Vardaan
Class 9 Maths • Chapter 10 • NCERT + RS Aggarwal

Heron's Formula

Vardaan Learning Institute  |  School-Exam Focused Notes

📐 1. Area of a Triangle (Basic Rule)

The standard formula for the area of a right-angled triangle or any triangle where the height (altitude) is known:
Area of Triangle = ½ × Base × Height
Note: The height must be perpendicular to the base chosen.

🧮 2. Heron's Formula (For Scalene Triangles)

When the height of a triangle is not given, but the lengths of all three sides are known, we use Heron's Formula.

B C A a (Opposite A) b c
Area = √(s(s-a)(s-b)(s-c))
Where:
a, b, c are the lengths of the sides of the triangle.
s is the semi-perimeter (half of the perimeter) of the triangle.
s = (a + b + c) / 2
Calculation Tip for Square Roots When finding the area using Heron's formula, DO NOT multiply $s(s-a)(s-b)(s-c)$ into one giant number. Instead, prime factorize each term and pair them up to easily take them out of the square root!
Example: $\sqrt{21 \times 7 \times 8 \times 6}$ = $\sqrt{(3\times 7) \times 7 \times (2\times 2\times 2) \times (2\times 3)}$.
Pair them: $(7\times 7) \times (3\times 3) \times (2\times 2) \times (2\times 2) \implies 7 \times 3 \times 2 \times 2 = 84$.

🔺 3. Derived Formulas for Specific Triangles

Type of Triangle Properties Formula for Area Height (Altitude)
Equilateral All 3 sides equal ($a=b=c$) $\frac{\sqrt{3}}{4} a^2$ $h = \frac{\sqrt{3}}{2} a$
Isosceles Two sides equal ($a=b$) $\frac{b}{4} \sqrt{4a^2 - b^2}$
*(b is unequal base)*		 Using Pythagoras
Right-Angled One angle = 90° $\frac{1}{2} \times base \times perpendicular$ ---
Worked Example: Find the area of a triangle with sides 13cm, 14cm, and 15cm.
1. Find semi-perimeter (s) = $\frac{13 + 14 + 15}{2} = \frac{42}{2} = 21$ cm.
2. $(s-a) = 21-13 = 8$ cm.
3. $(s-b) = 21-14 = 7$ cm.
4. $(s-c) = 21-15 = 6$ cm.
5. Area = $\sqrt{21 \times 8 \times 7 \times 6}$ = $\sqrt{7056} = \mathbf{84 cm^2}$.

🔲 4. Application: Area of Quadrilaterals

Heron's formula is heavily used to find the area of quadrilaterals by dividing them into two triangles.
Step 1: Join one diagonal to split the quadrilateral into 2 triangles.
Step 2: Use Heron's formula (or ½ base × height if it's right-angled) to find the area of Triangle 1.
Step 3: Use Heron's formula to find the area of Triangle 2.
Step 4: Total Area = Area(Triangle 1) + Area(Triangle 2).
✏️ Practice Questions — Heron's Formula (25 Questions)

Section A — 1 & 2 Mark Direct Qs Easy

Q1. Find the semi-perimeter of triangle with sides 8m, 11m, 13m.
Q2. The perimeter of a triangle is 60cm. If sides are in ratio 1:2:3, what is 's'?
Q3. Find the area of an equilateral triangle with side 10cm. ($\sqrt{3} \approx 1.73$)
Q4. Sides are 3cm, 4cm, 5cm. Find area. (Hint: Right triangle or Heron).
Q5. Find the area of a triangle with sides 15cm, 11cm and semi-perimeter 16cm.
Q6. The height of an equilateral triangle is $6\sqrt{3}$ cm. Find its area.
Q7. If the area of an equilateral triangle is $16\sqrt{3}$, find its perimeter.

Section B — Ratio & Cost Problems Medium

Q8. The sides of a triangular plot are in ratio 3:5:7 and perimeter is 300m. Find area.
Q9. A triangle has sides 35cm, 54cm, and 61cm. Find its longest altitude. (Hint: Longest altitude is on the shortest base).
Q10. The perimeter of an isosceles triangle is 32cm. The ratio of equal side to its base is 3:2. Find area.
Q11. A park in shape of quad ABCD has $\angle C = 90°$, AB=9, BC=12, CD=5, AD=8. How much area does it occupy?
Q12. Find the cost of leveling a triangular field at ₹40 per $m^2$ if its sides are 20m, 21m, 29m.
Q13. An umbrella is made by stitching 10 triangular pieces of cloth of two different colors, each piece measuring 20cm, 50cm, and 50cm. How much cloth of each colour is required?

Section C — NCERT Deep Dive Hard

  1. Q14. A kite in the shape of a square with a diagonal 32cm and an isosceles triangle of base 8cm and sides 6cm each is to be made of 3 different shades. How much paper of each shade?
  2. Q15. A field is in the shape of a trapezium whose parallel sides are 25m and 10m. The non-parallel sides are 14m and 13m. Find the area of the field. (Very Important!).
  3. Q16. Students of a school staged a rally for cleanliness. They walked in two groups through lanes forming triangle ABC and triangle ACD. If $AB = 9m, BC = 40m, CD = 15m, DA = 28m$ and $\angle B = 90°$. Which group cleaned more area and by how much? Find total area.
  4. Q17. A floral design on a floor is made up of 16 tiles which are triangular. Sides of the triangle are 9cm, 28cm and 35cm. Find cost of polishing tiles at 50p per $cm^2$.
  5. Q18. Radha made a picture of an aeroplane with coloured paper. Find total area of paper used (NCERT aeroplane problem - break it into 5 regions: triangle, rectangle, trapezium, 2 triangles).

Section D — Board Level Variants Hard

  1. Q19. If each side of a triangle is doubled, then find the ratio of area of the new triangle thus formed and the given triangle.
  2. Q20. Every side of a triangle is increased by 50%. Find the percentage increase in the area.
  3. Q21. A rhombus shaped field has green grass for 18 cows to graze. If each side of the rhombus is 30m and its longer diagonal is 48m, how much area of grass field will each cow be getting?
  4. Q22. The lengths of sides of a triangle are 5cm, 12cm and 13cm. Find the length of perpendicular from opposite vertex to the side whose length is 13cm.
  5. Q23. Find the area of a quadrilateral ABCD where $AB=7cm, BC=6cm, CD=12cm, DA=15cm, AC=9cm$.
  6. Q24. Base of a right triangle is 8cm and hypotenuse is 10cm. Find area.
  7. Q25. In the isosceles triangle, if the unequal side is 8cm and area is $12 cm^2$, find the equal sides.
✅ Key Checkpoints: Q1: s=16 | Q3: $43.25 cm^2$ | Q5: perimeter=32, third side=6. s=16, area = $\sqrt{16 \times 1 \times 5 \times 10} = 20\sqrt{2}$. | Q8: sides are 60,100,140. Area=$1500\sqrt{3}$ | Q11: BD=13 (Pythagoras). Area1=30, Area2=35.5. Total Area = 65.5 | Q15: Draw line parallel to non-parallel side. Inner triangle sides: 13,14,15. Area=84. Height=11.2. Trap Area=196. | Q19: Ratio is 4:1 | Q20: 125% increase.