Question

Derive the formula for height and area of an equilateral triangle.

Answer 1

Hint: We will first construct the figure of an equilateral triangle. As we know that equilateral triangles have sides of all equal length and equal angles of \[60^\circ \]. To determine the height, we can draw an altitude to one of the sides in order to split the triangle into two equal triangles. Let the side be \[a\] which is the hypotenuse of the obtained triangle. From here we have the sides of the triangle as \[x,x\sqrt 3, 2x\] where \[x\sqrt 3 \] represent the height so we will evaluate the value of \[x\] as \[a = 2x\] and then determine the value of height.
Next, we have to derive the area of an equilateral triangle, we will use the basic formula of triangle that is \[A = \dfrac{1}{2} \times {\text{base}} \times {\text{height}}\]. As we have already evaluated the height and base is the side we have let that is \[a\] so, we will substitute it in the formula and find the area of an equilateral triangle.

Complete step by step solution: We will first construct the figure of an equilateral triangle whose side is represented as \[a\].

As we know that equilateral triangles have sides of all equal length and angles of \[60^\circ \]. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equals \[30 - 60 - 90\] triangles.
Now, as we have let the side of the original equilateral triangle as \[a\] which shows the hypotenuse of the \[30 - 60 - 90\] triangle. As \[30 - 60 - 90\] triangle is a special type of triangle so, the sides of the triangle is shown as \[x,x\sqrt 3 \] and \[2x\] where \[2x\] shows the hypotenuse and \[x\sqrt 3 \] shows the height of the triangle.
Thus, from this, we get that \[a = 2x\]
Which further gives us \[x = \dfrac{a}{2}\]
Now, we will substitute it into the height of the triangle.
Thus, we get,
\[H = \dfrac{{a\sqrt 3 }}{2}\]
Next, we will derive the formula for the area of an equilateral triangle.
The sides of an equilateral triangle are shown by \[a\] units.
As we know that the area of triangle is given by: \[A = \dfrac{1}{2} \times {\text{base}} \times {\text{height}}\]
We already have the derived formula for the height of the triangle, and the base of the triangle is given as \[a\] units.
Thus, we get,
\[
   \Rightarrow A = \dfrac{1}{2} \times a \times \dfrac{{a\sqrt 3 }}{2} \\
   \Rightarrow A = \dfrac{{\sqrt 3 {a^2}}}{4} \\
 \]

Hence, we can conclude that the height is derived as \[\dfrac{{\sqrt 3 a}}{2}\] and area is derived as \[\dfrac{{\sqrt 3 {a^2}}}{4}\].

Note: Remember the basic formula for the area of the triangle. Do not get confused in the special triangle that is \[30 - 60 - 90\] triangle as when we have divided the triangle with an altitude so, the top angle gets divided into half each that is \[30^\circ \] and the base at which perpendicular drops make the \[90^\circ \] angle and the third one is of \[60^\circ \].