Area Of A Triangle

Most of us are already acquainted with the term area. It is defined as the region occupied inside the boundary of a flat object or figure. The measurement is done in square units with the standard unit being square meters (m2). For the computation of area, there are pre-defined formulas for squares, rectangles, circle, triangles, etc. In case of a triangle, the area of a triangle are defined as the total space that is enclosed by that particular triangle. In this article, we would discuss the area of triangles for the right-angled triangle, equilateral triangle, and isosceles triangle. For this, we would use the formula for the area of a triangle on different types of triangles and see if we are able to obtain their area.

Area of a Triangle

Perimeter and Area of Triangle

The area of the triangle  is given by the formula mentioned below:

    • Area of a Triangle = A = ½ (b × h)

 

  • Perimeter of a triangle = P = A + B + C, where A,B,C are the three sides of a triangle

 

This is the general area formula for triangles (right-angled and isosceles). Now, the area formulas for all the different types of triangles are explained below.

Area of a Triangle with Three Sides

The area of triangle can be found using Heron’s formula if all the three sides of the triangle are given. Heron’s formula needs the measurement of all three sides of the triangle as it is used to find the semi perimeter first and then using the same in the main formula to find the area of the triangle.

How to calculate area of triangle

Use the following formulas to calculate the area of triangle. There are different formulas for calculating the area of an equilateral triangle, right-angled triangle, and isosceles triangle

Triangle Type Formula for the Area
Right Angled Triangle A = ½ (b × h)
Equilateral Triangle A = (√3)/4 × a2
Isosceles Triangle A = ½ (b × h)

Area of a Right Triangle

When we consider a right-angled triangle, its area could be computed directly. Two of its sides that we consider is the base and the height. So, the formula for the area of a right-angled triangle is calculated as:

  • Area of a Right-angled Triangle = A = ½ × Base × Height

Example: Find the area of a right-angled triangle having two sides equal to 3 cm and 4 cm.

Solution:

For determining the area of a right-angled triangle, it is not essentially required to know which side is base and which one is height. We can easily determine the area if we know the length of both sides. Considering any of the ones to be base and other to be height, we have

Area of Triangle = A = ½ (b × h)

Area =(½) × 4 × 3 = 6 cm2

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

Solution:

We know,

A = (½) × b × h

⇒ A = (½) ×(15 in) × (4 in)

⇒ A = (½) × (60 in2)

⇒ A = 30 in2

Check all the right triangle formulas from this article which includes the formulas to calculate perimeter, area, and hypotenuse of a right-angled triangle along with solved example questions.

Equilateral Triangle Area

The formula for the equilateral triangle is given by

  • Area of an Equilateral Triangle = A = (√3)/4 × a2, where a is the side of the triangle.

For an equilateral triangle area, the height of the triangle is to be found first. For doing so, we drop a perpendicular from one of its vertices (we know that height of an equilateral triangle is the same as the perpendicular bisector of one of its sides). In the ∆ABC, CD represents the height.

Area of an Equilateral Triangle

Given- Side of an equilateral triangle to be 8 cm. For calculating CD, we consider the right-angled ∆ACD.

We have,

AC2 = AD2 + CD2

Or,

AD2 = AC2 – CD2

⇒ AD2 = 64 – 16 = 48

AD = √48 = 4√3 cm

Thus,

Area of ∆ ABC =

Area = (½) × base × height = 12 × 8 × 4√3 cm2

Area = 16√3 cm2

Check out more formulas of an equilateral triangle which includes the area, perimeter, semi-perimeter, and the height of an equilateral triangle along with solved example questions.

Isosceles Triangle Area:

The formula for an isosceles triangle area is similar to that of a right triangle.

  • Area of an Isosceles Triangle = A = ½ (b × h), where b is base and h is the height of the triangle.

All the formulas for an isosceles triangle can be checked from here and the derivation of the area is given below.

Similar to the equilateral triangle, we find the height of the triangle first to compute its area. Note that the height of the ABC could be obtained by drawing the perpendicular line AD from A which bisects the base AC at right angles. Now, in the right-angled ∆ABD, we find AD by,

AD2 = AB2 – BD2

An isosceles triangle has two equal sides. Consider sides AB = AC to be 5 cm and BC to be 6 cm.

AD2 = 25 – 9 = 16

AD = √16 = 4 cm

Thus, area of ∆ABC = (½) × Base × Height = 12 × 6 × 4 = 24 cm2

We have seen that the area of special triangles could be obtained using the formula. However, for a triangle with the sides being given usually, calculation of height would not be simple. For the same reason, we rely on Heron’s Formula to calculate the area of the triangles with unequal lengths i.e. the equilateral triangles. So, the area of an isosceles triangle formula is given as:

  • Area of an Isosceles Triangle = A = ½ (b × h)

Additional Lessons on Triangle Formulas

Practice Questions from Triangle Areas:

Example 1: Find the area of the acute triangle with a base of 13 inches and a height of 5 inches.

Solution:

A = (½)× b × h

⇒ A = (½) × (13 in) × (5 in)

⇒ A = (½) × (65 in2)

⇒ A = 32.5 in2

Example 2: Find the area of the right angled triangle with a base of 7cm and a height of 8cm.

Solution:

A = (½) × b × h

⇒ A = (½) × (13 in) × (5 in)

⇒ A = (½) × (65 in2)

⇒ A = 32.5 in2

Example 3: Find the area of the right angled triangle with a base of 7cm and a height of 8cm.

Solution:

A = (½) × b × h

⇒ A = (½) × (7 cm) × (8 cm)

⇒ A = (½) × (56 cm)

⇒ A = 28 cm2

Example 4: Find the area of obtuse-angled triangle with a base of 4cm and a height 7cm.

Solution:

A = (½) × b × h

⇒ A = (½) × (4 cm) × (7 cm)

⇒ A = (½) × (28 cm)

⇒ A = 14 cm2

Example 5: The area of the triangular shaped field is 24 square feet and its height is 6 feet. Find the base.

Solution:

⇒ A = (½) × b × h

⇒ 24 ft2 = 12 × b × 6 ft

On multiplying both sides of the equation by 2, we get:

48 ft2 = b × 6 ft

Dividing both sides of the equation by 6 ft, we get:

4 ft = b

Commuting this equation, we get: b = 4 ft

Therefore, Base = 4 feet

More Triangles Related Articles

To learn more about the computation of the area of the triangle using Heron’s Formula, keep visiting BYJU’S.


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