Area of Equilateral Triangle
An equilateral triangle is a triangle in which all three sides are equal and all three angles are 60°. Its area can be calculated using a special formula derived from Heron's Formula or from the standard base-height method.
The formula for the area of an equilateral triangle with side a is (√3/4)a². This is one of the most frequently used results in Class 9 and Class 10 Mathematics.
This formula eliminates the need to calculate the height separately. It is derived using Heron's Formula (as covered in the Class 9 NCERT chapter) and can also be obtained using the Pythagorean theorem to find the altitude.
What is Area of Equilateral Triangle?
Definition: An equilateral triangle is a triangle with all three sides of equal length. All interior angles are 60°.
Area Formula:
Area = (√3 / 4) × a²
Where:
- a = length of each side
- √3 ≈ 1.732
Related formulas:
- Perimeter: P = 3a
- Height (altitude): h = (√3 / 2) × a
- Semi-perimeter: s = 3a/2
Important:
- The altitude of an equilateral triangle bisects the base and passes through the centroid.
- In an equilateral triangle, the centroid, circumcentre, incentre, and orthocentre all coincide at the same point.
- The altitude is also the median, angle bisector, and perpendicular bisector.
Area of Equilateral Triangle Formula
Key Formulas for an Equilateral Triangle with side a:
1. Area:
Area = (√3 / 4) × a²
2. Height (altitude):
h = (√3 / 2) × a
3. Perimeter:
P = 3a
4. Circumradius (R):
- R = a / √3 = a√3 / 3
5. Inradius (r):
- r = a / (2√3) = a√3 / 6
6. Relationship between R and r:
- R = 2r (circumradius is twice the inradius)
7. Finding side from area:
- a = √(4 × Area / √3) = 2 × √(Area / √3)
Derivation and Proof
Derivation of the Area Formula
Method 1: Using Pythagoras Theorem
- Consider equilateral ▵ABC with side a.
- Draw altitude AD from A to BC. Since the triangle is equilateral, D is the midpoint of BC.
- BD = BC/2 = a/2.
- In right ▵ABD, apply Pythagoras theorem:
- AB² = AD² + BD²
- a² = AD² + (a/2)²
- a² = AD² + a²/4
- AD² = a² − a²/4 = 3a²/4
- AD = a√3 / 2 (altitude)
Now apply the standard area formula:
- Area = (1/2) × base × height
- Area = (1/2) × a × (a√3/2)
- Area = (√3/4) × a² ■
Method 2: Using Heron's Formula
- All three sides = a.
- Semi-perimeter: s = (a + a + a)/2 = 3a/2
- s − a = 3a/2 − a = a/2 (same for all three sides)
- Area = √[s(s − a)(s − b)(s − c)]
- Area = √[(3a/2)(a/2)(a/2)(a/2)]
- Area = √[3a&sup4;/16]
- Area = (a²/4) × √3
- Area = (√3/4) × a² ■
Types and Properties
Different Ways to Calculate the Area:
1. When side length is given
- Directly apply: Area = (√3/4) × a²
- This is the most common scenario.
2. When perimeter is given
- Find side: a = Perimeter / 3
- Then apply the area formula.
3. When altitude (height) is given
- Find side from height: a = 2h / √3
- Then apply the area formula.
- Or directly: Area = h² / √3
4. When area is given, find the side
- a² = 4 × Area / √3
- a = 2 × √(Area / √3)
5. When circumradius R is given
- a = R√3
- Area = (√3/4) × 3R² = (3√3/4) × R²
6. When inradius r is given
- a = 2r√3
- Area = (√3/4) × 12r² = 3√3 × r²
Solved Examples
Example 1: Example 1: Finding area from side length
Problem: Find the area of an equilateral triangle with side 8 cm.
Solution:
Given: a = 8 cm
Using the formula:
- Area = (√3/4) × a²
- Area = (√3/4) × 8²
- Area = (√3/4) × 64
- Area = 16√3
- Area = 16 × 1.732 = 27.71 sq cm
Answer: Area = 16√3 ≈ 27.71 sq cm.
Example 2: Example 2: Finding area from perimeter
Problem: The perimeter of an equilateral triangle is 36 cm. Find its area.
Solution:
Given: Perimeter = 36 cm
Finding the side:
- a = 36/3 = 12 cm
Finding the area:
- Area = (√3/4) × 12²
- Area = (√3/4) × 144
- Area = 36√3
- Area = 36 × 1.732 = 62.35 sq cm
Answer: Area = 36√3 ≈ 62.35 sq cm.
Example 3: Example 3: Finding the side from area
Problem: The area of an equilateral triangle is 25√3 sq cm. Find the side length.
Solution:
Given: Area = 25√3 sq cm
Using the formula:
- 25√3 = (√3/4) × a²
- a² = 25√3 × 4/√3
- a² = 100
- a = 10 cm
Answer: The side length is 10 cm.
Example 4: Example 4: Finding the altitude
Problem: Find the altitude of an equilateral triangle with side 14 cm.
Solution:
Given: a = 14 cm
Using the formula:
- h = (√3/2) × a
- h = (√3/2) × 14
- h = 7√3
- h = 7 × 1.732 = 12.12 cm
Answer: The altitude is 7√3 ≈ 12.12 cm.
Example 5: Example 5: Verifying with Heron's Formula
Problem: Find the area of an equilateral triangle with side 6 cm using (a) the direct formula and (b) Heron's Formula. Verify both give the same result.
Solution:
(a) Direct formula:
- Area = (√3/4) × 6² = (√3/4) × 36 = 9√3 ≈ 15.59 sq cm
(b) Heron's Formula:
- s = (6 + 6 + 6)/2 = 9
- s − a = s − b = s − c = 9 − 6 = 3
- Area = √(9 × 3 × 3 × 3) = √243 = √(81 × 3) = 9√3 ≈ 15.59 sq cm
Both methods give 9√3 sq cm ✔
Answer: Area = 9√3 ≈ 15.59 sq cm.
Example 6: Example 6: Cost of tiling
Problem: A triangular park in the shape of an equilateral triangle has a side of 20 m. Find the cost of planting grass at Rs 8 per square metre.
Solution:
Given: a = 20 m; cost = Rs 8 per sq m
Finding the area:
- Area = (√3/4) × 20²
- Area = (√3/4) × 400
- Area = 100√3 ≈ 173.2 sq m
Cost = 173.2 × 8 = Rs 1,385.60
Answer: The cost is approximately Rs 1,385.60.
Example 7: Example 7: Finding side from altitude
Problem: The altitude of an equilateral triangle is 12√3 cm. Find the side and the area.
Solution:
Given: h = 12√3 cm
Finding the side:
- h = (√3/2) × a
- 12√3 = (√3/2) × a
- a = 12√3 × 2/√3 = 24 cm
Finding the area:
- Area = (√3/4) × 24² = (√3/4) × 576 = 144√3 ≈ 249.41 sq cm
Answer: Side = 24 cm; Area = 144√3 ≈ 249.41 sq cm.
Example 8: Example 8: Ratio of areas
Problem: Two equilateral triangles have sides 5 cm and 10 cm. Find the ratio of their areas.
Solution:
Area of first triangle:
- A&sub1; = (√3/4) × 5² = 25√3/4
Area of second triangle:
- A&sub2; = (√3/4) × 10² = 100√3/4
Ratio:
- A&sub1; : A&sub2; = (25√3/4) : (100√3/4) = 25 : 100 = 1 : 4
Note: When sides are in ratio 1 : 2, areas are in ratio 1 : 4 (areas scale as the square of side ratio).
Answer: Ratio = 1 : 4.
Example 9: Example 9: Inscribed circle
Problem: An equilateral triangle has side 12 cm. Find the radius of the inscribed circle (inradius).
Solution:
Given: a = 12 cm
Using the inradius formula:
- r = a / (2√3)
- r = 12 / (2√3)
- r = 6/√3
- r = 6√3/3 = 2√3
- r ≈ 3.46 cm
Verification using Area = s × r:
- Area = (√3/4) × 144 = 36√3
- s = 18
- r = Area/s = 36√3/18 = 2√3 ✔
Answer: Inradius = 2√3 ≈ 3.46 cm.
Example 10: Example 10: Finding area using coordinates
Problem: An equilateral triangle has vertices at A(0, 0), B(6, 0), and C(3, 3√3). Verify the area using both the formula and the coordinate method.
Solution:
Side length:
- AB = √(36 + 0) = 6
Method 1 (formula):
- Area = (√3/4) × 6² = 9√3 ≈ 15.59 sq units
Method 2 (coordinate formula):
- Area = (1/2)|x&sub1;(y&sub2; − y&sub3;) + x&sub2;(y&sub3; − y&sub1;) + x&sub3;(y&sub1; − y&sub2;)|
- = (1/2)|0(0 − 3√3) + 6(3√3 − 0) + 3(0 − 0)|
- = (1/2)|0 + 18√3 + 0|
- = 9√3 ≈ 15.59 sq units
Both methods match ✔
Answer: Area = 9√3 ≈ 15.59 sq units.
Real-World Applications
Applications of the Equilateral Triangle Area Formula:
- Land measurement: Triangular plots with equal sides are calculated quickly using the direct formula.
- Construction and design: Equilateral triangles appear in truss structures, geodesic domes, and architectural patterns. The area formula is used for material estimation.
- Tiling and tessellation: Equilateral triangles tessellate the plane. The area formula determines how many tiles are needed to cover a surface.
- Traffic signs: Warning signs are equilateral triangles. The formula determines the sheet metal area needed.
- Hexagonal patterns: A regular hexagon is composed of six equilateral triangles. Hexagon area = 6 × equilateral triangle area.
- Crystallography: Many crystal structures have equilateral triangular cross-sections. The area formula is used in material science calculations.
Key Points to Remember
- Area of an equilateral triangle = (√3/4) × a², where a is the side length.
- Altitude (height) = (√3/2) × a.
- The formula is derived using either Pythagoras theorem or Heron's Formula.
- In an equilateral triangle, the centroid, circumcentre, incentre, and orthocentre all coincide.
- The altitude is also the median, angle bisector, and perpendicular bisector.
- Circumradius R = a/√3; Inradius r = a/(2√3); and R = 2r.
- If sides are in ratio m : n, areas are in ratio m² : n².
- All angles of an equilateral triangle are 60°.
- A regular hexagon has area = 6 × equilateral triangle area (with side equal to hexagon side).
- This formula is extensively used in Class 9 Heron's Formula chapter and in mensuration problems.
Practice Problems
- Find the area of an equilateral triangle with side 15 cm.
- The perimeter of an equilateral triangle is 48 cm. Find its area.
- The area of an equilateral triangle is 64√3 sq cm. Find the side length.
- Find the altitude of an equilateral triangle whose area is 100√3 sq cm.
- An equilateral triangle has the same perimeter as a square of side 9 cm. Which has a greater area?
- Find the area of an equilateral triangle whose altitude is 9 cm.
- The ratio of sides of two equilateral triangles is 3 : 5. Find the ratio of their areas.
- A regular hexagon has a side of 10 cm. Find its area using the equilateral triangle area formula.
Frequently Asked Questions
Q1. What is the formula for the area of an equilateral triangle?
Area = (√3/4) × a², where a is the length of each side. For example, if a = 10 cm, Area = (√3/4) × 100 = 25√3 ≈ 43.30 sq cm.
Q2. How is the formula derived?
Drop an altitude from one vertex to the opposite side. The altitude bisects the base (half = a/2). By Pythagoras: height = √(a² − a²/4) = a√3/2. Then Area = (1/2) × a × a√3/2 = (√3/4)a².
Q3. Can Heron's Formula be used for an equilateral triangle?
Yes. With all sides = a: s = 3a/2, each (s − a) = a/2. Area = √[(3a/2)(a/2)(a/2)(a/2)] = √(3a⁴/16) = (√3/4)a². Both methods give the same result.
Q4. What is the height of an equilateral triangle?
Height = (√3/2) × a. For example, if side = 12 cm, height = (√3/2) × 12 = 6√3 ≈ 10.39 cm.
Q5. How do you find the side if the area is given?
From Area = (√3/4)a², rearrange: a² = 4 × Area / √3, so a = 2√(Area / √3). For example, if Area = 16√3, then a² = 64, a = 8 cm.
Q6. What is the inradius of an equilateral triangle?
The inradius r = a / (2√3) = a√3/6. It can also be found using r = Area / s, where s is the semi-perimeter.
Q7. What is the circumradius of an equilateral triangle?
The circumradius R = a / √3 = a√3/3. The circumradius is always twice the inradius (R = 2r) in an equilateral triangle.
Q8. How does the area of an equilateral triangle compare to a square of the same perimeter?
The square has a larger area. For perimeter P: equilateral triangle area = (√3/4)(P/3)² = √3P²/36, square area = (P/4)² = P²/16. Since 1/16 > √3/36, the square's area is greater.
Q9. Is this formula in the NCERT Class 9 syllabus?
Yes. The area formula for an equilateral triangle is derived as a special case of Heron's Formula in NCERT Class 9 Chapter 12 (Heron's Formula).
Q10. How is the equilateral triangle formula used for hexagons?
A regular hexagon with side a is composed of 6 equilateral triangles, each with side a. Hexagon area = 6 × (√3/4)a² = (3√3/2)a².
