Class 9 — Mathematics (NCERT)

Heron's Formula

Class 9

✓To understand the concept of Heron's Formula for calculating the area of a triangle.
✓To calculate the area of a triangle when the lengths of its three sides are given using Heron's Formula.
✓To apply Heron's Formula to find the area of quadrilaterals by dividing them into triangular parts.
✓To solve practical problems involving the area of triangles and quadrilaterals.

Key concepts

Area of a Triangle (Review)

We know that the area of a triangle is half the product of its base and corresponding height. That is, Area = (1/2) × base × height. This formula is useful when the height of the triangle is known or can be easily found. However, if only the lengths of the three sides are known, finding the height can be a complex task.

Area = (1/2) × base × height

Semi-perimeter of a Triangle

The perimeter of a triangle is the sum of the lengths of its three sides. If the sides of a triangle are denoted by 'a', 'b', and 'c', then its perimeter is a + b + c. The semi-perimeter is half of the perimeter and is usually denoted by 's'.

s = (a + b + c) / 2

Heron's Formula

Heron's Formula provides a direct method to calculate the area of a triangle when the lengths of all three sides are known, without needing to find the height. Let 'a', 'b', and 'c' be the lengths of the sides of a triangle, and let 's' be its semi-perimeter.

Area = √[s(s - a)(s - b)(s - c)]

Application of Heron's Formula to Quadrilaterals

To find the area of a quadrilateral using Heron's Formula, we divide the quadrilateral into two triangular parts by drawing one of its diagonals. We then calculate the area of each triangle separately using Heron's Formula (or (1/2) × base × height if it's a right-angled triangle) and add them up to find the total area of the quadrilateral. It is essential to know the lengths of all sides of both triangles, including the diagonal.

Key facts to remember

1Heron's Formula is used to find the area of a triangle when the lengths of all three sides (a, b, c) are known.
2The semi-perimeter (s) of a triangle is half its perimeter: s = (a + b + c) / 2.
3Heron's Formula states: Area = √[s(s - a)(s - b)(s - c)].
4The unit of area is always in square units (e.g., cm², m²).
5To find the area of a quadrilateral using Heron's Formula, divide it into two triangles by drawing a diagonal. Calculate the area of each triangle and sum them up.
6For a right-angled triangle, the area can also be found using the formula (1/2) × base × height, which might be simpler.

Worked examples

Example 1

Find the area of a triangle whose sides are 13 cm, 14 cm, and 15 cm.

ILet the sides of the triangle be a = 13 cm, b = 14 cm, and c = 15 cm.

IIFirst, calculate the semi-perimeter (s):

IIIs = (a + b + c) / 2

IVs = (13 + 14 + 15) / 2

Vs = 42 / 2

VIs = 21 cm

VIINow, apply Heron's Formula:

VIIIArea = √[s(s - a)(s - b)(s - c)]

9Area = √[21(21 - 13)(21 - 14)(21 - 15)]

10Area = √[21 × 8 × 7 × 6]

11Area = √[(3 × 7) × (2 × 2 × 2) × 7 × (2 × 3)]

12Area = √[2 × 2 × 2 × 2 × 3 × 3 × 7 × 7]

13Area = √[2⁴ × 3² × 7²]

14Area = 2² × 3 × 7

15Area = 4 × 3 × 7

16Area = 84 cm²

Answer

The area of the triangle is 84 cm².

Factorising the numbers under the square root into prime factors makes the calculation easier.

Example 2

A park is in the shape of a quadrilateral ABCD, where ∠C = 90°, AB = 9 m, BC = 12 m, CD = 5 m and AD = 8 m. Find the area of the park.

IThe quadrilateral ABCD can be divided into two triangles, ΔBCD and ΔABD, by drawing the diagonal BD.

II**For ΔBCD:**

IIIGiven ∠C = 90°, so ΔBCD is a right-angled triangle.

IVBase BC = 12 m, Height CD = 5 m.

VArea(ΔBCD) = (1/2) × base × height

VIArea(ΔBCD) = (1/2) × 12 × 5

VIIArea(ΔBCD) = 30 m²

VIIINow, we need the length of the diagonal BD to find the area of ΔABD. Using Pythagoras theorem in ΔBCD:

9BD² = BC² + CD²

10BD² = 12² + 5²

11BD² = 144 + 25

12BD² = 169

13BD = √169 = 13 m

14**For ΔABD:**

15The sides are a = AB = 9 m, b = AD = 8 m, c = BD = 13 m.

16Semi-perimeter (s) = (a + b + c) / 2

17s = (9 + 8 + 13) / 2

18s = 30 / 2

19s = 15 m

20Apply Heron's Formula for ΔABD:

21Area(ΔABD) = √[s(s - a)(s - b)(s - c)]

22Area(ΔABD) = √[15(15 - 9)(15 - 8)(15 - 13)]

23Area(ΔABD) = √[15 × 6 × 7 × 2]

24Area(ΔABD) = √[(3 × 5) × (2 × 3) × 7 × 2]

25Area(ΔABD) = √[2 × 2 × 3 × 3 × 5 × 7]

26Area(ΔABD) = √[2² × 3² × 5 × 7]

27Area(ΔABD) = 2 × 3 × √[5 × 7]

28Area(ΔABD) = 6√35 m²

29Total Area of the park ABCD = Area(ΔBCD) + Area(ΔABD)

30Total Area = 30 + 6√35 m²

31To get a numerical value, we can approximate √35 ≈ 5.916

32Total Area ≈ 30 + 6 × 5.916

33Total Area ≈ 30 + 35.496

34Total Area ≈ 65.496 m²

Answer

The area of the park is (30 + 6√35) m² (approximately 65.5 m²).

When one of the triangles formed by the diagonal is a right-angled triangle, its area can be calculated using (1/2) × base × height, which might be simpler than Heron's Formula.

Example 3

The perimeter of a triangular field is 300 m and its sides are in the ratio 3:5:7. Find the area of the field.

ILet the sides of the triangular field be 3x, 5x, and 7x metres.

IIThe perimeter is given as 300 m.

III3x + 5x + 7x = 300

IV15x = 300

Vx = 300 / 15

VIx = 20

VIISo, the lengths of the sides are:

VIIIa = 3x = 3 × 20 = 60 m

9b = 5x = 5 × 20 = 100 m

10c = 7x = 7 × 20 = 140 m

11Calculate the semi-perimeter (s):

12s = (a + b + c) / 2

13s = (60 + 100 + 140) / 2

14s = 300 / 2

15s = 150 m

16Now, apply Heron's Formula:

17Area = √[s(s - a)(s - b)(s - c)]

18Area = √[150(150 - 60)(150 - 100)(150 - 140)]

19Area = √[150 × 90 × 50 × 10]

20Area = √[(15 × 10) × (9 × 10) × (5 × 10) × 10]

21Area = √[15 × 9 × 5 × 10⁴]

22Area = √[(3 × 5) × (3 × 3) × 5 × 10⁴]

23Area = √[3² × 5² × 3 × 10⁴]

24Area = 3 × 5 × 10² × √3

25Area = 15 × 100 × √3

26Area = 1500√3 m²

Answer

The area of the field is 1500√3 m².

Always find the actual side lengths before calculating the semi-perimeter and applying Heron's Formula.

Common mistakes

✗**Calculation Errors**: Making mistakes while calculating the semi-perimeter or simplifying the expression under the square root.
✗**Incorrect Semi-perimeter**: Forgetting to divide the perimeter by 2 to get the semi-perimeter (s).
✗**Units**: Not writing the correct square units (e.g., m², cm²) for the area in the final answer.
✗**Simplification of Square Roots**: Not simplifying the square root completely or making errors in factorisation.
✗**Quadrilateral Application**: Trying to apply Heron's Formula directly to a quadrilateral instead of dividing it into two triangles first.

Exam tips

★**Draw a Diagram**: For problems involving quadrilaterals, always draw a neat diagram to visualise the problem and identify the triangles clearly.
★**Factorise for Simplification**: When calculating the square root, factorise the numbers under the radical into prime factors to easily find pairs and simplify the square root.
★**Check Semi-perimeter**: Double-check the calculation of the semi-perimeter (s) as an error here will lead to an incorrect final answer.
★**Show All Steps**: Present your solution step-by-step, clearly showing the calculation of 's', (s-a), (s-b), (s-c), and then the application of the formula. Remember to write units at each relevant step and in the final answer.

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