Practice Problems on Heron’s Formula

Heron’s formula is also known as Hero’s formula. It calculates the area of triangles or quadrilaterals based on the lengths of their sides. It does not consider the angles of the shapes, only their side lengths. The formula includes the semi-perimeter, which is symbolized by “s”, which is half the perimeter of the shape. This concept can be extended to find the area of quadrilaterals too.

Heron’s Formula for Area of Triangle

As we know Heron’s formula is used to calculate the area of a triangle,

Heron’s Formula for Area of Triangle

Therefore, Heron’s Formula is

Area of Triangle ABC = √s(s-a)(s-b)(s-c)

where,

• s = Perimeter/2 = (a + b + c)/2

Area of Triangle Formulas

Formula to calculate different types of triangles is different. In the table given below, we have summarized the formulas for all the Types of Triangles.

Types of Triangles	Area Formulas
Equilateral Triangle	[Tex]A = \frac{\sqrt{3}}{4} \times a^2[/Tex] where a is the length of one side of the equilateral triangle.
Scalene Triangle	[Tex]A = \sqrt{s(s-a)(s-b)(s-c)}[/Tex]
Isosceles Triangle	[Tex]A = \sqrt{s(s-a)(s-b)(s-b)} [/Tex] where: a is the length of Unequal Side b is the length of Equal Sides

Practice Problems on Heron’s Formula

Question 1. Find the area of a quadrilateral with side lengths of 6 cm, 8 cm, 10 cm, and 12 cm using Heron’s Formula (considering it as two triangles).

Question 2. A triangle has sides of lengths 3x, 4x, and 5x. Find the area of the triangle using Heron’s Formula in terms of x.

Question 3. Calculate the area of a triangle with sides of lengths 13 cm, 14 cm, and 15 cm using Heron’s Formula.

Question 4. An equilateral triangle has an area of 25√3 square units. Find the length of one of its sides using Heron’s Formula.

Question 5. If the sides of a triangle measure 17 cm, 25 cm, and 28 cm, what is its area using Heron’s Formula?

Practice Problems on Heron’s Formula with Solution

Problem 1: In a triangle with side lengths of 5 cm, 12 cm, and 13 cm, find the area using Heron’s Formula.

Solution:

Given,

• a = 5cm
• b= 12 cm
• c = 13 cm

According to Heron’s Formula: [Tex]A = \sqrt{s(s-a)(s-b)(s-c)}[/Tex]

s = (a+b+c)/2

s = (5+12+13)/2

s = 30/2 = 15

[Tex]A = \sqrt{s(s-a)(s-b)(s-c)}[/Tex]

A = [Tex]\sqrt{15(15-5)(15-12)(15-13)}[/Tex]

A = [Tex]\sqrt{15 \times 10 \times 3 \times 2}[/Tex]

A = √900

A = 30 cm²

∴ Area of Triangle is 30 cm².

Problem 2: Calculate the area of an equilateral triangle with a side length of 10 meters using Heron’s Formula.

Solution:

Given,

a = b = c = 10 cm For Equilateral Triangle

s = (a+b+c)/2

s = (10+10+10)/2

s = 30/2 = 15 meters

According to Heron’s Formula: [Tex]A = \sqrt{s(s-a)(s-b)(s-c)}[/Tex]

Putting s = 15 in the formula we get,

A = [Tex]\sqrt{15(15-10)(15-10)(15-10)} [/Tex]

A = [Tex]\sqrt{15 \times 5 \times 5 \times 5}[/Tex]

A = √1875 ≈ 43.3 meters²

∴ Area of Equilateral Triangle with a side length of 10 meters is approximately 43.3 m²

Problem 3: A scalene triangle has side lengths of 7 cm, 10 cm, and 12 cm. Find its area using Heron’s Formula.

Solution:

Given,

• a = 7 cm
• b = 10 cm
• c = 12 cm

We know s = (a+b+c)/2

s = (7+10+12)/2

s = 29/2 = 14.5 cm

Putting value of s in the formula we get,

[Tex]A = \sqrt{s(s-a)(s-b)(s-c)}[/Tex]

A = [Tex]\sqrt{14.5(14.5-7)(14.5-10)(14.5-12)}[/Tex]

A = \sqrt{14.5 \times 7.5 \times 4.5 \times 2.5}

A = √1912.8125 cm²

A ≈ 43.74 cm²

∴ Area of Scalene Triangle with side lengths 7 cm, 10 cm, and 12 cm is approximately 43.74 cm².

Problem 4: Find the area of triangle if its two sides are 8 cm and 5 cm resecpectivelu and its perimeter is 20 cm.

Solution:

Let a, b and c be the three sides of the triangle.

a = 8 cm

b = 5 cm

c = ?

Perimeter = 20 cm

Perimeter equals to the sum of the length of three sides of a triangle.

Perimeter = (a + b + c)

⇒ 20 = 8 + 5 + c

⇒ c = 20 – 13

⇒ c = 7 cm

s = perimeter / 2

⇒ s = 20 / 2

⇒ s = 10 cm

As, a = 8 cm, b = 5 cm, c = 7 cm, s = 10 cm

Thus, A = √(s(s – a)(s – a)(s – a)

⇒ A = √(10(10 – 8)(10 – 5)(10 – 7)

⇒ A = 17.32 cm²

Problem 5: Find area of an equilateral triangle with a side of 9 cm.

Solution:

Given,

Side = 9 cm

Area of Equilateral Triangle = √3 / 4 × a₂

⇒ Area of Equilateral Triangle = √3 / 4 × (9)²

⇒ Area of Equilateral Triangle = 20.25 √3 cm²

Frequently Asked Questions

What is a Triangle?

A triangle is the simplest polygon with three sides.

What does S stand for in Heron’s formula?

S in Heron’s Formula stands for semi-perimeter and is calculated as: S = (a + b + c)/2

Is Heron’s formula accurate?

Yes, in general Heron’s formula is accurate but in case of very thin triangles, calculating area of triangle using Heron’s formula may not give accurate results.