Area of Triangle Formula

Area of a Triangle

A triangle is a polygon, a 2-dimensional object with 3 sides and 3 vertices. The area of a triangle is determined by using a simple formula to be used while solving problems or questions. You must know the length of sides, the type of triangle, the height of the triangle in order to find the area of a triangle. The area of a triangle is a measurement of the area covered by the triangle. We can express the area of a triangle in the square units. The area of a triangle is determined by two formulas i.e. the base multiplies by the height of a triangle divided by 2 and second is Heron’s formula for area of a triangle. So, let us study, What is the formula for the area of a triangle? This article will answer all your queries related to the area of triangles. 

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Area of Triangle: Definition & Types of Triangle

A triangle is a two-dimensional shape having 3 sides and 3 angles. An area of a triangle is the region enclosed inside the triangle. A triangle can be of 4 types depending upon the length of its sides or angles. The four types of triangles are:

  1. Right Angled Triangle- (angle is of 90 degrees).

  2. Isosceles Triangle- (two sides are equal.).

  3. Equilateral Triangle-(all the three sides are equal and each angle is of 60°)

  4. Scalene triangle- (all the three sides are unequal).

Area of Triangle Formula

The area of triangle formula is given as

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

where b is the base and h is the height of the triangle.

The area of a triangle depends upon the type of triangle. The area formulas for all the different types of triangles equilateral triangle, right-angled triangle, an isosceles triangle are given below. Also, how to find the area of a triangle with 3 sides using heron's theorem


Area of Right Angled Triangle Formula

A right-angled triangle is a triangle having one of its angles of 90°. The side opposite to the right angle is said to be the hypotenuse.

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In the right-angled triangle given, we have the perpendicular height as ‘h’ and base as ‘b’ So the formula for the area of a right-angled triangle can be given by:

Area of a right-angled triangle = 1/2 x b x h 


Area of Isosceles Triangle Formula

We know that an isosceles triangle has two sides of equal lengths. In the figure given below, we have an isosceles triangle with two equal sides, ‘a’ and the base as ‘b’. AD is the perpendicular is drawn which divides the base into 2 equal parts.

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Hence, by the formula, A = 1/2 x b x h, we can derive the formula for the area of the isosceles triangle by the formula given below:

Area of isosceles triangle = 1/4 x b x √4a²-b²


Area of Equilateral Triangle

We know that an equilateral triangle has three sides of equal lengths. And, all the 3 angles of this triangle will be equal to 600. In the below figure, an equilateral triangle with equal sides as ‘a’ is given. AD is the perpendicular drawn from A to D which divides the base into 2 equal parts.

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Hence, by the formula, A = 1/2 x b x h, the formula for the area of the equilateral triangle can be derived as:

Area of Equilateral triangle = √3/4 x a²


Heron’s Formula for Area of a triangle

A scalene triangle is a triangle in which all three sides are unequal. To find the area of a scalene triangle and even other triangles, we use Heron’s Formula. Heron’s formula for area of a triangle is also known as Hero’s formula.

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From the figure given above, a scalene triangle is given with 3 sides as ‘a’, ‘b’ and ‘c’. The Heron’s formula is stated as:

Area of the triangle = \[\sqrt{s(s-a)(s-b)(s-c)}\]

where a, b and c are the sides of the given triangle, 

and s = semi-perimeter which is given by-

s = (a + b+ c) / 2

We can also determine the area of a triangle by some other methods. Some of them are mentioned in the below table.

Properties Given

Area of Triangle Formula

Base and Height

A = ½ bh

Where b = base, h = height

Three Sides

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

Where a, b and c are the lengths of the sides and s = ½ (a+b+c) (half the perimeter)

Two Sides and Including Angles

A = ½ ab sinC

Where a, b are two sides and C is the angle between them

Equilateral Triangle

A = \[\frac{s^{2}\sqrt{3}}{4}\]

Where s = sides

Three Vertices on the Coordinate Plane

A = \[\pm \frac{1}{2}\begin{vmatrix} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{vmatrix}\]

Where (x1, y1), (x2, y2), (x3, y3), are the coordinates of the three vertices

Two Vectors From One Vertex

A = \[\frac{1}{2}\left \| \overrightarrow{\upsilon } \times \overrightarrow{\omega } \right \|\]

Where \[\overrightarrow{\upsilon }\] and \[\overrightarrow{\omega }\] are the vectors that from the sides


Solved Examples

  1. Find the Area of a Right-Angled Triangle With a Base of 8cm and a Height of 8cm.

Solution:

Area of right-angled triangle = (½) × b × h sq.units

⇒ A = (½) × (8 cm) × (8 cm)

⇒ A = (½) × (64 cm2)

⇒ A = 32 cm2


  1. Find the Area of a Triangle, Which Has Three Sides 10 Cm, 12cm and 14 Cm Respectively.

Solution: Here we have sides of the triangle a = 12 cm and b = 11 cm and c = 14cm.

We have Perimeter s = a + b+ c / 2

= 10 + 12 + 14 / 2

= 36/2

=18

Area of the triangle = \[\sqrt{s(s-a)(s-b)(s-c)}\]

= \[\sqrt{18(18-10)(18-12)(18-14)}\]

= \[\sqrt{18 \times 8 \times 6 \times 4}\]

=\[\sqrt{3456}\]

= 58.78

Quiz Time

Example 1: Find the area of a triangle, two sides of which are 10 cm and 14 cm and the perimeter is 50 cm.


Example 2: Find the area of a right-angled triangle with a base of 8 cm and a height of 5 cm.

FAQ (Frequently Asked Questions)

1. What is the Difference Between Area and Perimeter?

As we have understood what is area and what is perimeter let us understand the relationship between area and perimeter.

The area is defined as the space occupied by the shape. While perimeter id defined as the distance around the shape(the boundary of the shape)

Shapes with the same area can have different perimeters and the shapes with the same perimeter can have different areas. The area is measured in square units and the perimeter is measured in linear units. The area can be measured for 2 - dimensional objects while the perimeter is measured for one-dimensional shapes.

The below figure represents the difference between area and perimeter.

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2. What is Heron’s Theorem in Math?

In geometry, Hero's formula also known as the heron mathematician heron's theorem, is named after the Hero of Alexandria, which gives the area of a triangle when the length of all three sides are known. You can find the area of a triangle by just using the sides instead of the height of the triangle. Heron's theorem makes your life easier when you don't have the height of the triangle. Just plug your side lengths into Heron's formula, and you can find your area. Heron's formula consists of two parts. The first part is the half perimeter, s, which is equal to the sum of the sides divided by two.