Chapter 5 · Section 3Textbook pages 92–96

Areas of Triangles ത്രികോണപ്പരപ്പുകൾ

Numbers like √2 and √3 were invented to name lengths that are not fractions. Now we find they are needed for areas too: even when every side of a triangle is a whole number, its area is often an irrational number. This section ends with one of the most famous formulas in mathematics – Heron's formula – which gives the area of any triangle straight from its three sides.

📖 What you already know

Area of a triangle = ½ × base × height (page 4).
Pythagoras: (hypotenuse)² = (leg)² + (leg)² (page 4).
In an isosceles triangle the height from the apex bisects the base (Class 8, തുല്യത്രികോണങ്ങൾ).
√x × √y = √(xy), √(x/y) = √x/√y, and simplifying √(k²m) = k√m (pages 5 and 7).
(x + y)(x − y) = x² − y² (pages 4 and 7).

1. The equilateral triangle of side 2 m

In Section 1 we saw that an equilateral triangle with side 2 m has height √3 m. What is its area?

🎬 Animation: area = ½ × base × height

2. Any equilateral triangle: side a

The same method works for every equilateral triangle. Let the side be a (in centimetres, metres – any unit).

🎬 Animation: deriving the height and the area

⭐ Key idea 5 (textbook page 93)

ഒരു സമഭുജത്രികോണത്തിന്റെ ഉയരം, വശത്തിന്റെ പകുതിയുടെ √3 മടങ്ങും, പരപ്പളവ്, വശത്തിന്റെ പകുതിയുടെ വർഗത്തിന്റെ √3 മടങ്ങുമാണ്.

height = √3 × (half the side) = (√3/2)·a area = √3 × (half the side)² = (√3/4)·a²

Example: side 8 cm ⇒ half side 4 cm ⇒ height 4√3 cm ≈ 6.93 cm, area √3 × 4² = 16√3 sq cm ≈ 27.7 sq cm.

🎚️ Equilateral calculator

side a

Keep the answers in the form k√3 – that is exact. Turn them into decimals only at the end.

3. An isosceles triangle: sides 6, 6 and 4

Isosceles triangles (സമപാർശ്വത്രികോണം) work the same way, because the height from the apex bisects the base (the two halves are congruent right triangles).

🎬 Animation

4. A general triangle: sides 8, 7 and 3

When no two sides are equal, the height no longer lands at the midpoint of the base. The textbook shows how a little algebra – the difference of two squares – still finds the height.

🎬 Animation: finding where the height lands

💡 The trick inside the calculation

Two Pythagoras equations, one for each right triangle, both contain h². Subtracting kills h² and leaves x² − (8 − x)², a difference of two squares, which factorises as (x + 8 − x)(x − 8 + x) = 8(2x − 8). That is why the identity x² − y² = (x + y)(x − y) from Class 8 matters here.

5. Heron's formula ഹെറോൺ സൂത്രവാക്യം

Doing that algebra for every triangle is tiring. Doing it once in general (with sides a, b, c instead of 8, 7, 3) gives a formula that skips the height entirely. The derivation is in the textbook's appendix – you will see it step by step on page 10 of this guide.

⭐ Key idea 6 – Heron's formula (textbook page 95)

വശങ്ങളുടെ നീളം a, b, c ആയ ത്രികോണത്തിൽ s = ½(a + b + c) എന്നെടുത്താൽ, പരപ്പളവ്

area = √s(s − a)(s − b)(s − c), where s = ½(a + b + c) is half the perimeter

Check with the 8-7-3 triangle: s = ½(8 + 7 + 3) = 9, s − a = 1, s − b = 2, s − c = 6, so area = √(9 × 1 × 2 × 6) = √108 = √(36 × 3) = 6√3 sq cm – exactly what the long calculation gave.

🏛️ Heron of Alexandria

Heron (or Hero) lived in Alexandria, Egypt, in the 1st century AD. He was an engineer as much as a mathematician: he described a steam-powered toy, automatic doors and a coin-operated machine. His formula let surveyors compute the area of a triangular field from three tape measurements alone – no need to find a height at all.

🧮 Heron calculator – any three sides

a b c

If two sides together are not longer than the third, no triangle exists – the calculator tells you. Try 5-12-13 (a right triangle), 6-6-4, 13-14-15 (a whole-number area!), and 4-5-7.

6. Measuring land സ്ഥലപ്പരപ്പ്

A plot of land with straight boundaries can be split into triangles from one corner. Measure every side and every diagonal with a tape, find each triangle's area by Heron's formula, and add them up. Click the triangles (or the buttons) in the textbook's plot below.

🗺️ Area of a seven-sided plot (lengths in metres)

Click a triangle to compute its area.

Sides: 13, 15, 13, 9, 11, 10, 12 m; diagonals from the hub corner: 15, 17, 17, 16 m. Each triangle's three sides are known, so Heron's formula works without a single height being measured.

7. Check yourself

🧭 Summary of Section 3

Equilateral triangle, side a: height = (√3/2)a, area = (√3/4)a². Side 8 ⇒ 4√3 and 16√3.
Isosceles triangle: the height bisects the base; use Pythagoras on one half. Sides 6, 6, 4 ⇒ height 4√2, area 8√2.
Any triangle: name the foot of the height (x and a − x), write two Pythagoras equations, subtract, factorise – or skip all that with Heron.
Heron: s = ½(a + b + c), area = √(s(s−a)(s−b)(s−c)). For 8, 7, 3: √(9·1·2·6) = 6√3.
Surveyors divide a plot into triangles and add their Heron areas.

Next: Exercise 3 – all six questions solved, plus the proof of Heron's formula ▶