Heron's Formula MCQs

Mathematicians refer to Heron's formula as Heron's formula because Heron, a professor of mathematics in Alexandria, Egypt, was born approximately in 10 AD. A triangle's area can be calculated using Heron's formula. The area of a triangle can generally be found using formulas. However, those formulas may not be appropriate for all types of triangles. The following formula can be used to calculate the area of a triangle of any type by using the basic formula Heron provided:

Area of a triangle = √s (s-a) (s-b) (s-c)

The sides of a triangle are a, b, and c, and s refers to half the perimeter of a triangle, which is calculated as follows:

s = a+ b + c /2

Where the height of a triangle cannot be determined, this formula can be very useful.

The Applications of Heron's Formula

Heron's formula is used in two major ways:

• If the length of three sides of a triangle is given, find the area of the different types of triangles.

• Finding the area of a quadrilateral (given the lengths of the three sides).

What is Heron's Formula for Finding the Area?

Using Heron's formula, we can find the area of a triangle as follows:

• Find the perimeter of the triangle given

• Next, calculate the semi-perimeter of the given triangle; S = (a+b+c)/2

• Now use Heron’s formula to find the area of a triangle (√(s(s – a)(s – b)(s – c)))

To conclude, represent the area accurately with square units (such as m2, cm2, in2, etc.).

Using Heron's Formula to Find Equilateral Triangles

Equilateral triangles have equal sides, as we all know. In order to find the area of the equilateral triangle, let us first find the semi-perimeter of the triangle:

s = (a+a+a)/2

s = 3a/2

where, a is the length of the side.

As per the Heron's formula, we know;

Area=√s(s−a)(s−b)(s−c)

Since, a = b = c

Therefore, 

A = √[s(s-a)3] is the required formula.

Heron's Formula for the Isosceles Triangle

Triangles that have two equal sides and congruent angles are known as isosceles triangles. We can derive the following formula from the heron's formula to calculate the area of an isosceles triangle: 

Consider a as the length of the congruent sides and b as the length of the base.

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

s = (2a + b)/2

Using the heron’s formula for a triangle,

Area of triangle = √[s(s – a)(s – b)(s – c)]

When the sides of an isosceles triangle are substituted,

Area of triangle = √[s(s – a)(s – a)(s – b)]

= √[s(s – a)^2(s – b)]

Or

= (s – a)√[s(s – b)]

This is the formula to find the area for a given isosceles triangle.

Heron's Formula for the Scalene Triangle

Using the original Heron's formula, we can determine the area of a scalene triangle because it has all three unequal sides.

Area of scalene triangle = √s(s-a)(s-b)(s-c),

In this equation, s = (a + b + c)/2

The Heron's Formula For Quadrilaterals

This page will teach you Heron's formula for finding the area of a quadrilateral.

If ABCD is a quadrilateral, where AB||CD and AC & BD are the diagonals.

(Image will be uploaded soon)

AC divides the quad.ABCD into two triangles ADC and ABC.

Now we have two triangles here.

Area of quadrilateral, ABCD = Area of ∆ADC + Area of ∆ABC

The area of a quadrilateral can be found by using Heron's formula if we know the lengths of all four sides and the diagonal AC.

Then, we will use Heron's formula to calculate the area of ∆ADC and area of ∆ABC and then add them together to arrive at the final value.

Examples Based On Heron’S Formula

1. Calculate the area of a triangle whose sides are 40m, 24m, and 32 m?

Solution:

According to the questions, sides of a triangle are given below:

a= 40 m 

b= 24 m 

c= 32 m 

First, we find the value of s which can be find as:

s = a+ b + c /2

= 40 + 24 + 32 / 2

= 96/2 m 

= 48 m 

Now we will calculate future value:

S - a = (48-40) m = 8m

S - b = (48-24) m = 24 m 

S - c = (48-32) m = 16 m 

Therefore, the area of a triangle=√s (s-a) (s-b) (s-c)          

= √48 8 24 16 m²

= 384 m²

2. 24 cm and 10 cm are the perimeter and hypotenuse of the right triangle respectively. Find the other two sides of a triangle. Find the area of a triangle using the formula: Area of the right triangle and verify the results by using Heron’s formula.

Solution: We can assume that x and y are the two lines of a right triangle.

AB= x cm 

BC= y cm

AC= 10 cm (given)

According to the question:

Perimeter = 24 cm (given)

x + y + 10=24

x + y= 14 cm    ------------(1)                               

Using Pythagoras theorem (PT)

AB² + BC² = 10²

x² + y² = 100    ----------------(2)

From equation 1: 

(x+y)² = 14²

x² + y² + 2xy = 196

Therefore, 100 + 2xy = 196 ( from eq 2)

2xy = 196 - 100

xy = 96/2

xy = 48 cm                         

Area of triangle ABC= ½ x y

= ½ 48 

= 24 cm²     ------------------- (3)

Again, (x-y)² = (x+y)² - 4xy

= 14² - 4 48

x-y = +2, -2

(1) When (x-y) = 2 and (x+y) = 14 then 2x= 16

x = 16/2 

x = 8

And (8 - y) = 2

y= 6 

(2) When (x-y)= -2 and (x+y)= 14 then 2x = 12

x= 12/2

x= 6

And y= 8

Verification by using Heron's formula:

a= 6 cm

b= 8 cm 

c= 10 cm 

Now by applying Heron's formula :

Area =√s(s-a)(s-b)(s-c)

Here, s = a+b+c/2

S = 6+ 8 + 10 /2

s = 12 cm  

Now: (s-a) = 12 -6 = 6 cm 

(s-b) = 12 -8 = 4 cm 

(s-c) = 12 - 10 = 2cm 

Area= √12 * 6 * 4 * 2

= 24 cm²             ------------------ (4) 

Therefore from both the formulas, the area of a triangle is the same which can be observed from equations 3 and 4.

Hence Verified.

Class 9 Math Chapter 12 Heron's Formula MCQ:

Students can refer to the Vedantu site, for the class 9 heron’s formula MCQ, a worksheet with detailed solutions. The ncert class 9 heron's formula worksheet with answers are also available on the Vedantu site. The worksheets are available also in the form of the pdfs. The students can refer to them as well as practice the questions.

1. The Edges of a Triangular Board are Given as  6cm, 8cm, and 10 cm. What will be the Cost of Painting it at the Rate of 9 Paisa/ cm ² is 

1. Rs 3.00

2. Rs 2.16

3. Rs 5.00

4. Rs.2.92

Ans: Rs 2.16

2. The Sides of a Triangle are Given as 35 cm, 54 cm, and 61 cm Respectively. What will be the Length of its Longest Altitude?

1. 16 

2. 24√5

3. 28

4. 40√2

Ans: 24 √5

3. The Sides of a Triangle are 56 cm, 60 cm, and 52 cm Long. Then the Area of the Triangle is 

1. 1311 cm²

2. 1344 cm²

3. 1392 cm²

4. 1322 cm²

Ans: 1344 cm²