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 $$A={\displaystyle \frac{1}{2}}\times \text{base}\times \text{height}$$. 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 $$x={\displaystyle \frac{a}{2}}$$
Now, we will substitute it into the height of the triangle.
Thus, we get,
$$H={\displaystyle \frac{a\sqrt{3}}{2}}$$
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: $$A={\displaystyle \frac{1}{2}}\times \text{base}\times \text{height}$$
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,
$$\begin{gathered} \Rightarrow A={\displaystyle \frac{1}{2}}\times a\times {\displaystyle \frac{a\sqrt{3}}{2}} \\ \Rightarrow A={\displaystyle \frac{\sqrt{3}{a}^2}{4}} \end{gathered}$$

Hence, we can conclude that the height is derived as $${\displaystyle \frac{\sqrt{3}a}{2}}$$ and area is derived as $${\displaystyle \frac{\sqrt{3}{a}^2}{4}}$$.

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 $${60}^{\circ }$$.