A triangle is a two-dimensional geometrical shape with three sides. It is a polygon of three sides. It also has three vertices, three sides and three angles.
The shape of traffic signals, a clothes hanger, Light wooden clothes hanger etc. are examples of the shape of a triangle.
The triangles that we see around us or what we study in the classroom are classified into two types The first type is based on the sides, and the second type is based on the angles of a triangle.
Types of triangles based on sides:
There are three types of triangles based on the sides:
Scalene triangle: A triangle in which the three sides are unequal is called a scalene triangle.
Isosceles triangle: A triangle in which two sides are equal and the third side is of a different length is called an isosceles triangle. The two base angles of an isosceles triangle are also equal.
Equilateral triangle: A triangle with three equal sides is called an equilateral triangle. The three angles of an equilateral triangle are also equal in measure.
1) A triangle has three vertices, three sides and three angles.
2) The sum of the length of any two sides of a triangle is always greater than the length of the third side of the triangle.
3) The angle sum or the summation of the three angles is always equal to two right angles or \(180^{\circ }\)
The area of a triangle is the space occupied by the sides of a triangle in a two-dimensional plane. It is measured in square units like square inches, square meters, square yards, etc. For any triangle, the area is calculated as half of the product of the base and the height of the triangle.
The area of a triangle can also be calculated from the area of the parallelogram. We draw a parallelogram and cut the parallelogram into two identical triangles along one of the diagonals .The area of one of these triangles is half the area of the parallelogram.
Area of triangle = \(\frac{1}{2}\times \text{b}\times \text{h}\)
The area (A) of a triangle is one-half of the product of its base (b) and height (h). The area of the triangle formula in an algebraic form is:
Area of triangle (A) = \(\frac{1}{2}\times \text{b}\times \text{h}\)
Where “b” is the base of a triangle and “h” is the height of a triangle.
Example 1: Find the area of the triangle whose base is 8 inches long and height is 12 inches.
Solution:
Area of triangle = \(\frac{1}{2}\times \text{b}\times \text{h}\)
= \(\frac{1}{2}\) (8) (12) [Substitute values of b and h]
= \(\frac{1}{2}\)(96) [Multiply]
= 48 [Multiply]
The area of the triangle is 48 square inches.
Example 2:
Find the area of the triangle shown in the figure.
Solution:
The height of the triangle is 10 feet and the base is 8 feet. Therefore, we use the area of a triangle formula to calculate the area,
Area of triangle = \(\frac{1}{2}\times \text{b}\times \text{h}\)
= \(\frac{1}{2}\) (8) (10) [Substitute values of b and h]
= \(\frac{1}{2}\)(80) [Multiply]
= 40 [Multiply]
The area of the triangle is 40 square feet.
Example 3:
The area of the triangle is 48 square inches. Find the height of the triangle if the base is 8 inches long.
Solution:
The area of the triangle is 48 square inches and the base is 8 inches. Therefore, we use the area of a triangle formula to form an equation to find height
Area of triangle = \(\frac{1}{2}\times \text{b}\times \text{h}\)
\(\frac{1}{2}\) (8) (h) = 48 [Substitute values of b and h]
(8) (h) = 96 [Multiply by 2 both sides]
h = 12 [Divide both sides by 8]
The height of the triangle is 12 inches.
Example 4: Larry has brought a plot in New York. The plot is in the shape of a triangle as shown in the diagram. Find the area of the plot.
Solution:
The plot is in the shape of a triangle. The base of the triangle is 50 yards and the height of the triangle is 100 yards. We use the area of a triangle formula to calculate the area,
Area of triangle = \(\frac{1}{2}\times \text{b}\times \text{h}\)
= \(\frac{1}{2}\) (50) (100) [Substitute values of b and h]
= \(\frac{1}{2}\) (5000) [Multiply]
= 2500 [Multiply]
The area of the triangular plot is about 2500 square yards.
Example 5: There is a triangular wall poster on a street wall. Find the area of the poster.
Solution:
The poster is in the shape of a triangle. The base of the triangle is 20 inches and the height of the triangle is about 50 inches. We use the area of a triangle formula to calculate the area,
Area of triangle = \(\frac{1}{2}\times \text{b}\times \text{h}\)
= \(\frac{1}{2}\) (20) (50) [Substitute values of b and h]
= \(\frac{1}{2}\) (1000) [Multiply]
= 500 [Multiply]
The area of the poster is about 500 square inches.
Heron’s formula is applied to find the area of a triangle when the lengths of all the three sides of a triangle are known.
If a, b, and c are sides of a triangle,
Area of the triangle (A) = \( \sqrt{\text{s(s-a)(s-b)(s-c)}}\)
Here S = \( \frac{\text{a+b+c}}{2}\) is the semi-perimeter of the triangle.
The area of a equilateral triangle formula is,
A = \(\sqrt{\frac{3}{4}}\times a^{2}\)
Where ‘a’ is the side of the equilateral triangle.
The area formula for a right angled triangle is,
Area of triangle = \(\frac{1}{2}\times \text{b}\times \text{h}\)
Where ‘h’ is the altitude or height of the triangle and ‘b’ is the base of the triangle.
However the area for any other triangle when the three sides are given is,
Area of triangle A = \(\sqrt{s(s-a)(s-b)(s-c)}\)
in which “s” is the semi-perimeter and\(s=\frac{a+b+c}{2}\), and a,b,c are the lengths of the sides of the triangle.