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HERON'S FORMULA - Q&A

Exercise 10.1

1. A traffic signal board, indicating 'SCHOOL AHEAD', is an equilateral triangle with side 'a'. Find the area of the signal board, using Heron's formula. If its perimeter is 180 cm, what will be the area of the signal board?

Step 1: Find the area of the equilateral triangle with side 'a' using Heron's formula.
Let the sides of the triangle be a, a, and a.
Semi-perimeter (s) = (a + a + a) / 2 = 3a/2.

Using Heron's formula, Area = √[s(s - a)(s - b)(s - c)]
Here, a = b = c = a.
Area = √[3a/2 (3a/2 - a) (3a/2 - a) (3a/2 - a)]
Area = √[3a/2 (a/2) (a/2) (a/2)]
Area = √[3a⁴ / 16]
Area = (√3 / 4) a².

Step 2: Find the area when perimeter is 180 cm.
Perimeter = 180 cm.
Since it is an equilateral triangle, 3a = 180 cm.
Side a = 180 / 3 = 60 cm.

Substitute a = 60 in the area formula:
Area = (√3 / 4) (60)²
Area = (√3 / 4) × 3600
Area = 900√3 cm².

2. The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m and 120 m (see Fig. 10.6). The advertisements yield an earning of ₹ 5000 per m² per year. A company hired one of its walls for 3 months. How much rent did it pay?

Step 1: Find the area of one triangular wall.
The sides are a = 122 m, b = 22 m, c = 120 m.
Semi-perimeter s = (122 + 22 + 120) / 2 = 264 / 2 = 132 m.

Using Heron's Formula:
Area = √[s(s - a)(s - b)(s - c)]
Area = √[132 (132 - 122) (132 - 22) (132 - 120)]
Area = √[132 × 10 × 110 × 12]
Area = √[(11 × 12) × 10 × (11 × 10) × 12]
Area = √[11 × 11 × 12 × 12 × 10 × 10]
Area = 11 × 12 × 10
Area = 1320 m².

Step 2: Calculate the rent.
Rate = ₹ 5000 per m² per year.
Rent for 1 year (12 months) for 1 m² = ₹ 5000.
Rent for 3 months for 1 m² = (5000 × 3) / 12 = ₹ 1250.

Total rent for 1320 m² for 3 months:
Total Rent = 1320 × 1250
Total Rent = ₹ 16,50,000.

3. There is a slide in a park. One of its side walls has been painted in some colour with a message "KEEP THE PARK GREEN AND CLEAN" (see Fig. 10.7). If the sides of the wall are 15 m, 11 m and 6 m, find the area painted in colour.

Step 1: Identify the sides of the triangle.
a = 15 m, b = 11 m, c = 6 m.

Step 2: Calculate semi-perimeter (s).
s = (15 + 11 + 6) / 2 = 32 / 2 = 16 m.

Step 3: Calculate the area using Heron's Formula.
Area = √[s(s - a)(s - b)(s - c)]
Area = √[16 (16 - 15) (16 - 11) (16 - 6)]
Area = √[16 × 1 × 5 × 10]
Area = √[16 × 50]
Area = √[16 × 25 × 2]
Area = 4 × 5 × √2
Area = 20√2 m².

Answer: The area painted in colour is 20√2 m².

4. Find the area of a triangle two sides of which are 18cm and 10cm and the perimeter is 42cm.

Step 1: Find the third side (c).
Perimeter = a + b + c
42 = 18 + 10 + c
42 = 28 + c
c = 42 - 28 = 14 cm.

Step 2: Calculate semi-perimeter (s).
s = Perimeter / 2 = 42 / 2 = 21 cm.

Step 3: Calculate area.
Area = √[s(s - a)(s - b)(s - c)]
Area = √[21 (21 - 18) (21 - 10) (21 - 14)]
Area = √[21 × 3 × 11 × 7]
Area = √[(3 × 7) × 3 × 11 × 7]
Area = √[3 × 3 × 7 × 7 × 11]
Area = 3 × 7 × √11
Area = 21√11 cm².

5. Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540cm. Find its area.

Step 1: Find the actual lengths of the sides.
Let the sides be 12x, 17x, and 25x.
Perimeter = 12x + 17x + 25x = 540
54x = 540
x = 10.
So, the sides are:
a = 12 × 10 = 120 cm
b = 17 × 10 = 170 cm
c = 25 × 10 = 250 cm

Step 2: Calculate semi-perimeter (s).
s = 540 / 2 = 270 cm.

Step 3: Calculate area.
Area = √[s(s - a)(s - b)(s - c)]
Area = √[270 (270 - 120) (270 - 170) (270 - 250)]
Area = √[270 × 150 × 100 × 20]
Area = √[27 × 10 × 15 × 10 × 100 × 2 × 10]
Area = √[(9 × 3) × (3 × 5) × (2) × 100000]
Area = √[9 × 9 × 100 × 100 × 100]
Let's simplify differently:
Area = √[270 × 150 × 100 × 20]
Area = √[81,000,000]
Area = 9000 cm².

6. An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.

Step 1: Find the third side.
Let equal sides a = 12 cm, b = 12 cm. Let third side be c.
Perimeter = a + b + c = 30
12 + 12 + c = 30
24 + c = 30
c = 6 cm.

Step 2: Calculate semi-perimeter (s).
s = 30 / 2 = 15 cm.

Step 3: Calculate area.
Area = √[s(s - a)(s - b)(s - c)]
Area = √[15 (15 - 12) (15 - 12) (15 - 6)]
Area = √[15 × 3 × 3 × 9]
Area = √[15 × 9 × 9]
Area = 9√15 cm².

Answer: The area of the triangle is 9√15 cm².

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Quick Review Flashcards - Click to flip and test your knowledge!

Question

What is the standard formula for the area of a triangle when the height is known?

Answer

The area is calculated as $\frac{1}{2} \times \text{base} \times \text{height}$.

Question

Under what condition is Heron's formula typically used to find the area of a triangle?

Answer

It is used when the lengths of all three sides are known but the height is not easily determined.

Question

According to the text, in which ancient city did Heron possibly live and work?

Answer

Heron worked in Alexandria, Egypt.

Question

Term: Heron of Alexandria

Answer

Definition: An encyclopaedic writer born around 10 C.E. who worked extensively in applied mathematics and mensuration.

Question

How many books comprise Heron's geometrical works on mensuration?

Answer

His geometrical works on mensuration are written in three books.

Question

Which specific book by Heron contains the derivation of the formula for the area of a triangle in terms of its three sides?

Answer

The derivation is found in Book I.

Question

Besides triangles, name three other shapes discussed in Heron's Book I.

Answer

Book I covers squares, rectangles, trapezoids (trapezia), regular polygons, circles, cylinders, cones, and spheres.

Question

What is an alternative name for Heron's formula mentioned in the text?

Answer

It is also known as Hero's formula.

Question

State Heron's formula for the area of a triangle.

Answer

The area is $\sqrt{s(s-a)(s-b)(s-c)}$.

Question

In Heron's formula, what do the variables $a$, $b$, and $c$ represent?

Answer

These variables represent the lengths of the three sides of the triangle.

Question

In Heron's formula, what does the variable $s$ represent?

Answer

The variable $s$ represents the semi-perimeter of the triangle.

Question

Formula: Semi-perimeter ($s$)

Answer

$s = \frac{a + b + c}{2}$.

Question

Concept: Semi-perimeter

Answer

Definition: It is a value equal to half the perimeter of a polygon.

Question

If a triangular park has sides of $40\text{ m}$, $24\text{ m}$, and $32\text{ m}$, what is its semi-perimeter ($s$)?

Answer

The semi-perimeter is $48\text{ m}$.

Question

For a triangle with $s = 48\text{ m}$ and sides $a = 40\text{ m}$, $b = 24\text{ m}$, and $c = 32\text{ m}$, calculate the value of $(s - a)$.

Answer

The value of $(s - a)$ is $8\text{ m}$.

Question

For a triangle with $s = 48\text{ m}$ and sides $a = 40\text{ m}$, $b = 24\text{ m}$, and $c = 32\text{ m}$, calculate the value of $(s - b)$.

Answer

The value of $(s - b)$ is $24\text{ m}$.

Question

For a triangle with $s = 48\text{ m}$ and sides $a = 40\text{ m}$, $b = 24\text{ m}$, and $c = 32\text{ m}$, calculate the value of $(s - c)$.

Answer

The value of $(s - c)$ is $16\text{ m}$.

Question

What is the area of a triangular park with sides $40\text{ m}$, $24\text{ m}$, and $32\text{ m}$?

Answer

The area is $384\text{ m}^2$.

Question

How can the sides $32$, $24$, and $40$ be used to verify if a triangle is right-angled?

Answer

By checking if the square of the longest side equals the sum of the squares of the other two sides ($32^2 + 24^2 = 40^2$).

Question

If the sides of a triangle are $32\text{ m}$, $24\text{ m}$, and $40\text{ m}$, which side acts as the hypotenuse?

Answer

The side measuring $40\text{ m}$ is the hypotenuse.

Question

For an equilateral triangle with side length $10\text{ cm}$, what is the value of the semi-perimeter $s$?

Answer

The semi-perimeter is $15\text{ cm}$.

Question

Using Heron's formula, calculate the area of an equilateral triangle with a side of $10\text{ cm}$.

Answer

The area is $25\sqrt{3}\text{ cm}^2$.

Question

In an isosceles triangle with equal sides of $5\text{ cm}$ and an unequal side of $8\text{ cm}$, what is the semi-perimeter $s$?

Answer

The semi-perimeter is $9\text{ cm}$.

Question

What is the area of an isosceles triangle with sides $5\text{ cm}$, $5\text{ cm}$, and $8\text{ cm}$?

Answer

The area is $12\text{ cm}^2$.

Question

Two sides of a triangle are $8\text{ cm}$ and $11\text{ cm}$, and the total perimeter is $32\text{ cm}$. What is the length of the third side?

Answer

The third side is $13\text{ cm}$.

Question

If the perimeter of a triangle is $32\text{ cm}$, what is its semi-perimeter $s$?

Answer

The semi-perimeter is $16\text{ cm}$.

Question

Calculate the area of a triangle with sides $8\text{ cm}$, $11\text{ cm}$, and $13\text{ cm}$.

Answer

The area is $8\sqrt{30}\text{ cm}^2$.

Question

A triangular park has sides $120\text{ m}$, $80\text{ m}$, and $50\text{ m}$. What is the total perimeter?

Answer

The total perimeter is $250\text{ m}$.

Question

A triangular park has sides $120\text{ m}$, $80\text{ m}$, and $50\text{ m}$. What is the semi-perimeter $s$?

Answer

The semi-perimeter is $125\text{ m}$.

Question

What is the area of a triangular park with side lengths $120\text{ m}$, $80\text{ m}$, and $50\text{ m}$?

Answer

The area is $375\sqrt{15}\text{ m}^2$.

Question

If a $3\text{ m}$ space is left for a gate in a fence surrounding a park with a $250\text{ m}$ perimeter, what is the total length of the fencing wire needed?

Answer

The length of wire needed is $247\text{ m}$.

Question

How is the total cost of fencing calculated given a perimeter (minus gate) of $247\text{ m}$ and a rate of $£20$ per metre?

Answer

The cost is $247 \times 20$, which equals $£4940$.

Question

If the sides of a triangular plot are in the ratio $3:5:7$ and the perimeter is $300\text{ m}$, how is the common factor $x$ determined?

Answer

It is determined by solving the equation $3x + 5x + 7x = 300$.

Question

In a triangle with sides in the ratio $3:5:7$ and a perimeter of $300\text{ m}$, what is the value of the common factor $x$?

Answer

The common factor $x$ is $20$.

Question

What are the specific side lengths of a triangle with sides in ratio $3:5:7$ and a perimeter of $300\text{ m}$?

Answer

The side lengths are $60\text{ m}$, $100\text{ m}$, and $140\text{ m}$.

Question

What is the semi-perimeter $s$ of a triangle with sides $60\text{ m}$, $100\text{ m}$, and $140\text{ m}$?

Answer

The semi-perimeter is $150\text{ m}$.

Question

What is the area of a triangular plot with sides $60\text{ m}$, $100\text{ m}$, and $140\text{ m}$?

Answer

The area is $1500\sqrt{3}\text{ m}^2$.

Question

A traffic signal board is an equilateral triangle with side '$a$'. Express its semi-perimeter in terms of '$a$'.

Answer

The semi-perimeter is $\frac{3a}{2}$.

Question

If a traffic signal board is an equilateral triangle with a perimeter of $180\text{ cm}$, what is the length of each side?

Answer

Each side is $60\text{ cm}$.

Question

The side walls of a flyover are $122\text{ m}$, $22\text{ m}$, and $120\text{ m}$. What is the semi-perimeter $s$ of these walls?

Answer

The semi-perimeter is $132\text{ m}$.

Question

If the perimeter of a triangle is $42\text{ cm}$ and two sides are $18\text{ cm}$ and $10\text{ cm}$, what is the length of the third side?

Answer

The third side is $14\text{ cm}$.

Question

A triangle has sides in the ratio $12:17:25$ and a perimeter of $540\text{ cm}$. What is the value of the constant ratio $x$?

Answer

The constant ratio $x$ is $10$.

Question

What are the side lengths of a triangle with ratio $12:17:25$ and perimeter $540\text{ cm}$?

Answer

The side lengths are $120\text{ cm}$, $170\text{ cm}$, and $250\text{ cm}$.

Question

An isosceles triangle has a perimeter of $30\text{ cm}$ and equal sides of $12\text{ cm}$. What is the length of the unequal side?

Answer

The unequal side is $6\text{ cm}$.

Question

The _____ of a triangle is defined as half of its perimeter.

Answer

semi-perimeter

Question

The formula $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$ is known as _____ formula.

Answer

Heron's

Question

In the context of Heron's formula, the term $2s$ is equivalent to the _____ of the triangle.

Answer

perimeter

Question

Heron's formula is particularly helpful for calculating the area of a _____ triangle when the height is unknown.

Answer

scalene

Question

Heron's Book I deals with problems on _____, covering areas of various polygons and surfaces of solids.

Answer

mensuration

Question

What is the semi-perimeter of a triangle with sides $15\text{ m}$, $11\text{ m}$, and $6\text{ m}$?

Answer

The semi-perimeter is $16\text{ m}$.

Question

If $s = 16\text{ m}$ and the sides of a triangle are $15\text{ m}$, $11\text{ m}$, and $6\text{ m}$, what is the calculated area?

Answer

The area is $20\sqrt{2}\text{ m}^2$.

Question

How do you calculate the semi-perimeter $s$ if you are given the perimeter $P$ directly?

Answer

The semi-perimeter is calculated as $s = \frac{P}{2}$.

Question

If the sides of a triangle are $122\text{ m}$, $22\text{ m}$, and $120\text{ m}$, find the area.

Answer

The area is $1320\text{ m}^2$.