Geometry - Equilateral Triangle
- What is an Equilateral Triangle?
- Properties of Equilateral Triangles
- Examples of Equilateral Triangles in Real Life
- Area and Perimeter of Equilateral Triangles
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What is an Equilateral Triangle?
A triangle having all sides of equal measure is called an Equilateral Triangle. All the angles inside an equilateral triangle are equal to `60^\circ`.
The word equilateral is derived from two words equi meaning equal and lateral meaning sides. Therefore, an equilateral triangle is a triangle with all sides equal.
Properties of Equilateral Triangles
`1`. All the sides of an equilateral triangle have the same length, `a`.
`2`. All the internal angles of an equilateral triangle are equal to `60°`.
`3`. An equilateral triangle is a symmetrical figure (if an equilateral triangle is divided into two halves, the two parts would overlap each other.)
Examples of Equilateral Triangles in Real Life
`1`. A traffic sign board that can be seen on the roads resembles an equilateral triangle.
`2` The most popular snack in the world Nachos are in the shape of an equilateral triangle.
`3` The faces of the great pyramids of Egypt are also in the shape of an equilateral triangle.
Area and Perimeter of Equilateral Triangles
Perimeter: The perimeter of an equilateral triangle can be found by adding all its sides together. If the equilateral triangle has the length of the side as ‘`a`’ then the perimeter of the equilateral triangle is `a + a + a = 3a`.
Area: The area of an equilateral triangle of side ‘`a`’ is found by using the formula : `\sqrt{3}/4 \times a^2`
Solved Examples
Example `1`: Find the perimeter of an equilateral triangle having its sides equal to `16` cm.
Solution:
Perimeter of an equilateral triangle `= 3 \times ` side length `= 3 \times 16 = 48`
Therefore, the perimeter of an equilateral triangle is `48` cm.
Example `2`: Find out the area of an equilateral triangle with its sides measuring `30` cm.
Solution:
Side length = `30` cm.
\(\begin{align*}
\text{Area of the equilateral triangle} & = \frac{\sqrt{3}}{4} \times \text{Side length}^2 \\
& = \frac{\sqrt{3}}{4} \times 30^2 \\
& = \frac{\sqrt{3}}{4} \times 900 \\
& = \frac{900\sqrt{3}}{4}\\
& = 225\sqrt{3}
\end{align*}\)
Therefore, the area of the equilateral triangle is `225\sqrt{3}` `\text{cm}^2`.
Practice Problems
Q`1`. What is the area of an equilateral triangle having measurement of the side equal to `9` cm?
- \( \frac{27\sqrt{3}}{4} \, \text{cm}^2 \)
- \( \frac{36\sqrt{3}}{4} \, \text{cm}^2 \)
- \( \frac{81\sqrt{3}}{4} \, \text{cm}^2 \)
- \( \frac{45\sqrt{3}}{4} \, \text{cm}^2 \)
Answer: c
Q`2`. Find out the area of an equilateral triangle having a perimeter equal to `33` cm.
- \( \frac{99\sqrt{3}}{4} \, \text{cm}^2 \)
- \( \frac{121\sqrt{3}}{4} \, \text{cm}^2 \)
- \( \frac{165\sqrt{3}}{4} \, \text{cm}^2 \)
- \( \frac{198\sqrt{3}}{4} \, \text{cm}^2 \)
Answer: b
Q`3`. Find the perimeter of an equilateral triangle having a side equal to `8` cm.
- \(24 \, \text{cm}\)
- \(16 \, \text{cm}\)
- \(32 \, \text{cm}\)
- \(12 \, \text{cm}\)
Answer: a
Frequently Asked Questions
Q1. What is an equilateral triangle?
Answer: An equilateral triangle is a type of triangle that has all three sides of equal length. In other words, all three angles and sides are identical in measurement.
Q`2`. How do you find the area of an equilateral triangle?
Answer: To find the area of an equilateral triangle, you can use the formula:
\( \text{Area} = \frac{\sqrt{3}}{4} \times \text{side length}^2 \)
Q`3`. What is the relationship between the side length and the height of an equilateral triangle?
Answer: In an equilateral triangle, the height (distance from any vertex to the midpoint of the opposite side) is equal to \( \frac{\sqrt{3}}{2} \) times the side length.
Q`4`. Can an equilateral triangle also be an isosceles triangle?
Answer: Yes, an equilateral triangle is a special case of an isosceles triangle where all sides are equal. So, every equilateral triangle is also an isosceles triangle.
Q`5`. What is the sum of interior angles in an equilateral triangle?
Answer: The sum of interior angles in any triangle is always `180^\circ`. In an equilateral triangle, each angle measures `60^\circ`, so the sum of all three interior angles is `180^\circ`.