10. Equilateral Triangle Formulas
An equilateral triangle is a triangle in which all three sides are equal.
All three interior angles of an equilateral triangle are also equal, and each angle is 60°.
If the length of each side is represented by a, the important formulas include its area, perimeter, height, inradius and circumradius.
What Is an Equilateral Triangle?
a = side length | h = height
1. Important Properties of an Equilateral Triangle
- All three sides are equal.
- All three angles are equal.
- Each interior angle is 60°.
- The height divides the triangle into two equal right-angled triangles.
- The height is perpendicular to the opposite side.
2. Area of an Equilateral Triangle
The area formula given in the formula sheet is:
Where a is the length of each side.
Example 1: Side = 4 cm
Find the area of an equilateral triangle whose side is 4 cm.
= (√3 / 4) × 4²
= (√3 / 4) × 16
A = 4√3 cm²
Answer: 4√3 cm²
Example 2: Side = 6 cm
Find the area of an equilateral triangle whose side is 6 cm.
= (√3 / 4) × 36
A = 9√3 cm²
Approximately:
3. Perimeter of an Equilateral Triangle
Since all three sides are equal:
Example
An equilateral triangle has a side of 12 cm. Find its perimeter.
= 3 × 12
P = 36 cm
Answer: 36 cm
4. Height of an Equilateral Triangle
The height of an equilateral triangle is:
or:
Why Is the Height √3a / 2?
When the height is drawn, it divides the equilateral triangle into two right-angled triangles.
The hypotenuse is a and half of the base is a/2.
Using the Pythagorean theorem:
Therefore:
Hence:
Example
Find the height of an equilateral triangle whose side is 10 cm.
= √3 × 10 / 2
h = 5√3 cm
Approximately:
5. Inradius of an Equilateral Triangle
The inradius is the radius of the circle that fits inside the triangle and touches all three sides.
The formula given in the PDF is:
This can also be written as:
Example
Find the inradius of an equilateral triangle whose side is 12 cm.
= 12 / (2√3)
= 6 / √3
r = 2√3 cm
Approximately:
6. Circumradius of an Equilateral Triangle
The circumradius is the radius of the circle passing through all three vertices of the triangle.
The formula given in the PDF is:
It can also be written as:
Example
Find the circumradius of an equilateral triangle whose side is 9 cm.
= 9 / √3
R = 3√3 cm
Approximately:
7. Relation Between Inradius and Circumradius
For an equilateral triangle:
Therefore:
This is useful in competitive-exam questions when either the inradius or circumradius is given.
Example
If the inradius of an equilateral triangle is 5 cm, find its circumradius.
= 2 × 5
R = 10 cm
Answer: 10 cm
8. Finding the Side from the Perimeter
From:
we get:
Example
The perimeter of an equilateral triangle is 45 cm. Find the side.
= 45 / 3
a = 15 cm
Answer: 15 cm
9. Finding the Side from the Height
From:
we can rearrange it to find the side:
Example
The height of an equilateral triangle is 6√3 cm. Find its side.
= 2 × 6√3 / √3
a = 12 cm
Answer: 12 cm
10. Combined Exam Example
An equilateral triangle has a side of 12 cm. Find:
- Perimeter
- Height
- Area
- Inradius
- Circumradius
Step 1: Perimeter
= 3 × 12
P = 36 cm
Step 2: Height
= √3 × 12 / 2
h = 6√3 cm
Step 3: Area
= (√3 / 4) × 12²
= (√3 / 4) × 144
A = 36√3 cm²
Step 4: Inradius
= 12 / (2√3)
r = 2√3 cm
Step 5: Circumradius
= 12 / √3
R = 4√3 cm
Final Answers:
- Perimeter = 36 cm
- Height = 6√3 cm
- Area = 36√3 cm²
- Inradius = 2√3 cm
- Circumradius = 4√3 cm
11. Equilateral Triangle Formula Chart
| Quantity | Formula |
|---|---|
| Area | (√3 / 4)a² |
| Perimeter | 3a |
| Height | √3a / 2 |
| Inradius | a / (2√3) |
| Circumradius | a / √3 |
| Relation | R = 2r |
12. Common Mistakes in Equilateral Triangle Questions
Mistake 1: Using the wrong area formula
For an equilateral triangle, remember:
Mistake 2: Confusing height and side
The height is not equal to the side.
Mistake 3: Confusing inradius and circumradius
For an equilateral triangle:
Mistake 4: Forgetting that all three sides are equal
If one side is a, all three sides are a.
13. Equilateral Triangle Quick Revision
14. Practice Questions
Question 1: Find the area of an equilateral triangle whose side is 8 cm.
Question 2: Find the perimeter of an equilateral triangle whose side is 15 cm.
Question 3: Find the height of an equilateral triangle whose side is 14 cm.
Question 4: Find the inradius of an equilateral triangle whose side is 6 cm.
Question 5: Find the circumradius of an equilateral triangle whose side is 12 cm.
Question 6: The perimeter of an equilateral triangle is 72 cm. Find its side.
Question 7: The height of an equilateral triangle is 9√3 cm. Find its side.
Question 8: If the inradius of an equilateral triangle is 7 cm, find its circumradius.
15. Mensuration Exam Tip
When a question says equilateral triangle, immediately remember that all three sides are equal and every angle is 60°.
For a direct area question, use:
For height:
For perimeter:
For circle-related questions:
The formulas in this section are based on the Equilateral Triangle section of the provided mensuration formula sheet.
Part 11 — General Triangle Formulas will cover semiperimeter, Heron's formula, area using ½ × base × height, area using two sides and included angle, perimeter and inradius, with diagrams, explanations and solved examples.