Maths · Suggested Grade 9
Class 9 Heron’s Formula: Worked Examples & Practice
You know all three sides of a triangular garden, but no height. Heron’s formula finds its area—after you first check that those sides can form a triangle.
Original explanations and practice by BananaLearn · Published 31 October 2026
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Choose Heron’s formula when three sides are known
For a triangle with side lengths a, b and c, let s = (a + b + c)/2. Heron’s formula gives area = √[s(s − a)(s − b)(s − c)]. The letter s is the semiperimeter, half of the boundary length. It is not a side and it is not the area.
You need addition, multiplication, square roots and consistent units. If a base and its perpendicular height are already known, area = ½ × base × height is often simpler. Both formulas describe the same area. Choose the formula from the information supplied rather than applying the longest method automatically.
Check whether a triangle is possible
Each pair of side lengths must add to more than the third. For positive sides arranged from smallest to largest, checking that the two smaller sides sum to more than the largest is sufficient. The lengths 4, 7 and 12 fail because 4 + 7 is only 11.
The lengths 4, 7 and 11 make a flattened, zero-area arrangement, not a non-degenerate triangle. Do not treat the zero inside the square root as proof of a valid triangle. Negative or zero side lengths also fail before the formula is used. All measurements must refer to the same unit.
Worked example: sides 13 cm, 14 cm and 15 cm
The two smaller sides sum to 27 cm, greater than 15 cm. The perimeter is 42 cm, so s = 21 cm. The four factors are 21, 21 − 13 = 8, 21 − 14 = 7 and 21 − 15 = 6. Their product is 7,056. Its square root is 84, so the area is 84 cm².
Keep the subtractions visible until you have checked them. The product inside the root has fourth-power length units; taking its square root gives square units. Writing 84 cm would report a length, not an area. A tidy calculation should include the semiperimeter and all three differences, not only the final square root.
Worked example: recover a perpendicular height
For that same 13–14–15 triangle, choose the 14 cm side as the base. Since 84 = ½ × 14 × height, the height is 168 ÷ 14 = 12 cm. The height is perpendicular to that chosen base; it need not be one of the three sides.
Changing the chosen base changes the corresponding height but not the area. With base 15 cm, height = 168 ÷ 15 = 11.2 cm. A sketch that appears to show a vertical side is not evidence that the side is an altitude. Mark a right angle only when it is given or established.
Worked example: a square root that stays exact
An equilateral triangle has side 6 cm. Then s = 9 cm, and area = √(9 × 3 × 3 × 3) = √243 = 9√3 cm². A decimal approximation is about 15.59 cm², but 9√3 is the exact value.
Do not remove the root from each factor by dividing by two. Instead, identify a square factor: 243 = 81 × 3, so √243 = 9√3. Rounding before multiplying can create avoidable error. Follow the question’s instruction about exact form or decimal places.
Worked example: a field and a cost
A triangular patch has sides 10 m, 10 m and 12 m. Its semiperimeter is 16 m, and area = √(16 × 6 × 6 × 4) = 48 m². If covering the patch costs ₹35 per square metre, the cost is 48 × 35 = ₹1,680, assuming the question excludes wastage and additional charges.
Fencing would use the perimeter 32 m instead. Read whether the task concerns the boundary or the surface before multiplying by a rate. The cost is not an official quote; the numbers are an original teaching example. A real purchase would need its actual rates and conditions.
Common errors and an efficient checking order
A frequent mistake uses the full perimeter in place of s. Another subtracts the wrong side or omits a factor. Make four labelled boxes: s, s − a, s − b and s − c. Calculate each once, multiply, then take the root. For valid positive sides, every difference should be positive.
To check understanding, ask the learner to explain why a longer perimeter does not by itself determine the area. Triangles can have equal perimeters and different shapes and areas. A parent can check the triangle condition, units and substitution even without doing the final square-root simplification. Try an invalid set as well as easy perfect-square examples.
Copy this Heron calculation frame
Copy the record into a notebook or use it to organise your own answer. Blank spaces are for your work.
| a, b, c and unit | Triangle check | s | s − a, s − b, s − c | Area with square unit |
|---|---|---|---|---|
| 13, 14, 15 cm | ___ | ___ | ___ | ___ |
| 10, 10, 12 m | ___ | ___ | ___ | ___ |
| Your example | ___ | ___ | ___ | ___ |
10 questions with explained answers
Try the questions before opening the answers. Explain the reason, then try a changed example to check your understanding.
Find s for sides 8, 9 and 11 cm.
Show answer and explanation
14 cm
(8 + 9 + 11) ÷ 2 = 14.Can 3, 5 and 9 cm form a triangle?
Show answer and explanation
No
3 + 5 < 9.Find the area for sides 5, 5 and 6 cm.
Show answer and explanation
12 cm²
s = 8; √(8 × 3 × 3 × 2) = 12.Find the area for sides 6, 8 and 10 cm.
Show answer and explanation
24 cm²
s = 12; √(12 × 6 × 4 × 2) = 24.Find the area for sides 13, 14 and 15 cm.
Show answer and explanation
84 cm²
s = 21; the product under the root is 7,056.Use that area with base 14 cm to find the height.
Show answer and explanation
12 cm
height = 2 × 84 ÷ 14.Find the exact area of an equilateral triangle of side 4 cm.
Show answer and explanation
4√3 cm²
s = 6; √(6 × 2 × 2 × 2) = √48.Is 4, 7, 11 a non-degenerate triangle?
Show answer and explanation
No
The smaller sides sum exactly to the largest; the area would be zero.A 10–10–12 m patch costs ₹35 per m² to cover. Find the cost.
Show answer and explanation
₹1,680
Area 48 m² multiplied by 35.If all side lengths double, what happens to the area?
Show answer and explanation
It becomes four times as large
Each of the four factors under the root doubles, so the root scales by √16 = 4.
Keep the ideas connected
- Keep boundary length and area separate
- Connect area to a known right triangle
- Review roots and powers
- Explore Class 9 school Maths support
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The examples, diagrams and practice are original BananaLearn work. Grade labels suggest a learning level; teaching sequences and school requirements can differ. Official sources below support the stated facts and resource links. These activities are independent resources, not official exam questions or endorsed campaign materials.
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