Area of a Triangle Formula – with Solved Examples

The Area of a triangle is the region cordoned off by it, in a flattened plane. As we know, a triangle is a padlocked shape that has three sides and three apexes. Thus, the area of a triangle formula is the entire space occupied inside the three sides of a triangle. The broad procedure to find the area of the triangle formula is given by half of the product of its base and height.

In broad-spectrum, the term “area” is described as the region occupied inside the margin of a flat entity or figure. The measurement is done in square entities with the average unit being square meters (m²). For the calculation of area, there are pre-defined formulas for squares, rectangles, circles, triangles, etc. In this blog, we will study the area of triangle formulas for diverse types of triangles, along with some illustration problems.

How Do You Define the Area of a Triangle?

The area of a triangle formula can be described as the total region that is cordoned off by the three sides of any specific triangle.

Therefore, to find the area of a tri-sided polygon, we must know the base (b) and height (h). It is relevant to all categories of triangles, whether it is scalene, isosceles, or equilateral. Point to be emphasized- the base and height of the triangle are at right angles to each other. Then the unit of area is calculated in square units (m², cm²).

Illustration: Can you find the area of a triangle with base b = 3 cm and height h = 4 cm?
Using the formula,

The Area of a Triangle, A = 1/2 × b × h = 1/2 × 4 cm × 3 cm = 2 cm × 3 cm = 6 cm²

Apart from the overhead formula, we have Heron’s method to calculate the triangle’s area, when we identify the length of its three sides. Similarly, trigonometric functions are used to discover the area when we know two sides and the angle designed between them in a triangle. We will analyze the area for all the situations given here.

Area of a Triangle Formula

The area of the triangle formula is cited below:

The Area of a Triangle formula = A = ½ (b × h) square units

Where b and h are the base and height of the triangle, correspondingly.

Now, let’s understand how to compute the area of a triangle using the given formula. Correspondingly, how to find the area of a triangle with 3 sides using Heron’s formula with samples.

The Area of a Right-Angled Triangle
A ninety-degree triangle, also called a right triangle has one angle at 90°, and the additional two acute angles sum to 90°. As a result, the height of the triangle will be the length of the perpendicular side.
Area of a Right Triangle = A = ½ × Base × Height (Perpendicular expanse)

The Area of an Equilateral Triangle

An equilateral triangle is a triangle where everything on all sides is equivalent. The bolt-upright drawn from the top of the triangle to the base divides the base into two identical parts. To estimate the area of the equilateral triangle, we have to know the dimensions of its sides.

• Area of an Equilateral Triangle = A = (√3)/4 × side²

Area of an Isosceles Triangle
An isosceles triangle has two of its sides equivalent and the angles opposite the equal sides are alike.

• Area of an Isosceles Triangle = 1/4 b√(4a² – b²)

The perimeter of a Triangle
The boundary of a triangle is the distance enclosed around the triangle and is calculated by tallying all three sides of a triangle.

• The perimeter of a triangle = P = (a + b + c) units
Where a, b, and c are the margins of the triangle.

Heron’s Formula- Area of Triangle with Three Sides

The part of a triangle with 3 sides of altered measures can be found using Heron’s formula. Heron’s formula consists of two important steps. The initial step is to find the semi-perimeter of a triangle by totaling all three sides of a triangle and dividing it by 2. The following step is to, apply the semi-perimeter of triangle value in the central formula called “Heron’s Formula” to find the area of a triangle.

At present, the question emanates, when we know the two sides of a triangle and an angle comprised between them, then how to find its area.
So, if any two margins and the angle between them are given, then the methods to compute the area of a triangle are given by:
Area (∆ABC) = ½ bc sin A
Area (∆ABC) = ½ ab sin C
Area (∆ABC) = ½ ca sin B
These formulas are very easy to summon up and also to analyze.
For instance, If, in ∆ABC, A = 30° and b = 2, c = 4 in units. Then the area will be;
Area (∆ABC) = ½ bc sin A
= ½ (2) (4) in 30
= 4 x ½ (since sin 30 = ½)
= 2 sq. unit.

Area of a Triangle Solved Examples

Illustration 1:
Discover the area of an acute triangle with a base of 13 inches and a height of 5 inches.
Explanation:
A = (½) × b × h sq. Units
⇒ A = (½) × (13 in) × (5 in)
⇒ A = (½) × (65 in²)
⇒ A = 32.5 in²

Illustration 2:
Find the area of a ninety-degree triangle with a base of 7 cm and a height of 8 cm.
Explanation:
A = (½) × b × h square Units
⇒ A = (½) × (7 cm) × (8 cm)
⇒ A = (½) × (56 cm²)
⇒ A = 28 cm²

Illustration 3:
Discover the area of an obtuse-angled triangle with a base of 4 cm and a height of 7 cm.
Explanation:
A = (½) × b × h square units
⇒ A == (½) × (4 cm) × (7 cm)
⇒ A = (½) × (28 cm²)
⇒ A = 14 cm²

Area of Triangle Formulas

You can learn about the area of triangle formulas for numerous varied types of triangles like the equilateral triangle, right-angled triangle, and isosceles triangle below.

• Area of a Right-Angled Triangle
A ninety-degree triangle also called a right triangle, has a single angle equal to 90°, and the other two acute angles sum up to 90°. For that reason, the height of the triangle is the length of the side of the right angle.
The Area of a Right Triangle = A = 1/2 × B × H

• Area of an Equilateral Triangle
An equilateral triangle is a triangle where all the sides are identical. The perpendicular drawn from the apex of the triangle to the base divides the base into two identical parts. To estimate the area of the equilateral triangle, we must know the dimensions of its sides.

Area of an Equilateral Triangle = A = (√3)/4 × side²

• Area of an Isosceles Triangle
An isosceles triangle has dual of sides equal and the angles opposite the equal sides are also equivalent.
Area of an Isosceles Triangle = A = 1/4 ×b√4a²−b²4a²−b²
Where ‘b’ is the base and ‘a’ is the degree of one of the identical sides.

Note the table specified underneath which abridges all the formulas for the area of a triangle.

Given Dimensions	Area of Triangle Formula
While the base and height of a triangle are specified.	A = 1/2 (base × height)
While the sides of a triangle are specified as a, b, and c.	(Heron’s formula)Area of a scalene triangle = √s(s−a)(s−b)(s−c)s(s−a)(s−b)(s−c) where a, b, and c are the sides and ‘s’ is the semi-perimeter; s = (a + b + c)/2
While two sides and the included angle are specified.	A = 1/2 × plane 1 × plane 2 × sin(θ)where θ is the angle amid the given two sides
While base and height are known.	Area of a right-angled triangle = 1/2 × Base × Height
While it is an equilateral triangle and one side is specified.	The area of an equilateral triangle = (√3)/4 × plane2
While it is an isosceles triangle and an equivalent side and base are known.	Area of an isosceles triangle = 1/4 ×b√4a2−b24a2−b2where ‘b’ is the given base and ‘a’ is the provided length of an equal side.

The Area of a Triangle

You calculate the area of a triangle by applying various methods. For example, there’s the basic formula that the area of a triangle is part of the base times the height. This method only works, of course, while you identify the height of the triangle.

Additional is Heron’s formula which provides the area in terms of the three sides of the triangle, explicitly, as the square root of the product s(s – a)(s – b)(s – c) where s is the semi perimeter of the triangle, that is, s = (a + b + c)/2.
Now, we’ll contemplate a formula for the area of a triangle when you identify two sides and the included angle of the triangle. Presume we recognize the values of the two sides a and b of the triangle, and the involved angle C.

Drop at right angles AD from the vertex A of the triangle to the side BC, and tag this height h. Then the particular triangle ACD is a right triangle, so sin C will equal h/b. Consequently, h = b sin C. Subsequently the area of the triangle is half the base times the height h, hence the area also equals half of the ab sin C. Even though the figure is an acute triangle, you can understand from the argument in the previous section that h = b sin C holds when the triangle is right or obtuse as well. As a result, we get the general formula

How to Discover the Surface Area of Triangles

A triangle is a polygon with three sides that may be identical or unequal. The surface area of a triangle is the entire area of the surface inside the boundaries of the triangle. Surface area is stated in square units, such as square centimeters or square inches. Computing the surface area of a triangle is a common geometry task.

Measure the three flanks of the triangle. The lengthiest side is the base of the triangle. If the triangle is on paper, you can brand the base with the measurement; or else, write your base length on a writing pad.

Assess the height of the triangle. The height is the expanse from the base to the highest corner of the triangle. The height line is vertical to the base and intersects the opposite corner of the triangle. Draw this height line on your triangle, if conceivable, and tag the measurement. The height line will run from end to end in the interior of the triangle.
Increase the base length by height. For instance, if your base measurement is 10 cm and the height is 6 cm, the base increased by the height would be 60 square cm.

Divide the result of the base times height by two to fix the surface area. In the illustration, when you divide 60 square cm by two, you have a concluding surface area of 30 square cm.

To find the area of a triangle, multiply the base by the height, and then divide by 2. The division by 2 originates from the fact that a parallelogram can be divided into 2 triangles.

Since the part of a parallelogram is A = B * H, the area of a triangle must be one-half the area of a parallelogram. As a result, the formula for the area of a triangle is:

A= 1/2.b.h Or A= b.h/2

Where b is the base, h is the height and · means multiply.
The base and height of a triangle must be at right angles to each other. In each of the illustrations underneath, the base is a side of the triangle. Nevertheless, depending on the triangle, the height may or may not be a side of the triangle. For instance, in the right triangle in Illustration 2, the height is a side of the triangle since it is perpendicular to the base. In the triangles in Illustrations 1 and 3, the lateral sides are not at right angles to the base, so a dotted line is drawn to signify the height.

Illustration 1: Find the area of an acute triangle with a base of 15 inches and a height of 4 inches.
Answer:

A= 1/2 .b.h
A= 1/2· (15 in) · (4 in)
A= 1/2· (60 in²)
A= 30 in²
________________________________________
Illustration 2: Find the area of a right triangle with a base of 6 centimeters and a height of 9 centimeters.

Answer:

A= 1/2.b.h

A= 1/2 ·(6 cm) · (9 cm)
A= 1/2.(54 cm²)
A= 27 cm²
________________________________________
Illustration 3: Find the area of an obtuse triangle with a base of 5 inches and a height of 8 inches.

Answer:

A=1/2.b.h
A= 1/2 (5 in) · (8 in)
A= 1/2(40 in²)
A= 20 in²
________________________________________
Illustration 4: A triangle-shaped mat has an area of 18 square feet and the base is 3 feet. Find the height.

Answer:
In this illustration, we have specified the area of a triangle and one dimension, and we are asked to work backward to find the other measurement.

A= 1/2.b.h

18 ft² = 1/2\B7 (3 ft) · h
By multiplying individually the two sides of the equation by 2, we arrive at:
36 ft² = (3 ft) · h
By dividing individually the two sides of the equation by 3 ft., we arrive at:
12 ft = h
Calculating this equation, we get:
h = 12 ft
________________________________________
Synopsis: Specify the base and the height of a triangle, and we can discover the area. Given the area and the base or the height of a triangle, we can discover the other dimension. The formula for the area of a triangle is:
A=1/2bh  or A=b.h/2 where b is the base and h is the height.

What’s the Area of a Triangle Formula?

We all are acquainted with the that a triangle is a polygon, which has three sides. The area of a triangle is a dimension of the area covered by the triangle. We arrive at the area of a triangle in the square units. The area of a triangle can be arrived at by using the following two formulas i.e. the base increases by the height of a triangle divided by 2 and the second is Heron’s method. Let us discuss the Area of a Triangle formula in point.

Area of a Triangle Formula

What is an Area of a Triangle?

The area of a polygon is the number of square units covered by the polygon. The area of a triangle is decided by multiplying the base of the triangle and the height of the triangle and then dividing it by 2. The division by 2 is prepared for the reason that the triangle is part of a parallelogram that can be divided into 2 triangles.

The Area of a parallelogram = Base × Height
Where,

B	the base of the parallelogram
H	The height of the parallelogram

Equally, a triangle is one-half of the parallelogram, so the area of a triangle is:
A= 12×b×h
Where,

B	The base of the triangle
H	the height of the triangle

Heron’s Method for Area of a Triangle

Herons formula is a method for computing the area of a triangle when the lengths of all three sides of the triangle are specified.

Let a, b, and c are the lengths of the sides of a triangle.
The area of the triangle is:
Area=s(s−a)(s−b)(s−c)− √
Where s is half the perimeter,
s= a+b+c2
We can also determine the area of a triangle by the subsequent procedures:

1. In this method two Sides, one including Angle is exacting
Area= 12×a×b×sinc
In the above formula a, b, and c are to be considered as the lengths of the sides of a triangle

2. In this technique we find the area of an Equilateral Triangle
Area= 3√×a24

3. In this way we find the area of a triangle on a coordinate plane by Matrices
12×⎡⎣⎢x1x2x3y1y2y3111⎤⎦⎥
Where, (x1, y1), (x2, y2), (x3, y3) are the directs of the three vertices

4. In this technique, we find the area of a triangle in which two vectors from one vertex are at hand.
Area of triangle = 12(u→×v→)

Solved Illustrations

Q.1: Consider the sides of a right triangle ABC to be of the following dimensions; 5 cm, 12 cm, and 13 cm. respectively
Answer: In △ABC in which base= 12 cm and height= 5 cm
Area of △ABC=12×B×H
A = 12×12×5
A = 30 cm²

Q.2: Discover the area of a triangle, which has two sides 12 cm and 11 cm and the perimeter is 36 cm.
Answer: Here we have a perimeter of the triangle = 36 cm, a = 12 cm and b = 11 cm.
Third side c = 36 cm – (12 + 11) cm = 13 cm
Therefore, we get 2s = 36, i.e., s = 18 cm,
s – a = (18 – 12) cm = 6 cm,
s – b = (18 – 11) cm = 7 cm,
and, s – c = (18 – 13) cm = 5 cm.
Area of the triangle = s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−√
A= 18×6×7×5−−−−−−−−−−−√
A= 6105−−−√ cm²

Frequently Asked Questions

1. What do you understand by the area of a triangle?

The area of the triangle is the area cordoned off by its boundary or the three sides of the triangle.

2. What will be the area when two sides of a triangle and the included angle are known?

The area will be equivalent to half times the product of two given sides and sine of the comprised angle.

3. How do you find out the area of a triangle in which three sides are known?

While the values of the three sides of the triangle are known, then we will be able to find the area of that triangle using Heron’s Formula.

4. How can you find the area of a triangle using vectors?

Presume vectors u and v are creating a triangle in space. Now, the area of this triangle is equivalent to half of the amount of the product of these two vectors, such that, A = ½ |u × v|.

5. How do you calculate the area of a triangle formula?

For an assumed triangle, where the base of the triangle is b and the height is h, the area of the triangle can be premeditated by the formula, for example; A = ½ (b × h) Sq. Unit