GANITA MANJARI · GRADE 9 · PART I · CHAPTER 6

SIDES — Heron, Brahmagupta and Generalisation

Think first. Use AI purposefully. Verify always.

DISCOVER

Start with a question.

Can you determine area from side lengths alone? Does the same answer hold for triangles and quadrilaterals?

Understand

Heron’s formula applies to valid triangles. Brahmagupta’s formula applies to cyclic quadrilaterals. A general quadrilateral’s sides alone do not determine its area.

Key Ideas

A = \sqrt{s (s - a) (s - b) (s - c)}

For a valid triangle, s = $\frac{(a + b + c)}{2}$ . For a cyclic quadrilateral only:

s = \frac{a + b + c + d}{2}

A = \sqrt{(s - a) (s - b) (s - c) (s - d)}

Do not use Brahmagupta for an arbitrary quadrilateral. Heron gives no non-degenerate triangle when the largest side equals the sum of the others.

Visualise

Side a (units): 3

Side b (units): 4

Side c (units): 5

The figure is computed from the selected side lengths.

Move one vertex along the circumcircle. All four vertices must stay on that circle.

Moving vertex angle (degrees): 65

The circumcircle certifies the cyclic condition; four arbitrary side lengths alone are not enough.

Method

Check triangle inequalities first. Compute semiperimeter, substitute, and retain the radical exactly when possible. For four sides, first establish that the quadrilateral is cyclic.

Worked Example

Given

Sides 3, 4 and 5 units (Example 5).

To Find

Find the area in two ways.

Formula / Property / Theorem / Reasoning

Heron’s formula and the converse of the Baudhāyana–Pythagoras theorem.

Solution

s = $\frac{(3 + 4 + 5)}{2}$ = 6.
A = $\sqrt{6 \times 3 \times 2 \times 1}$ = 6 square units.
Because $3^{2}$ + $4^{2}$ = $5^{2}$ , the triangle is right-angled; $\frac{(3 \times 4)}{2}$ = 6.

Result

Both methods give 6 square units.

AIMS Method

A — Attempt independently. Make a prediction or start a sketch.
I — Investigate the idea. Identify the boundary, region and given conditions.
M — Make connections. Relate the figure to a known property or rearrangement.
S — Scrutinise your reasoning. Check conditions, units and whether the result is exact.

Think first. Use AI purposefully. Verify always.

Focused Practice

Can sides 2, 3 and 6 form a triangle? Explain before using a formula.

Your reasoning

Check & Feedback

What extra condition allows Brahmagupta’s formula for four side lengths?

Explain your conclusion

Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.8.1 · Examples 3–7, Figures 6.23–6.29 · printed pages 134–140.