What is a Triangle?
A triangle is a closed figure with three sides. The sides of the triangle may or may not be equal. In our daily lives, you might have seen objects that are triangular in shape, like sandwiches, faces of pyramids, and clothes hangers.
Equivalent Triangle
A triangle in which all sides are congruent and the angles are equiangular is known as an equilateral triangle. The measurement of each angle in such triangles is 60°.
Height of an Equilateral Triangle
The perpendicular drawn from any vertex to the opposite side is known as the height of an equilateral triangle. The height also divides the opposite sides into two equal parts. We can calculate the height of an equilateral triangle by using the Pythagorean theorem.
From the triangle in the image, \(\triangle \)XWZ is a right triangle.
\(XZ^2=XW^2~+~WZ^2\) Pythagorean Theorem
\(a^2=h^2~+~\left(\frac{a}{2}\right )^2\) Substitute the values
\(h^2=a^2~-~\frac{a^2}{4}\) Simplify for \(h^2\)
\(h^2=\frac{3a^2}{4}\) Subtract
\(h=\frac {\sqrt 3}{2}a\) Taking square root each side
So, height h of equilateral triangle is, \(h=\frac {\sqrt 3}{2}a\), where a is the side length of the triangle.
Perimeter Formula of an Equilateral Triangle
The total measurement of the boundary covered by an equilateral triangle is known as the perimeter of an equilateral triangle. All sides of the equilateral triangle are congruent, so it is also a regular polygon of three sides. Therefore, the perimeter of the equilateral triangle will be three times of the side length.
Let the side length of the equilateral triangle is ‘a’ units then,
Perimeter of an equilateral triangle, P = a + a + a = 3a units.
Area of an Equilateral Triangle Formula
The surface or region covered by an equilateral triangle is known as the area of an equilateral triangle.
Area of a equilateral triangle: \(A=\frac {\sqrt 3}{4}a^2\) square units.
Rapid Recall
- Height of equilateral triangle, \(h=\frac {\sqrt 3}{2}a\) units.
- Perimeter of equilateral triangle, P = a + a + a = 3a units.
- Area of a equilateral triangle, \(A=\frac {\sqrt 3}{4}a^2\) square units.
Solved Examples
Example 1: Find the height of the equilateral triangle in the image.
Solution:
The side of an equilateral triangle is 7 cm.
\(h=\frac {\sqrt 3}{2}a\) Height of an equilateral triangle formula
\(h=\frac {\sqrt 3}{2}~\times~7\) Substitute 7 for a
h = 6.06 cm
So, the height of the equilateral triangle is 6.06 centimeters.
Example 2: John visited Egypt to see the pyramids. He was looking at a pyramid whose face was in the shape of an equilateral triangular and he was told that the height of the face of the equilateral triangle is \(90\sqrt 3\) m. Find the perimeter of one of the faces of the pyramid.
Solution:
The height of the face of the pyramid in the shape of an equilateral triangle is \(90\sqrt 3\) m.
Use the formula of height of the equilateral triangle to find the side length of the face.
\(h=\frac {\sqrt 3}{2}a\) Height of an equilateral triangle formula
\(90\sqrt 3=\frac {\sqrt 3}{2}~\times~a\) Substitute
a = 180 Simplify
So, the length of the side of one of the faces is 180 m.
The total length of the boundary of a face of the pyramid is the perimeter of the face of the pyramid with side 180 m.
P = 3a Formula of the perimeter of an equilateral triangle
P = 3 x 180 Substitute
P = 540 Multiply
So, the perimeter of one face of the pyramid is 540 meters.
Example 3: The perimeter of a sign board that is shaped like an equilateral triangle, is 57 inches. Find the area of the sign board.
Solution:
P = 3a Formula for the perimeter of an equilateral triangle
57 = 3 x a Substitute 57 for P
a = 19 Multiply
Hence, the side length of the sign board is 19 inches.
\(A=\frac {\sqrt 3}{4}a^2\) Write the formula for area of an equilateral triangle
\(A=\frac {\sqrt 3}{4}~\times~(19)^2\) Substitute 19 for a
A = 156.31 Simplify
So, the area of the sign board is 156.31 square inches.
A triangle is a closed shape that has three sides and the sum of the interior angles is 180°.
On the basis of the measure of each side, triangles are classified into:
- Scalene triangles
- Isosceles triangles
- Equilateral triangles
A triangle in which all the sides and all angles are equal, is called an equilateral triangle.
Each angle of an equilateral triangle measures 60 degrees.