A triangular garden has sides measuring 8 cm, 11 cm, and 13 cm. What is the area of the garden using Heron's formula?

Correct Option: B. Solution: The semi-perimeter $$s$$ is $$\frac{8 + 11 + 13}{2} = 16$$ cm. The area is $$\sqrt{16(16-8)(16-11)(16-13)} = \sqrt{16 \times 8 \times 5 \times 3} = 44$$ cm².

A triangular park has sides measuring 120 m, 80 m, and 50 m. What is the area of the park using Heron's formula?

Correct Option: D. Solution: The semi-perimeter $$s$$ is $$\frac{120 + 80 + 50}{2} = 125$$ m. The area is $$\sqrt{125(125-120)(125-80)(125-50)} = 2400$$ m².

Heron’s Formula

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CBSE Learning Objectives – Key Concepts & Skills You Must Know

Understand and apply Heron's formula to calculate the area of a triangle given its three sides.
Calculate the area of triangles with specific side lengths and perimeters using Heron's formula.
Solve real-world problems involving triangular areas, such as calculating costs based on area.
Analyze the properties of triangles, including the relationship between side lengths and area.
Use Heron's formula to verify the area of right triangles and other types of triangles.

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter 10: Heron's Formula

10.1 Area of a Triangle - by Heron's Formula

The area of a triangle can be calculated using Heron's formula when the lengths of its sides are known.
Heron's Formula:

Area of triangle = √s(s - a)(s - b)(s - c)
where
s = (a + b + c) / 2

10.2 Summary

Heron's Formula:
- Area of triangle = √s(s - a)(s - b)(s - c)
- Where:
  - s = semi-perimeter = (a + b + c) / 2

Examples

Example 1:
Find the area of a triangle with sides 8 cm, 11 cm, and a perimeter of 32 cm.
- Solution:
  - Third side c = 32 - (8 + 11) = 13 cm
  - s = 32 / 2 = 16 cm
  - Area = √16(16 - 8)(16 - 11)(16 - 13) = √16 × 8 × 5 × 3 = 48√3 cm²
Example 2:
A triangular park ABC has sides 120 m, 80 m, and 50 m.
- Solution:
  - s = (120 + 80 + 50) / 2 = 125 m
  - Area = √125(125 - 120)(125 - 80)(125 - 50) = √125 × 5 × 45 × 75 = 3750 m²
Example 3:
The sides of a triangular plot are in the ratio of 3:5:7 and its perimeter is 300 m.
- Solution:
  - Sides are 60 m, 100 m, and 140 m.
  - s = (60 + 100 + 140) / 2 = 150 m
  - Area = √150(150 - 60)(150 - 100)(150 - 140) = √150 × 90 × 50 × 10 = 1500√3 m²

Important Notes

Heron's formula is particularly useful when the height of the triangle is not easily calculable.
The formula applies to all types of triangles, including scalene, isosceles, and equilateral.

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips

Common Pitfalls

Misapplication of Heron's Formula: Students often forget to calculate the semi-perimeter correctly before applying the formula. Ensure that you compute the semi-perimeter (s) as half the perimeter of the triangle.
Incorrect Side Lengths: When given the perimeter and two sides, students may miscalculate the third side. Always verify that the sum of the two known sides is less than the perimeter.
Units Confusion: Students sometimes forget to keep track of units when calculating area. Ensure that all measurements are in the same unit before performing calculations.

Tips for Success

Practice with Different Triangles: Familiarize yourself with various triangle types (isosceles, scalene, equilateral) and practice applying Heron's formula to each.
Draw Diagrams: Visualizing the triangle can help in understanding the relationships between the sides and can aid in avoiding mistakes.
Check Your Work: After calculating the area, double-check your calculations and ensure that the area makes sense given the side lengths.

CBSE Quiz & Practice Test – MCQs, True/False Questions with Solutions

Multiple Choice Questions

42 cm²

44 cm²

48 cm²

52 cm²

Correct Answer: B

Solution:

The semi-perimeter

s

\frac{8 + 11 + 13}{2} = 16

cm. The area is

\sqrt{16(16-8)(16-11)(16-13)} = \sqrt{16 \times 8 \times 5 \times 3} = 44

cm².

60 m

100 m

140 m

150 m

Correct Answer: C

Solution:

Let the sides be

3x

5x

, and

7x

. Then

3x + 5x + 7x = 300

, giving

15x = 300

and

x = 20

. Thus, the longest side is

7x = 140

54 m²

60 m²

72 m²

48 m²

Correct Answer: B

Solution:

The sides are 9 m, 12 m, and 15 m. The semi-perimeter

s = \frac{36}{2} = 18

. The area is

\sqrt{18(18-9)(18-12)(18-15)} = \sqrt{18 \times 9 \times 6 \times 3} = 54 \text{ m}^2

216 cm²

216√3 cm²

180 cm²

180√3 cm²

Correct Answer: A

Solution:

The semi-perimeter

s

is calculated as

s = \frac{18 + 24 + 30}{2} = 36

cm. Using Heron's formula, the area is

\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{36(36-18)(36-24)(36-30)} = \sqrt{36 \times 18 \times 12 \times 6} = 216

cm².

2000 m²

3000 m²

3750 m²

2500 m²

Correct Answer: C

Solution:

The semi-perimeter

s = \frac{120 + 80 + 50}{2} = 125

m. Using Heron's formula, the area is

\sqrt{125(125-120)(125-80)(125-50)} = \sqrt{125 \times 5 \times 45 \times 75} = 3750

m².

1500 m²

1800 m²

2000 m²

2200 m²

Correct Answer: A

Solution:

Using Heron's formula, the semi-perimeter

s

\frac{50 + 80 + 120}{2} = 125

m. The area is given by

\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{125(125-50)(125-80)(125-120)} = \sqrt{125 \times 75 \times 45 \times 5} = 1500 \text{ m}^2.

Yes, 384 m²

No, 384 m²

Yes, 480 m²

No, 480 m²

Correct Answer: A

Solution:

The sides satisfy the Pythagorean theorem: 40² = 32² + 24². Hence, it's a right triangle. The area is calculated as 1/2 * 32 * 24 = 384 m².

60 cm²

64 cm²

68 cm²

72 cm²

Correct Answer: A

Solution:

First, calculate the semi-perimeter:

s = \frac{8 + 15 + 17}{2} = 20

. Then, use Heron's formula:

\text{Area} = \sqrt{20(20-8)(20-15)(20-17)} = \sqrt{20 \times 12 \times 5 \times 3} = 60\, \text{cm}^2

60 m²

60√3 m²

60√2 m²

60√5 m²

Correct Answer: A

Solution:

The semi-perimeter

s

s = \frac{8 + 15 + 17}{2} = 20

m. Using Heron's formula, the area is

\sqrt{20(20-8)(20-15)(20-17)} = \sqrt{20 \times 12 \times 5 \times 3} = 60

m².

84 cm²

72 cm²

60 cm²

48 cm²

Correct Answer: B

Solution:

The third side is

42 - 18 - 10 = 14

cm. The semi-perimeter

s = \frac{42}{2} = 21

cm. Using Heron's formula, Area =

\sqrt{21(21-18)(21-10)(21-14)} = 72

cm².

2000 m²

3000 m²

4000 m²

5000 m²

Correct Answer: B

Solution:

Using Heron's formula, the semi-perimeter

s = \frac{50 + 80 + 120}{2} = 125

m. The area is

\sqrt{125(125-50)(125-80)(125-120)} = 3000

m².

60 m²

56 m²

64 m²

72 m²

Correct Answer: A

Solution:

The semi-perimeter

s = \frac{8 + 15 + 17}{2} = 20

. The area is

\sqrt{20(20-8)(20-15)(20-17)} = \sqrt{20 \times 12 \times 5 \times 3} = 60 \text{ m}^2

84 cm²

36 cm²

72 cm²

54 cm²

Correct Answer: B

Solution:

The third side is

42 - 18 - 10 = 14

cm. The semi-perimeter

s = \frac{42}{2} = 21

cm. Using Heron's formula, the area is

\sqrt{21(21-18)(21-10)(21-14)} = \sqrt{21 \times 3 \times 11 \times 7} = 36

cm².

3000 m²

4200 m²

1500√3 m²

4500 m²

Correct Answer: C

Solution:

The semi-perimeter

s

is calculated as

s = \frac{60 + 100 + 140}{2} = 150

m. Using Heron's formula, the area is

\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{150(150-60)(150-100)(150-140)} = \sqrt{150 \times 90 \times 50 \times 10} = 1500\sqrt{3}

m².

42 cm²

84 cm²

105 cm²

126 cm²

Correct Answer: A

Solution:

First, calculate the semi-perimeter:

s = \frac{7 + 24 + 25}{2} = 28

. Then, use Heron's formula:

\text{Area} = \sqrt{28(28-7)(28-24)(28-25)} = \sqrt{28 \times 21 \times 4 \times 3} = 42\, \text{cm}^2

₹1,500,000

₹2,250,000

₹750,000

₹1,000,000

Correct Answer: C

Solution:

Using Heron's formula, the semi-perimeter

s = 132

m. The area is

\sqrt{132(132-122)(132-22)(132-120)} = 1320

m². Rent for 3 months is

\frac{1320 \times 5000}{4} = ₹750,000

180 cm²

180√3 cm²

180√2 cm²

180√5 cm²

Correct Answer: A

Solution:

The semi-perimeter

s

s = \frac{9 + 40 + 41}{2} = 45

cm. Using Heron's formula, the area is

\sqrt{45(45-9)(45-40)(45-41)} = \sqrt{45 \times 36 \times 5 \times 4} = 180

cm².

1320 m²

2640 m²

3960 m²

5280 m²

Correct Answer: B

Solution:

The semi-perimeter

s = \frac{122 + 22 + 120}{2} = 132

m. The area is

\sqrt{132(132-122)(132-22)(132-120)} = 2640 \text{ m}^2

₹1,500,000

₹1,250,000

₹1,000,000

₹750,000

Correct Answer: B

Solution:

The semi-perimeter is 132 m. Using Heron's formula, the area is √(132(132-122)(132-22)(132-120)) = 1320 m². Rent for 3 months is (1320 * 5000 * 3/12) = ₹1,250,000.

33 m²

30 m²

36 m²

28 m²

Correct Answer: A

Solution:

Using Heron's formula, the semi-perimeter is 16 m. The area is √16(16 - 15)(16 - 11)(16 - 6) = 33 m².

₹25,000

₹50,000

₹75,000

₹100,000

Correct Answer: C

Solution:

Using Heron's formula,

s = 132

m. The area is

\sqrt{132(132-122)(132-22)(132-120)} = 1320 \text{ m}^2.

Annual rent =

1320 \times 5000 = ₹6,600,000

. Rent for 3 months =

\frac{6,600,000}{4} = ₹1,650,000

1500√3 m²

1200√3 m²

1800√3 m²

1000√3 m²

Correct Answer: A

Solution:

The sides of the triangle are 60 m, 100 m, and 140 m. Using Heron's formula, the semi-perimeter is 150 m. The area is calculated as √150(150 - 60)(150 - 100)(150 - 140) = 1500√3 m².

1500 m²

1800 m²

2000 m²

2400 m²

Correct Answer: A

Solution:

The semi-perimeter

s

is calculated as

s = \frac{50 + 80 + 120}{2} = 125

m. Using Heron's formula, the area is

\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{125(125-50)(125-80)(125-120)} = \sqrt{125 \times 75 \times 45 \times 5} = 1500

m².

3000 m²

1500√3 m²

1500√2 m²

3000√2 m²

Correct Answer: B

Solution:

The semi-perimeter

s

is calculated as

s = \frac{60 + 100 + 140}{2} = 150

m. Using Heron's formula, the area is

\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{150(150-60)(150-100)(150-140)} = 1500\sqrt{3}

m².

1500√3 m²

1200√5 m²

1800√2 m²

1350√3 m²

Correct Answer: A

Solution:

The sides of the triangle are 60 m, 100 m, and 140 m. The semi-perimeter is 150 m. Using Heron's formula, the area is √(150(150-60)(150-100)(150-140)) = 1500√3 m².

84 cm²

70 cm²

60 cm²

96 cm²

Correct Answer: D

Solution:

The semi-perimeter

s = \frac{7 + 24 + 25}{2} = 28

. The area is

\sqrt{28(28-7)(28-24)(28-25)} = \sqrt{28 \times 21 \times 4 \times 3} = 84 \text{ cm}^2

18 cm²

24 cm²

30 cm²

36 cm²

Correct Answer: A

Solution:

The base of the triangle is

30 - 2 \times 12 = 6

cm. The semi-perimeter

s = 15

cm. The area is

\sqrt{15(15-12)(15-12)(15-6)} = 18 \text{ cm}^2

1500 m²

1500√3 m²

1500√2 m²

1500√5 m²

Correct Answer: B

Solution:

The sides are 60 m, 100 m, and 140 m. Using Heron's formula,

s = 150

. The area is

\sqrt{150(150 - 60)(150 - 100)(150 - 140)} = 1500\sqrt{3}

m².

84 cm²

90 cm²

91 cm²

92 cm²

Correct Answer: C

Solution:

The semi-perimeter

s = \frac{13 + 14 + 15}{2} = 21

. The area is

\sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \times 8 \times 7 \times 6} = 84 \text{ cm}^2

384 m²

400 m²

360 m²

420 m²

Correct Answer: A

Solution:

Using Heron's formula, the semi-perimeter is 48 m. The area is √48(48 - 40)(48 - 32)(48 - 24) = 384 m².

1320 m²

1320√3 m²

1320√2 m²

1320√5 m²

Correct Answer: A

Solution:

Using Heron's formula,

s = \frac{122 + 22 + 120}{2} = 132

. The area is

\sqrt{132(132 - 122)(132 - 22)(132 - 120)} = 1320

m².

60 cm²

84 cm²

72 cm²

96 cm²

Correct Answer: C

Solution:

The semi-perimeter

s = \frac{18 + 10 + 14}{2} = 21

cm. The area is

\sqrt{21(21-18)(21-10)(21-14)} = 72

cm².

84 cm²

84√3 cm²

84√2 cm²

84√5 cm²

Correct Answer: A

Solution:

The semi-perimeter

s

s = \frac{7 + 24 + 25}{2} = 28

cm. Using Heron's formula, the area is

\sqrt{28(28-7)(28-24)(28-25)} = \sqrt{28 \times 21 \times 4 \times 3} = 84

cm².

₹150,000

₹250,000

₹125,000

₹62,500

Correct Answer: C

Solution:

First, calculate the area using Heron's formula. The semi-perimeter

s = \frac{122 + 22 + 120}{2} = 132

m. The area is

\sqrt{132(132-122)(132-22)(132-120)}

. The calculation yields an area of 1320 m². The annual rent is ₹5000 per m², so for 3 months, the rent is

\frac{5000}{4} \times 1320 = ₹125,000

1800 m²

2160 m²

216√3 m²

180√3 m²

Correct Answer: B

Solution:

Let the sides be

3x

4x

, and

5x

. The perimeter is

3x + 4x + 5x = 12x = 180

m, giving

x = 15

m. The sides are 45 m, 60 m, and 75 m. The semi-perimeter

s = \frac{180}{2} = 90

m. The area is

\sqrt{90(90-45)(90-60)(90-75)} = \sqrt{90 \times 45 \times 30 \times 15} = 2160

m².

60 m²

72 m²

84 m²

96 m²

Correct Answer: C

Solution:

First, calculate the semi-perimeter:

s = \frac{10 + 14 + 18}{2} = 21

. Then, use Heron's formula:

\text{Area} = \sqrt{21(21-10)(21-14)(21-18)} = \sqrt{21 \times 11 \times 7 \times 3} = 84\, \text{m}^2

384 m²

512 m²

600 m²

720 m²

Correct Answer: A

Solution:

The semi-perimeter

s = \frac{40 + 32 + 24}{2} = 48

m. Using Heron's formula, the area is

\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{48(48-40)(48-32)(48-24)} = \sqrt{48 \times 8 \times 16 \times 24} = 384

m².

900√3 cm²

600√3 cm²

1800√3 cm²

1200√3 cm²

Correct Answer: A

Solution:

The semi-perimeter is 90 cm. Using Heron's formula, the area is √(90(90-60)(90-60)(90-60)) = 900√3 cm².

900√3 cm²

600√3 cm²

450√3 cm²

300√3 cm²

Correct Answer: C

Solution:

For an equilateral triangle with side

a

, the perimeter is

3a = 180

cm, so

a = 60

cm. The semi-perimeter

s = 90

cm. Using Heron's formula, Area =

\sqrt{90(90-60)(90-60)(90-60)} = 450\sqrt{3}

cm².

1800 cm²

3600 cm²

2700 cm²

5400 cm²

Correct Answer: B

Solution:

Let the sides be

12x

17x

, and

25x

. Then

12x + 17x + 25x = 540

, giving

x = 9

. The sides are 108 cm, 153 cm, and 225 cm. The semi-perimeter

s = 243

cm. The area is

\sqrt{243(243-108)(243-153)(243-225)} = 3600 \text{ cm}^2

900√3 cm²

600√3 cm²

300√3 cm²

150√3 cm²

Correct Answer: A

Solution:

Each side of the equilateral triangle is

\frac{180}{3} = 60

cm. The semi-perimeter

s = 90

cm. The area is

\sqrt{90(90-60)(90-60)(90-60)} = 900\sqrt{3} \text{ cm}^2

36 cm²

48 cm²

54 cm²

60 cm²

Correct Answer: B

Solution:

The third side is 14 cm. The semi-perimeter

s = 21

cm. The area is

\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{21(21-18)(21-10)(21-14)} = 48

cm².

54 m²

60 m²

72 m²

90 m²

Correct Answer: A

Solution:

First, calculate the semi-perimeter:

s = \frac{9 + 12 + 15}{2} = 18

. Then, use Heron's formula:

\text{Area} = \sqrt{18(18-9)(18-12)(18-15)} = \sqrt{18 \times 9 \times 6 \times 3} = 54\, \text{m}^2

900√3 cm²

600√3 cm²

300√3 cm²

450√3 cm²

Correct Answer: A

Solution:

Each side of the equilateral triangle is

\frac{180}{3} = 60

cm. The semi-perimeter

s = \frac{180}{2} = 90

cm. Using Heron's formula, the area is

\sqrt{90(90-60)(90-60)(90-60)} = \sqrt{90 \times 30 \times 30 \times 30} = 900\sqrt{3}

cm².

384 m²

512 m²

256 m²

128 m²

Correct Answer: A

Solution:

Using Heron's formula, the semi-perimeter

s = \frac{40 + 32 + 24}{2} = 48

m. The area is given by

\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{48 \times 8 \times 24 \times 16} = 384 \text{ m}^2

84 cm²

72 cm²

60 cm²

96 cm²

Correct Answer: A

Solution:

Using Heron's formula, the semi-perimeter is 21 cm. The area is √21(21 - 18)(21 - 10)(21 - 14) = 84 cm².

900 m²

1500 m²

1800 m²

2100 m²

Correct Answer: B

Solution:

Let the sides be

3x

5x

, and

7x

. The perimeter equation

3x + 5x + 7x = 300

gives

x = 20

. So, the sides are 60 m, 100 m, and 140 m. The semi-perimeter

s = 150

m. Using Heron's formula, the area is

\sqrt{150(150-60)(150-100)(150-140)} = 1500 \text{ m}^2.

24 cm²

30 cm²

36 cm²

40 cm²

Correct Answer: B

Solution:

The third side is 6 cm. The semi-perimeter is 15 cm. Using Heron's formula, the area is √(15(15-12)(15-12)(15-6)) = 30 cm².

1500 m²

1800 m²

2000 m²

2400 m²

Correct Answer: D

Solution:

The semi-perimeter

s

\frac{120 + 80 + 50}{2} = 125

m. The area is

\sqrt{125(125-120)(125-80)(125-50)} = 2400

m².

1500 m²

3000 m²

3750 m²

4500 m²

Correct Answer: C

Solution:

The sides are 60 m, 100 m, and 140 m. The semi-perimeter

s = 150

m. The area is

\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{150(150-60)(150-100)(150-140)} = 3750

m².

44 cm²

44√3 cm²

44√2 cm²

44√5 cm²

Correct Answer: A

Solution:

Using Heron's formula,

s = \frac{8 + 11 + 13}{2} = 16

. The area is

\sqrt{16(16 - 8)(16 - 11)(16 - 13)} = 44

cm².

36 cm²

48 cm²

54 cm²

60 cm²

Correct Answer: A

Solution:

The third side

c = 42 - (18 + 10) = 14

cm. The semi-perimeter

s = 21

cm. Using Heron's formula, the area is

\sqrt{21(21-18)(21-10)(21-14)} = 36 \text{ cm}^2.

54 cm²

45 cm²

72 cm²

36 cm²

Correct Answer: A

Solution:

The semi-perimeter

s = \frac{9 + 12 + 15}{2} = 18

. The area is

\sqrt{18(18-9)(18-12)(18-15)} = \sqrt{18 \times 9 \times 6 \times 3} = 54 \text{ cm}^2

384 m²

400 m²

350 m²

360 m²

Correct Answer: A

Solution:

Using Heron's formula, the semi-perimeter

s = \frac{40 + 32 + 24}{2} = 48

m. The area is

\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{48(48-40)(48-32)(48-24)} = 384

m².

6 cm

8 cm

10 cm

12 cm

Correct Answer: A

Solution:

Let the base be

b

. Then

12 + 12 + b = 30

, giving

b = 30 - 24 = 6

cm.

900√3 cm²

1200√3 cm²

1350√3 cm²

1500√3 cm²

Correct Answer: A

Solution:

For an equilateral triangle,

a = \frac{180}{3} = 60

cm. The semi-perimeter

s

is 90 cm. The area is

\sqrt{90(90-60)(90-60)(90-60)} = 900\sqrt{3}

cm².

True or False

Correct Answer: True

Solution:

The semi-perimeter is calculated as

s = \frac{18 + 10 + 14}{2} = 21

cm.

Correct Answer: True

Solution:

The semi-perimeter is calculated as

\frac{8 + 11 + 13}{2} = 16

cm.

Correct Answer: False

Solution:

The base of the triangle is 30 cm - 2(12 cm) = 6 cm. Using Heron's formula, the area is not 9 cm².

Correct Answer: True

Solution:

Heron's formula is specifically designed to calculate the area of a triangle using the lengths of its sides, which is useful when the height is not known.

Correct Answer: True

Solution:

Heron's formula can be used to find the area of any triangle given its side lengths. For sides 120 m, 80 m, and 50 m, the semi-perimeter

s = 125

m, and the area can be calculated using the formula.

Correct Answer: True

Solution:

Heron was known for his extensive works in applied mathematics and is considered an encyclopedic writer.

Correct Answer: True

Solution:

Heron's formula is used to find the area of a triangle when the side lengths are known, which can be derived from the given ratio and perimeter.

Correct Answer: True

Solution:

Using Heron's formula, the area is calculated as 384 m² for the given side lengths.

Correct Answer: True

Solution:

The semi-perimeter is calculated as

s = \frac{18 + 10 + 14}{2} = 21

cm.

Correct Answer: True

Solution:

The semi-perimeter is calculated as half the sum of the sides:

s = \frac{40 + 32 + 24}{2} = 48

Correct Answer: True

Solution:

Heron's formula is specifically designed to calculate the area of a triangle using the lengths of its three sides.

Correct Answer: True

Solution:

The semi-perimeter is calculated as half the perimeter:

s = \frac{10 + 15 + 25}{2} = 25

cm.

Correct Answer: True

Solution:

Using Heron's formula, the semi-perimeter is 48 m, and the area is calculated as 384 m².

Correct Answer: True

Solution:

Given the ratio 3:5:7 and perimeter 300 m, the sides are

3x, 5x, 7x

where

15x = 300

, so

x = 20

. Thus, the sides are 60 m, 100 m, and 140 m.

Correct Answer: False

Solution:

Heron's formula can be used for any triangle, including equilateral triangles, as it only requires the side lengths.

Correct Answer: True

Solution:

Heron's formula can be used for any triangle, including equilateral triangles, as it only requires the lengths of the sides.

Correct Answer: True

Solution:

The perimeter is the sum of the sides:

120 + 80 + 50 = 250

Correct Answer: True

Solution:

Heron was an encyclopedic writer in applied mathematics, known for his works on mensuration.

Correct Answer: True

Solution:

The base is calculated as 30 cm - 2(12 cm) = 6 cm.

Correct Answer: False

Solution:

Using Heron's formula, the area is calculated as

\sqrt{16(16-8)(16-11)(16-13)} = 48 \text{ cm}^2

Correct Answer: True

Solution:

Given the side lengths can be determined from the ratio and perimeter, Heron's formula can be used to find the area.

Correct Answer: True

Solution:

Heron's formula can be applied to any triangle when the lengths of all three sides are known, regardless of their specific measurements.

Correct Answer: True

Solution:

The semi-perimeter

s

is calculated as

s = \frac{a + b + c}{2} = \frac{40 + 24 + 32}{2} = 48

Correct Answer: True

Solution:

The sides are 3x, 5x, and 7x. Solving

3x + 5x + 7x = 300

gives

x = 20

, so the longest side is

7 \times 20 = 140

Heron’s Formula

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CBSE Learning Objectives – Key Concepts & Skills You Must Know

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Chapter 10: Heron's Formula

10.1 Area of a Triangle - by Heron's Formula

10.2 Summary

Examples

Important Notes

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips

Common Pitfalls

Tips for Success

Multiple Choice Questions

Q1.A triangular garden has sides measuring 8 cm, 11 cm, and 13 cm. What is the area of the garden using Heron's formula?

Correct Answer: B

Q2.A triangular plot has sides in the ratio 3:5:7 and its perimeter is 300 m. What is the length of the longest side?

Correct Answer: C

Q3.A triangular garden has sides in the ratio 3:4:5 and a perimeter of 36 m. What is the area of the garden?

Correct Answer: B

Q4.Consider a triangle with sides 18 cm, 24 cm, and 30 cm. Using Heron's formula, calculate the area of the triangle.

Correct Answer: A

Q5.A triangular park has sides 120 m, 80 m, and 50 m. If a gardener needs to plant grass inside, what is the area available for planting?

Correct Answer: C

Q6.A triangular garden has sides measuring 50 m, 80 m, and 120 m. If a gardener needs to plant grass inside the garden, how much area does she need to cover?

Correct Answer: A

Q7.A triangular park has sides 40 m, 32 m, and 24 m. Verify if the park forms a right triangle and calculate its area.

Correct Answer: A

Q8.The sides of a triangular plot are 8 cm, 15 cm, and 17 cm. Calculate the area of the plot using Heron's formula.

Correct Answer: A

Q9.A triangular plot of land has sides measuring 8 m, 15 m, and 17 m. Calculate its area using Heron's formula.

Correct Answer: A

Q10.Find the area of a triangle where two sides are 18 cm and 10 cm, and the perimeter is 42 cm.

Correct Answer: B

Q11.A triangular garden has sides measuring 50 m, 80 m, and 120 m. What is the area of the garden using Heron's formula?

Correct Answer: B

Q12.A triangular plot has sides measuring 8 m, 15 m, and 17 m. What is the area of the triangle using Heron's formula?

Correct Answer: A

Q13.A triangle has sides 18 cm, 10 cm, and a perimeter of 42 cm. Calculate its area using Heron's formula.

Correct Answer: B

Q14.A triangular garden has sides measuring 60 m, 100 m, and 140 m. Using Heron's formula, calculate the area of the garden.

Correct Answer: C

Q15.A triangular sign board has sides of lengths 7 cm, 24 cm, and 25 cm. What is the area of the sign board using Heron's formula?

Correct Answer: A

Q16.The sides of a triangle are 122 m, 22 m, and 120 m. If the area is used for advertisements at ₹5000 per m² per year, how much rent is paid for 3 months?

Correct Answer: C

Q17.For a triangle with sides 9 cm, 40 cm, and 41 cm, find the area using Heron's formula.

Correct Answer: A

Q18.A triangular side wall of a flyover is used for advertisements. The sides of the wall are 122 m, 22 m, and 120 m. What is the area of this triangle?

Correct Answer: B

Q19.A company hires a triangular wall for advertisement. The sides of the wall are 122 m, 22 m, and 120 m. If the rent is ₹5000 per m² per year, how much does the company pay for 3 months?

Correct Answer: B

Q20.A triangular side wall of a park has sides 15 m, 11 m, and 6 m. What is the area of the wall?

Correct Answer: A

Q21.A triangular side wall of a flyover used for advertisements has sides 122 m, 22 m, and 120 m. Calculate the rent paid for 3 months if the rent is ₹5000 per m² per year.

Correct Answer: C

Q22.A triangular plot has sides in the ratio 3:5:7 and its perimeter is 300 m. What is the area of the plot?

Correct Answer: A

Q23.A triangular garden has sides measuring 50 m, 80 m, and 120 m. If a gardener needs to plant grass inside, what is the area available for planting using Heron's formula?

Correct Answer: A

Q24.A triangular plot has sides measuring 60 m, 100 m, and 140 m. Using Heron's formula, what is the area of the plot?

Correct Answer: B

Q25.A triangular plot has sides in the ratio 3:5:7 and a perimeter of 300 m. Calculate the area of the plot using Heron's formula.

Correct Answer: A

Q26.Find the area of a triangle with sides 7 cm, 24 cm, and 25 cm using Heron's formula.

Correct Answer: D

Q27.An isosceles triangle has a perimeter of 30 cm, with each of the equal sides measuring 12 cm. What is the area of the triangle?

Correct Answer: A

Q28.The sides of a triangle are in the ratio 3:5:7 and its perimeter is 300 m. What is the area of the triangle?

Correct Answer: B

Q29.Calculate the area of a triangle with sides 13 cm, 14 cm, and 15 cm using Heron's formula.

Correct Answer: C

Q30.A triangular park has sides measuring 40 m, 32 m, and 24 m. What is the area of the park?

Correct Answer: A

Q31.A triangular plot has sides measuring 122 m, 22 m, and 120 m. Using Heron's formula, what is the area of the plot?

Correct Answer: A

Q32.If the sides of a triangle are 18 cm, 10 cm, and 14 cm, what is the area of the triangle using Heron's formula?

Correct Answer: C

Q33.A triangle has sides 7 cm, 24 cm, and 25 cm. Determine its area using Heron's formula.

Correct Answer: A