Area of Equilateral Triangle

What is an Equilateral Triangle?

An equilateral triangle is a triangle where all three sides are the same length.

Each interior angle is exactly 60 degrees, making it a perfectly symmetrical shape.

Because it is so uniform, it is often used in everyday things, like:

• Road signs (e.g., yield signs)

• Roof trusses in buildings

• Origami and geometric designs

• Molecular and atomic diagrams in science

 

Table of Content

• Properties of an Equilateral Triangle
• General Triangle Area Formula
• Derivation of Area Formula for Equilateral Triangle
• Worked Example
• Area Using Trigonometry
• Practice Questions
• Real-Life Applications
• Conclusion
• FAQs on Area of Equilateral Triangle

 

Properties of an Equilateral Triangle

Know the properties of an equilateral triangle to make solving geometry problems a piece of cake and also to enhance conceptual understanding. Here are the major properties, each of which is discussed in detail:

1. Three sides of equal length

• In an equilateral triangle, all three sides are equal in measurement.

• That means if one side is 5 cm, then the second and third sides are also 5 cm.

• This equal side length leads to the triangle being perfectly balanced and symmetrical in all directions.

 

2. All three internal angles are equal, each measuring 60°

• The sum of interior angles in any triangle is always 180°.

• Since all sides in an equilateral triangle are the same, the angles opposite those sides must also be the same.

• Therefore, each angle is:

180° ÷ 3 = 60°

• This property makes the triangle equiangular (equal angles), just like it's equilateral (equal sides).

 

3. All medians, altitudes, angle bisectors, and perpendicular bisectors are all equal and meet at the same point (the centroid)

In general triangles, these segments (median, altitude, etc.) vary.

But in an equilateral triangle:

• Median: Line from a vertex to the midpoint of the opposite side.

• Altitude: Perpendicular dropped from a vertex to opposite side.

• Angle bisector: Creates two equal portions of the angle.

• Perpendicular bisector: Splits a side into two equal halves perpendicularly.

In an equilateral triangle, all three of these are the same line from every vertex, because of its ideal symmetry.

They all meet at the centroid, which is the center of gravity or point of balance of the triangle.

 

4. Can be split into two 30°-60°-90° right triangles by dropping an altitude from the highest vertex

Constructing an altitude (height) from any vertex to the opposite side (base) produces two congruent right triangles.

Both right triangles have:

• One 90° angle (at the base upon which the height intersects it)

• One 60° angle

• One 30° angle

These 30°-60°-90° triangles are widely applicable in trigonometry and can be utilized to derive formulas such as height and area.

 

5. Has line and rotational symmetry and thus is a regular polygon

• Line symmetry: An equilateral triangle possesses 3 lines of symmetry, every one of which goes through a vertex and the center of the opposite side.

• Rotational symmetry: The triangle appears the same after rotating 120°, 240°, or 360° around its center.

• This symmetry makes the triangle a regular polygon, which means:

• All angles and sides are equal

• It is balanced from any angle

• It seems uniform and predictable in design and structure

 

General Triangle Area Formula

The general formula to calculate the area of any triangle is:

Area = ½ × base × height

In the case of an equilateral triangle, the height isn't explicitly given but can be obtained through geometry or trigonometry approaches.

 

Derivation of Area Formula for Equilateral Triangle

Let a be the length of every side of the equilateral triangle.

Drop a perpendicular from the vertex at the top onto the base. This cuts the triangle into two right-angled triangles.

Use the Pythagorean Theorem to calculate height (h):

Height (h) = (√3 / 2) × a

Now, plug in to the area formula:

Area = ½ × base × height

     = ½ × a × (√3 / 2) × a

     = (√3 / 4) × a²

 

Solved Example

Given: Side length = 6 cm

Solution:

Area = (√3 / 4) × a²

= (√3 / 4) × 6²

     = (√3 / 4) × 36

     = 9√3 cm² ≈ 15.59 cm²

 

Area Using Trigonometry

The area can also be found using the sine function:

Area = ½ × a × a × sin(60°)

Since sin(60°) = √3 / 2, it becomes:

Area = ½ × a² × (√3 / 2)

     = (√3 / 4) × a²

This ensures that the trigonometric method and geometric method both produce the same result.

 

Practice Questions

1. Calculate the area of an equilateral triangle with side length = 10 cm.

2. If the area is 16√3 cm², determine the side length.

3. Apply both the normal formula and the trigonometric approach to determine the area for side = 14 cm.

4. Obtain the formula for height by the Pythagorean Theorem.

5. If the perimeter of a triangle is 21 cm, calculate its area.

 

Real-Life Applications

• Architecture: Applied in trusses, roof construction, and bridges for supporting strength.

• Mathematics: Encountered in geometry, congruence, and symmetry.

• Engineering: In load-bearing frames, mechanical engineering.

• Design and Art: Frequent in logos, branding, mandalas, and ornaments.

• Science: Observed in molecular shapes and illustrations of chemical bonds.

 

Conclusion

An equilateral triangle possesses all three sides and angles identical.

• The standard formula is:

Area = (√3 / 4) × a²

• The height can be expressed as:

Height = (√3 / 2) × a

Both geometric and trig methods give the same answer.

This triangle is not only mathematically beautiful but also utilized in numerous real-life designs and models.

 

Frequently Asked Questions on Area of Equilateral Triangle

1. What is the formula for the area of an equilateral triangle?

Answer: Area = (√3 / 4) × a², where a is the length of a side.

 

2. What is the area of an equilateral triangle with side = √3 / 4?

Answer: Using the formula:
Area = (√3 / 4) × (√3 / 4)²
= (√3 / 4) × (3 / 16)
= 3√3 / 64 units²

 

3. What is the formula for the area of a triangle with 3 equal sides?

Answer: This is the same as the area of an equilateral triangle:
Area = (√3 / 4) × a²

 

4. What is the area of an isosceles triangle?

Answer: Area = ½ × base × height
If only side lengths are given, you can use Heron’s formula to calculate the area.

 

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