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Question:1
Find the area of a triangle whose sides are respectively 150 cm, 120 cm and 200 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a =150 cm
b=120 cm
c =200 cm
Here we will calculate s,
So the area of the triangle is:
Question:2
Find the area of a triangle whose sides are 9 cm, 12 cm and 15 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a = 9 cm, b = 12 cm, c = 15 cm
Here we will calculate s,
So the area of the triangle is:
Question:3
Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a = 18 cm
b = 10 cm, and perimeter = 42 cm
We know that perimeter = 2s, 
So 2s = 42
Therefore s = 21 cm
Page 2


Question:1
Find the area of a triangle whose sides are respectively 150 cm, 120 cm and 200 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a =150 cm
b=120 cm
c =200 cm
Here we will calculate s,
So the area of the triangle is:
Question:2
Find the area of a triangle whose sides are 9 cm, 12 cm and 15 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a = 9 cm, b = 12 cm, c = 15 cm
Here we will calculate s,
So the area of the triangle is:
Question:3
Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a = 18 cm
b = 10 cm, and perimeter = 42 cm
We know that perimeter = 2s, 
So 2s = 42
Therefore s = 21 cm
We know that , so
So the area of the triangle is:
Question:4
In a ?ABC, AB = 15 cm, BC = 13 cm and AC = 14 cm. Find the area of ?ABC and hence its altitude on AC.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by ‘Area’, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
AB = 15 cm, BC = 13 cm, AC = 14 cm
Here we will calculate s,
So the area of the triangle is:
Now draw the altitude from point B on AC which intersects it at point D.BD is the required altitude. So if you draw the figure, you will see, 
Here . So,
Question:5
The perimeter of a triangular field is 540 m and its sides are in the ratio 25 : 17 : 12. Find the area of the triangle.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle. If we denote area of the triangle by A, then the
area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given,  and 
Here,
Using these data we will find the sides of the triangle. Suppose the sides of the triangle are as follows,
Since , so
Page 3


Question:1
Find the area of a triangle whose sides are respectively 150 cm, 120 cm and 200 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a =150 cm
b=120 cm
c =200 cm
Here we will calculate s,
So the area of the triangle is:
Question:2
Find the area of a triangle whose sides are 9 cm, 12 cm and 15 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a = 9 cm, b = 12 cm, c = 15 cm
Here we will calculate s,
So the area of the triangle is:
Question:3
Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a = 18 cm
b = 10 cm, and perimeter = 42 cm
We know that perimeter = 2s, 
So 2s = 42
Therefore s = 21 cm
We know that , so
So the area of the triangle is:
Question:4
In a ?ABC, AB = 15 cm, BC = 13 cm and AC = 14 cm. Find the area of ?ABC and hence its altitude on AC.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by ‘Area’, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
AB = 15 cm, BC = 13 cm, AC = 14 cm
Here we will calculate s,
So the area of the triangle is:
Now draw the altitude from point B on AC which intersects it at point D.BD is the required altitude. So if you draw the figure, you will see, 
Here . So,
Question:5
The perimeter of a triangular field is 540 m and its sides are in the ratio 25 : 17 : 12. Find the area of the triangle.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle. If we denote area of the triangle by A, then the
area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given,  and 
Here,
Using these data we will find the sides of the triangle. Suppose the sides of the triangle are as follows,
Since , so
Now we know each side that is,
Now we know all the sides. So we can use Heron’s formula.
The area of the triangle is;
Question:6
The perimeter of a triangle is 300 m. If its sides are in the ratio 3 : 5 : 7. Find the area of the triangle.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle. If we denote area of the triangle by A, then the
area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given, and 
Here,
Using these data we will find the sides of the triangle. Suppose the sides of the triangle are as follows,
Since , so
Now we know each side that is,
Now we know all the sides. So we can use Heron’s formula.
The area of the triangle is;
A = v s(s -a)(s -b)(s -c) = v 150(150 -60)(150 -100)(150 -140) = v 150(90)(50)(10) = 100v 15 ×9 ×5 = 100v 5 ×3 ×3 ×3 ×5 = 100 ×3 ×5v 3 = 1500v 3 m
2
Question:7
The perimeter of a triangular field is 240 dm. If two of its sides are 78 cm and 50 dm, find the length of the perpendicular on the side of length 50 dm from the opposite vertex.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle. If we denote area of the triangle by A, then the
area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where,  
We are given two sides of the triangle and .
That is a = 78 dm, b = 50 dm
We will find third side c and then the area of the triangle using Heron’s formula.
Page 4


Question:1
Find the area of a triangle whose sides are respectively 150 cm, 120 cm and 200 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a =150 cm
b=120 cm
c =200 cm
Here we will calculate s,
So the area of the triangle is:
Question:2
Find the area of a triangle whose sides are 9 cm, 12 cm and 15 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a = 9 cm, b = 12 cm, c = 15 cm
Here we will calculate s,
So the area of the triangle is:
Question:3
Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a = 18 cm
b = 10 cm, and perimeter = 42 cm
We know that perimeter = 2s, 
So 2s = 42
Therefore s = 21 cm
We know that , so
So the area of the triangle is:
Question:4
In a ?ABC, AB = 15 cm, BC = 13 cm and AC = 14 cm. Find the area of ?ABC and hence its altitude on AC.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by ‘Area’, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
AB = 15 cm, BC = 13 cm, AC = 14 cm
Here we will calculate s,
So the area of the triangle is:
Now draw the altitude from point B on AC which intersects it at point D.BD is the required altitude. So if you draw the figure, you will see, 
Here . So,
Question:5
The perimeter of a triangular field is 540 m and its sides are in the ratio 25 : 17 : 12. Find the area of the triangle.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle. If we denote area of the triangle by A, then the
area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given,  and 
Here,
Using these data we will find the sides of the triangle. Suppose the sides of the triangle are as follows,
Since , so
Now we know each side that is,
Now we know all the sides. So we can use Heron’s formula.
The area of the triangle is;
Question:6
The perimeter of a triangle is 300 m. If its sides are in the ratio 3 : 5 : 7. Find the area of the triangle.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle. If we denote area of the triangle by A, then the
area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given, and 
Here,
Using these data we will find the sides of the triangle. Suppose the sides of the triangle are as follows,
Since , so
Now we know each side that is,
Now we know all the sides. So we can use Heron’s formula.
The area of the triangle is;
A = v s(s -a)(s -b)(s -c) = v 150(150 -60)(150 -100)(150 -140) = v 150(90)(50)(10) = 100v 15 ×9 ×5 = 100v 5 ×3 ×3 ×3 ×5 = 100 ×3 ×5v 3 = 1500v 3 m
2
Question:7
The perimeter of a triangular field is 240 dm. If two of its sides are 78 cm and 50 dm, find the length of the perpendicular on the side of length 50 dm from the opposite vertex.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle. If we denote area of the triangle by A, then the
area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where,  
We are given two sides of the triangle and .
That is a = 78 dm, b = 50 dm
We will find third side c and then the area of the triangle using Heron’s formula.
Now,
Use Heron’s formula to find out the area of the triangle. That is 
  
Consider the triangle ?PQR in which 
PQ=50 dm, PR=78 dm, QR=120 dm
Where RD is the desired perpendicular length
Now from the figure we have
Question:8
A triangle has sides 35 cm, 54 cm and 61 cm long. Find its area. Also, find the smallest of its altitudes.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given: a = 35 cm; b = 54 cm; c = 61 cm
The area of the triangle is:
Suppose the triangle is ?PQR and focus on the triangle given below,
In which PD1, QD2 and RD3 are three altitudes
Where PQ=35 cm, QR=54 cm, PR=61 cm
We will calculate each altitude one by one to find the smallest one.
Case 1 
In case of ?PQR:
Page 5


Question:1
Find the area of a triangle whose sides are respectively 150 cm, 120 cm and 200 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a =150 cm
b=120 cm
c =200 cm
Here we will calculate s,
So the area of the triangle is:
Question:2
Find the area of a triangle whose sides are 9 cm, 12 cm and 15 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a = 9 cm, b = 12 cm, c = 15 cm
Here we will calculate s,
So the area of the triangle is:
Question:3
Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
a = 18 cm
b = 10 cm, and perimeter = 42 cm
We know that perimeter = 2s, 
So 2s = 42
Therefore s = 21 cm
We know that , so
So the area of the triangle is:
Question:4
In a ?ABC, AB = 15 cm, BC = 13 cm and AC = 14 cm. Find the area of ?ABC and hence its altitude on AC.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by ‘Area’, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given:
AB = 15 cm, BC = 13 cm, AC = 14 cm
Here we will calculate s,
So the area of the triangle is:
Now draw the altitude from point B on AC which intersects it at point D.BD is the required altitude. So if you draw the figure, you will see, 
Here . So,
Question:5
The perimeter of a triangular field is 540 m and its sides are in the ratio 25 : 17 : 12. Find the area of the triangle.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle. If we denote area of the triangle by A, then the
area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given,  and 
Here,
Using these data we will find the sides of the triangle. Suppose the sides of the triangle are as follows,
Since , so
Now we know each side that is,
Now we know all the sides. So we can use Heron’s formula.
The area of the triangle is;
Question:6
The perimeter of a triangle is 300 m. If its sides are in the ratio 3 : 5 : 7. Find the area of the triangle.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle. If we denote area of the triangle by A, then the
area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given, and 
Here,
Using these data we will find the sides of the triangle. Suppose the sides of the triangle are as follows,
Since , so
Now we know each side that is,
Now we know all the sides. So we can use Heron’s formula.
The area of the triangle is;
A = v s(s -a)(s -b)(s -c) = v 150(150 -60)(150 -100)(150 -140) = v 150(90)(50)(10) = 100v 15 ×9 ×5 = 100v 5 ×3 ×3 ×3 ×5 = 100 ×3 ×5v 3 = 1500v 3 m
2
Question:7
The perimeter of a triangular field is 240 dm. If two of its sides are 78 cm and 50 dm, find the length of the perpendicular on the side of length 50 dm from the opposite vertex.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle. If we denote area of the triangle by A, then the
area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where,  
We are given two sides of the triangle and .
That is a = 78 dm, b = 50 dm
We will find third side c and then the area of the triangle using Heron’s formula.
Now,
Use Heron’s formula to find out the area of the triangle. That is 
  
Consider the triangle ?PQR in which 
PQ=50 dm, PR=78 dm, QR=120 dm
Where RD is the desired perpendicular length
Now from the figure we have
Question:8
A triangle has sides 35 cm, 54 cm and 61 cm long. Find its area. Also, find the smallest of its altitudes.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given: a = 35 cm; b = 54 cm; c = 61 cm
The area of the triangle is:
Suppose the triangle is ?PQR and focus on the triangle given below,
In which PD1, QD2 and RD3 are three altitudes
Where PQ=35 cm, QR=54 cm, PR=61 cm
We will calculate each altitude one by one to find the smallest one.
Case 1 
In case of ?PQR:
Case 2
Case 3
The smallest altitude is QD2.
The smallest altitude is the one which is drawn on the side of length 61 cm from apposite vertex.
Question:9
The lengths of the sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 144 cm. Find the area of the triangle and the height corresponding to the longest side.
Solution:
Whenever we are given the measurement of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.
If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by;
Where, 
We are given, and 
Here,
Using these data we will find the sides of the triangle. Suppose the sides of the triangle are as follows,
Since 2s=144, so
Now we know each side that is,
Now we know all the sides. So we can use Heron’s formula.
The area of the triangle is;
We are asked to fin out the height corresponding to the longest side of the given triangle. The longest side is c and supposes the corresponding height is H then,
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FAQs on Heron`s Formula RD Sharma Solutions - Mathematics (Maths) Class 9

1. What is Heron's formula?
Ans. Heron's formula is a mathematical formula used to find the area of a triangle when the lengths of its sides are known. It is named after Hero of Alexandria, a Greek mathematician who derived this formula.
2. How is Heron's formula derived?
Ans. Heron's formula is derived using the concept of semiperimeter of a triangle. The semiperimeter is calculated by adding the lengths of all three sides of the triangle and dividing it by 2. Then, using the semiperimeter, the area of the triangle is calculated using the formula A = √(s(s-a)(s-b)(s-c)), where A is the area, s is the semiperimeter, and a, b, and c are the lengths of the sides of the triangle.
3. Can Heron's formula be used for all types of triangles?
Ans. Yes, Heron's formula can be used to find the area of any type of triangle, whether it is equilateral, scalene, or isosceles. It is a general formula that works for all triangles, as long as the lengths of the sides are known.
4. How is Heron's formula helpful in real-life applications?
Ans. Heron's formula is used in various real-life applications, especially in fields like architecture, engineering, and surveying. It helps in calculating the area of irregularly shaped land or fields, determining the amount of materials required for construction, and in designing structures like bridges and buildings.
5. Are there any limitations or special cases when using Heron's formula?
Ans. Yes, there are a few limitations and special cases to consider when using Heron's formula. One limitation is that the lengths of the sides must form a valid triangle, i.e., the sum of any two sides must be greater than the third side. Additionally, when the lengths of the sides are very large, the calculations involved in the formula may result in rounding errors. Finally, if the triangle is degenerate (having collinear vertices), Heron's formula will not yield a valid area.
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