Area of an equilateral triangle | When we talk about triangles, the equilateral triangle is one of the most simple yet beautiful shapes in geometry. All its three sides are equal, and each angle is exactly 60°. Finding its area is an important concept in maths, and once you understand the logic, you can easily solve any related problem. In this post, we’ll learn everything about the area of equilateral triangle — from its formula to real-life uses, step-by-step examples, and some common mistakes students make.
Table of Contents
What is an Equilateral Triangle?
An equilateral triangle is a special type of triangle where all sides are of the same length, and all three interior angles are equal to 60°. Because of its perfect symmetry, it’s often used in designs, tiling patterns, and even in structures like towers and frames.
For example, if a triangle has all sides of 5 cm, then it’s equilateral.
This is different from an isosceles triangle, which has only two sides equal, and a scalene triangle, where no sides are equal.
Why Learn the Area of Equilateral Triangle?
You might wonder, why bother learning this formula when we already know how to find the area of any triangle?
Well, the equilateral triangle has its own simple and elegant formula — one that doesn’t even need height if you know the side length.
Knowing the area of equilateral triangle helps in:
- Solving geometry and mensuration problems quickly
- Designing patterns and shapes (tiles, logos, flags, etc.)
- Construction and architectural planning
- Understanding the base concept for other regular polygons (like hexagons)
Formula for Area of Equilateral Triangle
The standard formula to find the area is: Area=34a2\text{Area} = \frac{\sqrt{3}}{4} a^2Area=43a2
Here:
- aaa = side length of the triangle
- 3\sqrt{3}3 ≈ 1.732
- The result gives the area in square units (like cm², m², etc.)
Example:
If each side of an equilateral triangle is 6 cm, Area=34×62=1.7324×36=15.588 $ cm2\text{Area} = \frac{\sqrt{3}}{4} \times 6^2 = \frac{1.732}{4} \times 36 = 15.588 \text{ cm}^2Area=43×62=41.732×36=15.588 cm2 $
Hence, the area ≈ 15.6 cm²
Derivation of the Formula
Let’s see how this simple-looking formula comes from basic geometry.
Method 1: Using Height and Base
- Draw an equilateral triangle ABC with each side = aaa.
- Drop a perpendicular line AD from vertex A to base BC.
This line acts as the height (h) of the triangle. - It divides the triangle into two equal right triangles, ABD and ADC.
- Using Pythagoras theorem in one of the triangles:
$ h2+(a2)2=a2h^2 + \left(\frac{a}{2}\right)^2 = a^2h2+(2a)2=a2 h2=a2−a24=3a24h^2 = a^2 – \frac{a^2}{4} = \frac{3a^2}{4}h2=a2−4a2=43a2 h=32ah = \frac{\sqrt{3}}{2}ah=23a $
- Area of triangle:
Area=12×a×h=12×a×32a=34a2\text{Area} = \frac{1}{2} \times a \times h = \frac{1}{2} \times a \times \frac{\sqrt{3}}{2}a = \frac{\sqrt{3}}{4}a^2Area=21×a×h=21×a×23a=43a2
And that’s exactly our formula!
Method 2: Using Trigonometry
$ Area=12a2sin(60°)\text{Area} = \frac{1}{2} a^2 \sin(60°)Area=21a2sin(60°)
Since sin(60°)=32\sin(60°) = \frac{\sqrt{3}}{2}sin(60°)=23, Area=12×a2×32=34a2\text{Area} = \frac{1}{2} \times a^2 \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4}a^2Area=21×a2×23=43a2 $
So no matter which way you use, the formula stays the same.
Examples
Let’s solve some problems step-by-step.
Example 1:
$ Find the area of equilateral triangle whose side is 10 cm. Area=34a2=1.7324×102=43.3 cm2\text{Area} = \frac{\sqrt{3}}{4} a^2 = \frac{1.732}{4} \times 10^2 = 43.3 \text{ cm}^2Area=43a2=41.732×102=43.3 cm2 $
✅ Answer: 43.3 cm²
Example 2:
Find the area if the perimeter of an equilateral triangle is $ 18 cm. P=3a⇒a=183=6P = 3a \Rightarrow a = \frac{18}{3} = 6P=3a⇒a=318=6 Area=34×62=15.6 cm2\text{Area} = \frac{\sqrt{3}}{4} \times 6^2 = 15.6 \text{ cm}^2Area=43×62=15.6 cm2 $
✅ Answer: 15.6 cm²
Example 3:
If the height of an equilateral triangle is 8.66 cm, find its area. h=32a⇒a=2h3=2×8.661.732≈10h = \frac{\sqrt{3}}{2}a \Rightarrow a = \frac{2h}{\sqrt{3}} = \frac{2 \times 8.66}{1.732} \approx 10h=23a⇒a=32h=1.7322×8.66≈10 Area=34a2=1.7324×100=43.3 cm2\text{Area} = \frac{\sqrt{3}}{4} a^2 = \frac{1.732}{4} \times 100 = 43.3 \text{ cm}^2Area=43a2=41.732×100=43.3 cm2
✅ Answer: 43.3 cm²
Area in Terms of Perimeter
If you know the perimeter $ PPP, a=P3a = \frac{P}{3}a=3P $
Substitute this in the main formula:$ Area=34(P3)2=3P236\text{Area} = \frac{\sqrt{3}}{4} \left(\frac{P}{3}\right)^2 = \frac{\sqrt{3}P^2}{36}Area=43(3P)2=363P2 $
Common Mistakes Students Make
Here are a few small errors students often make while solving these:
- Using the wrong triangle (not all sides equal!)
- Forgetting to square the side length
- Missing the $ √3/4 $ factor
- Writing units incorrectly (area should be in square units)
- Rounding $ √3 $ too early, which leads to wrong decimal answers
Always double-check your steps to avoid silly mistakes.
Real-Life Applications
The concept of the equilateral triangle isn’t limited to textbooks!
You can find it in:
- Architecture: Roofs, bridges, and design elements
- Tiling and flooring: Triangular tiles for aesthetic look
- Logos and art: Many brand logos are based on equilateral geometry
- Mathematics and physics: Used in vector representation, trusses, etc.
Understanding the area of equilateral triangle helps you relate geometry with real-world design and creativity.
Frequently Asked Questions
Q1. What is the formula for area of equilateral triangle?
→ $ Area=34a2\text{Area} = \frac{\sqrt{3}}{4}a^2Area=43a2 $
Q2. What if sides are not equal?
→ Then it’s not equilateral; use Heron’s formula instead.
Q3. Can I find area from height?
→ $ Yes, use a=2h3a = \frac{2h}{\sqrt{3}}a=32h and then plug into the formula. $
Q4. What is the value of √3?
→ It’s approximately 1.732.
Q5. In what unit is area measured?
→ Always in square units (cm², m², etc.)
Summary
So now we know:
- An equilateral triangle has three equal sides and equal angles.
- The formula for its area is $ 34a2\frac{\sqrt{3}}{4}a^243a2. $
- It can also be derived using trigonometry or height.
- Always remember to include square units in your final answer.
Understanding this formula once makes solving geometry problems a lot easier later on!
Try It Yourself
Find the area of an equilateral triangle whose side is 15 cm.
(Hint: Plug into the formula and see what you get!)
Conclusion
The area of equilateral triangle is a simple but powerful formula every student should know. It’s not just about maths — it’s about visualising geometry around you. From simple tiles to huge constructions, this formula finds its place everywhere.
Keep practising, and soon you’ll remember it without even trying!
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