areas.
4. If one figure is a triangle and the other a parallelogram, use the half-area relationship.

For example, if triangle ABC and triangle DBC are on the same base BC and the line AD is parallel to BC, then their areas are equal. This lets us find one area when the other is given, without any measurements.

Quick Revision Tables

Table 1: Key Area Formulas

Table 2: Theorems on Equal Areas

Mind Map

Important Diagrams (SVG)

Diagram 1: Parallelograms on the Same Base and Between the Same Parallels

Diagram 2: Median Divides a Triangle into Equal Areas

Common Mistakes

1. Students often use the slanted side of a parallelogram as the height. The height is always the perpendicular distance between the parallel sides.
2. A common error is forgetting the factor 1/2 in the area of a triangle. The correct formula is (1/2) x base x height.
3. Students sometimes conclude that two triangles with the same area must be congruent. Equal area does not imply congruence; the shapes can be very different.
4. When comparing areas, students forget to check that the figures are between the same parallels; the theorem requires this condition.
5. Students apply the "triangle is half the parallelogram" rule without confirming that they are on the same base and between the same parallels.
6. A frequent mistake is saying that the median bisects the area but also the angles. The median bisects the area, but it does not necessarily bisect the angle.
7. Students forget that the area of a rhombus is half the product of its diagonals, confusing it with the parallelogram formula.

Exam Tips

1. Always identify and state the common base and the pair of parallel lines before applying any equal-area theorem; this reasoning carries the marks.
2. For median problems, write "median divides the triangle into two equal-area triangles" and quote the half-area relationship.
3. When the areas of two triangles on the same base are given as equal, immediately conclude that they lie between the same parallels.
4. Practise converting between area of a triangle and area of a parallelogram, using the relationship that the triangle area is half the parallelogram area when bases and parallels match.
5. Draw the perpendicular height clearly on the figure, since the height is perpendicular and not along the slanted side.
6. For comparing areas, prefer the "same base and same height" reasoning over actual computation, as it is faster and less error-prone.
7. Memorise the formula area of rhombus = (1/2) x product of diagonals, which is frequently tested separately.

Conclusion

In this chapter we learnt the basic area formulas for parallelograms, triangles, rectangles and squares, and the crucial idea that the height is the perpendicular distance. We proved the fundamental theorem that parallelograms on the same base and between the same parallels have equal areas, and the corresponding theorem for triangles. We also established that a triangle on the same base and between the same parallels as a parallelogram has half its area, and that a median divides a triangle into two triangles of equal area. These results allow us to compare and relate areas without lengthy calculations, and they form the theoretical foundation of mensuration. The concepts learned here will be used directly in the chapters on circles, Heron's formula and surface areas and volumes.