13. Equilateral Triangle Formulas
An equilateral triangle is a triangle in which all three sides are equal.
If each side of an equilateral triangle is a, then:
- All three sides are equal to a.
- Each interior angle is 60°.
- The altitude divides the triangle into two equal right-angled triangles.
The provided formula sheet gives the following important formulas:
- Area
- Perimeter
- Height
- Inradius
- Circumradius
1. Diagram of an Equilateral Triangle
a = side | h = height / altitude
2. Important Properties of an Equilateral Triangle
- All three sides are equal.
- All three angles are equal to 60°.
- The altitude is also a median and angle bisector.
- The altitude divides the triangle into two equal right-angled triangles.
- The perpendicular height is: h = (√3/2)a.
3. Symbols Used
| Symbol | Meaning |
|---|---|
| a | Side of the equilateral triangle |
| h | Height / altitude |
| r | Inradius |
| R | Circumradius |
4. Area of an Equilateral Triangle
The area formula given in the formula sheet is:
Here a is the length of one side.
Example
Find the area of an equilateral triangle whose side is 8 cm.
Substitute a = 8:
= (√3 / 4) × 8²
= (√3 / 4) × 64
= 16√3
Approximately:
5. Perimeter of an Equilateral Triangle
Since all three sides are equal:
Example
The side of an equilateral triangle is 12 cm. Find its perimeter.
= 3 × 12
Answer: 36 cm
6. Height of an Equilateral Triangle
The height formula given in the PDF is:
where a is the side length.
Why Does This Formula Work?
When the altitude is drawn, it divides the equilateral triangle into two equal right-angled triangles.
The hypotenuse of each smaller triangle is a, while half of the base is a/2.
Using Pythagoras theorem:
= a² − a²/4
= 3a²/4
Therefore:
Example
Find the height of an equilateral triangle whose side is 10 cm.
= (√3 / 2) × 10
= 5√3
Approximately:
7. Inradius of an Equilateral Triangle
The formula sheet gives the inradius as:
It also gives the equivalent relation:
Therefore:
Example 1: Using Side
Find the inradius of an equilateral triangle whose side is 12 cm.
= 12 / (2√3)
= 6/√3
= 2√3
Approximately:
Example 2: Using Height
If the height of an equilateral triangle is 9 cm, find its inradius.
= 9 / 3
Answer: 3 cm
8. Circumradius of an Equilateral Triangle
The circumradius formula given in the PDF is:
The PDF also gives the equivalent relation:
Therefore:
Example 1: Using Side
Find the circumradius of an equilateral triangle whose side is 6 cm.
= 6 / √3
= 2√3
Approximately:
Example 2: Using Height
The height of an equilateral triangle is 12 cm. Find its circumradius.
= (2 × 12) / 3
= 24 / 3
Answer: 8 cm
9. Relationship Between Inradius and Circumradius
From the formulas:
Therefore:
So, for an equilateral triangle, the circumradius is twice the inradius.
Example
If the inradius of an equilateral triangle is 5 cm, find its circumradius.
= 2 × 5
10. Area Using Base and Height
The general triangle area formula can also be applied to an equilateral triangle:
Since:
substituting gives:
Thus, both methods give the same area.
11. Complete Solved Example
An equilateral triangle has a side of 12 cm. Find:
- Perimeter
- Height
- Area
- Inradius
- Circumradius
Step 1: Perimeter
= 3 × 12
Step 2: Height
= (√3/2) × 12
Step 3: Area
= (√3/4) × 12²
= (√3/4) × 144
Step 4: Inradius
= 12/(2√3)
= 6/√3
Step 5: Circumradius
= 12/√3
Final Answers
- Perimeter = 36 cm
- Height = 6√3 cm
- Area = 36√3 cm²
- Inradius = 2√3 cm
- Circumradius = 4√3 cm
12. Which Formula Should You Use?
| Given Information | Required Quantity | Formula |
|---|---|---|
| Side a | Area | (√3/4)a² |
| Side a | Perimeter | 3a |
| Side a | Height | (√3/2)a |
| Side a | Inradius | a/(2√3) |
| Height h | Inradius | h/3 |
| Side a | Circumradius | a/√3 |
| Height h | Circumradius | 2h/3 |
13. Important Equilateral Triangle Relations
14. Common Mistakes
Mistake 1: Forgetting that all sides are equal
For an equilateral triangle:
Therefore the perimeter is simply:
Mistake 2: Using the wrong area formula
For an equilateral triangle, remember:
Mistake 3: Confusing inradius and circumradius
Also remember:
Mistake 4: Forgetting the 60° angles
Every interior angle of an equilateral triangle is 60°.
Mistake 5: Mixing height and side
The height is not equal to the side. Use:
15. Equilateral Triangle Formula Chart
| Quantity | Formula |
|---|---|
| Area | (√3/4)a² |
| Perimeter | 3a |
| Height | (√3/2)a |
| Inradius | a/(2√3) |
| Inradius using height | h/3 |
| Circumradius | a/√3 |
| Circumradius using height | 2h/3 |
| Relation | R = 2r |
16. Practice Questions
Question 1: Find the perimeter of an equilateral triangle whose side is 14 cm.
Question 2: Find the area of an equilateral triangle whose side is 10 cm.
Question 3: Find the height of an equilateral triangle whose side is 16 cm.
Question 4: Find the inradius of an equilateral triangle whose side is 6√3 cm.
Question 5: The height of an equilateral triangle is 12 cm. Find its inradius.
Question 6: The height of an equilateral triangle is 15 cm. Find its circumradius.
Question 7: If the inradius of an equilateral triangle is 7 cm, find its circumradius.
Question 8: An equilateral triangle has side 20 cm. Find its area and height.
17. Quick Revision
18. Competitive Exam Tip
For competitive exams, remember these three formulas first:
Then remember the radius relations:
A particularly useful shortcut is:
The formulas in this section are based on the EQUILATERAL TRIANGLE section of the provided mensuration formula sheet. The source lists area, perimeter, height, inradius and circumradius, along with the corresponding diagram.
Part 14 — Quadrilateral Formulas will cover the general quadrilateral formulas given in the provided formula sheet, including angle sum, area and perimeter, with diagrams and solved examples.