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Area of Triangle with 3 Sides

In order to find the area of triangle with 3 sides, we use the Heron’s Formula. Using this, the area of a triangle (A) with 3 sides a, b, and c is calculated using the formula A = √[s(s-a)(s-b)(s-c)], where ‘s’ is the semi-perimeter of the triangle given by s = (a + b + c)/2.

The area of a triangle can be calculated with the help of various formulas. The basic formula that is used to find the area of a triangle is ½ × Base × Height where “Base” is the side of the triangle on which the altitude is formed, and “Height” is the length of the altitude drawn to the “Base” from its opposite vertex. However, if the altitude of a triangle is not known, and we need to find the area of triangle with 3 different sides, the Heron’s formula is used. This formula was derived by a Greek mathematician known as the Heron of Alexandria.

1. Area of Triangle with 3 Sides Formula
2. Proof of Area of Triangle with 3 Sides Formula
3. How to Find Area of Triangle with Three Sides?
4. Area of Triangle With 3 Equal Sides
5. FAQs on Area of Triangle with 3 Sides

Area of Triangle with 3 Sides Formula

In order to find the area of a triangle with 3 sides, we use the Heron’s formula which says if a, b, and c are the three sides of a triangle, then its area is,

Area = √[s(s-a)(s-b)(s-c)]

Here, “s” is the semi-perimeter of the triangle, i.e., s = (a + b + c)/2.

Let us see how to find the area of a triangle with 3 sides given as: 3, 6, and 7. We know that a = 3, b = 6, and c = 7, the semi-perimeter is, s = (a + b + c)/2 = (3 + 6 + 7)/2 = 8. We will find the area of the triangle using the Heron’s formula.

A = √[s(s-a)(s-b)(s-c)]

= √[8(8-3)(8-6)(8-7)]

= √[8 × 5 × 2 × 1]

= √(80)

≈ 8.94

☛ Also Check: Triangle Area Calculator With 3 sides

Proof of Area of Triangle with 3 Sides Formula

The proof of the formula for the area of triangle with 3 sides can be derived in the following way.

Consider the triangle shown above with sides a, b, c, and the opposite angles to the sides as angle A, angle B, angle C.

Using law of cosines, cos A = (b2 + c2 – a2) / 2bc.

Using one of the Trigonometric identities,

sin2 A = 1 – cos2 A

\( \begin{align}\sin A &= \sqrt{1-\cos^2A}\\[0.2cm]\sin A &= \sqrt{1 – \dfrac{(b^2+c^2-a^2)^2}{4b^2c^2}} \\[0.2cm] \sin A &= \dfrac{\sqrt{4b^2c^2 – (b^2+c^2-a^2)^2}}{2bc} \\[0.2cm] \dfrac{1}{2} bc \sin A& = \dfrac{\sqrt{4b^2c^2 – (b^2+c^2-a^2)^2}}{4} \end{align}\)

We know that one of the formulas of the area of a triangle is ½ bc sin A. Thus, the area of triangle = \(\dfrac{\sqrt{4b^2c^2 – (b^2+c^2-a^2)}}{4}\).

Now, we will derive Heron’s formula using the above formula just by applying some algebraic techniques. The above formula can be written as:

\( \begin{align} &\text{Area }\\[0.2cm] &= \dfrac{\sqrt{(2bc)^2- (b^2+c^2-a^2)^2}}{4}\\[0.2cm] &= \dfrac{1}{4} \sqrt{[ (b^2+c^2+2bc) – a^2] [ a^2 – (b^2+c^2-2bc)]}\\[0.2cm] &= \dfrac{1}{4} \sqrt{[ (b+c)^2-a^2] [a^2-(b-c)^2]}\\[0.2cm] &= \dfrac{1}{4} \sqrt{(b+c+a)(b+c-a)(a+b-c)(a-b+c)}\\[0.2cm] \end{align} \)

We know that the perimeter of triangle = (a + b + c) and hence semi-perimeter is, s = (a + b + c)/2. From this,

a + b + c = 2s

b + c – a = 2s – 2a

a + b – c = 2s – 2c

a – b + c = 2s – 2b

Substituting all these values in the last step,

\( \begin{align}& \text{Area } \\[0.2cm]&= \dfrac{1}{4} \sqrt{2s (2s-2a)(2s-2c)(2s-2b)}\\[0.2cm] &= \dfrac{4}{4} \sqrt{s(s-a)(s-c)(s-b)}\\[0.2cm] &= \sqrt{s(s-a)(s-b)(s-c)} \end{align}\)

Hence, we proved the Heron’s formula.

How to Find Area of Triangle with Three Sides?

Consider a triangle with 3 sides given as a, b, and c. Follow the steps below to find its area:

  • Step 1: Find semi-perimeter (s) by using the formula s = (a+b+c)/2.
  • Step 2: Compute the differences (s – a), (s – b), and (s – c).
  • Step 3: Compute the square root of [s(s – a)(s – b)(s – c)].

Example: Calculate the area of triangle with three sides 3, 4, and 5.

Solution: Let a = 3, b = 4, and c = 5.

  • Step 1: s = (a + b + c)/2 = (3 + 4 + 5)/2 = 6.
  • Step 2: (s – a) = 6 – 3 = 3; (s – b) = 6 – 4 = 2; (s – c) = 6 – 5 = 1.
  • Step 3: Area = √[s(s-a)(s-b)(s-c)] = √(6 × 3 × 2 × 1) = 6.

Note: To simplify square roots with big products such as √(21 × 3 × 11 × 7) easily, do prime factorization of each number, then group pairs of numbers.

Area of Triangle With 3 Equal Sides

A triangle with 3 equal sides is known as an equilateral triangle. We know that the area of an equilateral triangle with side ‘a’ is √3/4 × a2. We can derive this formula by using the above formula of triangle area with three sides. Here, a = b = c = a. Using this, perimeter is 3a and semi-perimeter is s = 3a/2 and (s – a) = (3a/2) – a = a/2. Then the area by Heron’s formula is:

A = √[s(s – a)(s – b)(s – c)]

= √ s (s – a) (s – a) (s – a)

= √(3a/2) (a/2) (a/2) (a/2)

= √3a2/4

Hence, the formula for triangle area with 3 equal sides is derived.

☛ Also Check: Area of triangle with 3 Equal sides Calculator

Area of a Triangle With 3 Sides Examples

  1. Example 1: Three sides of a given triangle are 8 units, 11 units, and 13 units. Find its semi-perimeter and its area.

    Solution:

    We know that the formula that is used to find the area of a triangle with 3 sides is, Area =√[s(s-a)(s-b)(s-c)], where ‘a’, ‘b’, ‘c’ are the three sides and ‘s’ is the semi perimeter of the triangle. In this case, a = 8; b = 11, c = 13, and the semi-perimeter is, s = 8 + 11 + 13 = 32/2 = 16

    Let us calculate the area of a triangle with 3 sides, using Heron’s formula.

    A =√[s(s-a)(s-b)(s-c)]

    = √[16(16-8)(16-11)(16-13)]

    = √[16 × 8 × 5 × 3]

    = √16 × √8 × √5 × √3

    = 4 × 2√2 × √5 ×√3

    = 8 √30 = 43.817 unit2

    Answer: Area of the given triangle = 43.817 unit2

  2. Example 2:If the three sides of a triangle are 4 units, 6 units, and 8 units, respectively, find the area of the triangle.

    Solution:

    In order to find the area of a triangle with 3 sides given, we use the formula: A =√[s(s-a)(s-b)(s-c)]

    The sides of the given triangle are 4 units, 6 units, and 8 units.

    The semi-perimeter of the triangle is,

    s = (a + b + c)/2 = (4 + 6 + 8)/2 = 18/2 =9.

    Now, we will find the area of the triangle using Heron’s formula:

    A =√[s(s-a)(s-b)(s-c)]

    = √[9(9-4)(9-6)(9-8)]

    = √[9 × 5 × 3 × 1]

    = 3 √15 = 11.61 square units

    Answer: The area of the triangle = 11.61 square units.

  3. Example 3: Given a triangle with 3 sides a = 122 in, b = 22 in, and c = 120 in, find the altitude of the triangle that corresponds to the side b = 22 in.

    Solution:

    First, we will find the area of the given triangle. Here, s = (122 + 22 + 120) / 2 = 132 in.

    Then area is, A = √ 132 (132 – 122) (132 – 22) (132 – 120)

    = √ 132(10)(110)(12)

    = 1320 in2

    Now, let us assume that the altitude that corresponds to the side ‘b’ be ‘h’ inches. Then the area is

    A = (1/2) (bh)

    1320 = (1/2) (22) (h)

    h = (1320 × 2)/22 = 120 in

    Answer: The altitude of triangle for side ‘b’ is 120 in.

Practice Questions on Area of Triangle with 3 Sides

FAQs on Area of Triangle with 3 Sides

What is the Area of a Triangle With 3 Sides?

The area of a triangle with 3 sides can be calculated with the help of the Heron’s formula according to which, the area of a triangle is √[s(s-a)(s-b)(s-c)], where a, b, and c, are the three different sides and ‘s’ is the semi perimeter of the triangle. ‘s’ be calculated as follows: semi perimeter = (a + b + c)/2

What is the Area of Triangle with 3 Sides Equal Sides?

If a triangle has 3 equal sides, it is called an equilateral triangle. The area of an equilateral triangle can be calculated using the formula, Area = a2(√3/4), where ‘a’ is the side of the triangle. For example, if an equilateral triangle has a side of 6 units, its area will be calculated as follows. Area = a2(√3/4), Area = 62(√3/4) = 15.59 square units.

What is the Area of Triangle with 3 Sides and Height?

If we know the sides of a triangle along with its height, we can use the basic formula for the area of a triangle. Area of a triangle = 1/2 × base × height. For example, if the height (altitude) of a triangle = 8 units, and the side of the triangle on which the altitude is formed is given (base) = 7 units, we can find its area using the formula, Area of a triangle = 1/2 × base × height. Area = 1/2 × 7 × 8 = 28 square units.

What is the Area of a Triangle with three Sides and an Angle?

If the sides of a triangle are given along with an included angle, the area of the triangle can be calculated with the formula, Area = (ab × sin C)/2, where ‘a’ and ‘b’ are the two given sides and C is the included angle. This is also known as the “side angle side ” method. For example, if two sides of a triangle are 5 units and 7 units and the included angle is 60°, then, Area = (7 × 5 × sin 60)/2 = 15.15 square units.

What is the Area of a Triangle with Sides 3, 5, 7?

If the three sides of a triangle are given as 3, 5 and 7, its area can be calculated with the formula, area = √[s(s-a)(s-b)(s-c)]. In this case, a = 3, b = 5, c = 7, and s (semi perimeter) = 7.5. Substituting the values in the formula, √[7.5(7.5-3)(7.5-5)(7.5-7)] = √(7.5 × 4.5 × 2.5 × 0.5) = 6.49 unit2

What is an Irregular Triangle?

An irregular triangle is a triangle in which all three sides are of different lengths. It is also known as a scalene triangle (if all three sides are different).

How do you find the Area of an Irregular Triangle?

We use the Heron’s formula to find the area of an irregular triangle. An irregular triangle means a triangle whose sides are different in length. According to the Heron’s formula: Area = √[s(s-a)(s-b)(s-c)], where, a, b and c are the sides of the triangle, and ‘s’ is the semi-perimeter of the triangle.

How to Find the Length of the sides of a Triangle with 3 Angles Only?

Let us recall the fact that two similar triangles have the same angles but different sides (sides are in proportion). Thus, there are an infinite number of triangles with the same set of any three given angles. So it is not possible to find the sides of a triangle if we just know all 3 angles, we need to know at least one side to determine the other two sides.

What is Heron’s Formula Used For?

Heron’s formula is used to find the area of a triangle that has three different sides. Heron’s formula is written as, Area = √[s(s-a)(s-b)(s-c)], where a, b and c are the sides of the triangle, and ‘s’ is the semi-perimeter of the triangle.

Who is Heron’s Formula Named After?

Heron’s formula is named after the Heron of Alexandria, a Greek mathematician, who found the area of a triangle using the 3 sides. That is the reason why the formula √[s(s-a)(s-b)(s-c)] is known as Heron’s formula.

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