11. General Triangle Formulas
A general triangle is a triangle in which the three sides and three angles may be different.
Let the three sides of a triangle be a, b and c.
For general triangle problems, some of the most important formulas are:
- Semiperimeter
- Area using Heron's formula
- Area using two sides and included angle
- Area using base and height
- Perimeter
- Inradius
What Is a General Triangle?
a, b, c = sides | h = height
1. Symbols Used in Triangle Formulas
| Symbol | Meaning |
|---|---|
| a | First side |
| b | Second side |
| c | Third side |
| s | Semiperimeter |
| h | Height / altitude |
| A | Area of triangle |
| r | Inradius |
| C | Included angle between sides a and b |
2. Semiperimeter of a Triangle
The semiperimeter is half of the perimeter of the triangle.
The symbol s represents the semiperimeter.
Example
The sides of a triangle are 5 cm, 6 cm and 7 cm. Find its semiperimeter.
= (5 + 6 + 7) / 2
= 18 / 2
s = 9 cm
Answer: 9 cm
3. Heron's Formula
When all three sides of a triangle are known, its area can be found using Heron's formula.
Here:
- A = Area
- a, b, c = three sides
- s = semiperimeter
Important: First calculate the semiperimeter before applying Heron's formula.
Example: Heron's Formula
Find the area of a triangle whose sides are 5 cm, 6 cm and 7 cm.
Step 1: Find Semiperimeter
s = 9 cm
Step 2: Apply Heron's Formula
= √[9(9 − 5)(9 − 6)(9 − 7)]
= √(9 × 4 × 3 × 2)
= √216
A = 6√6 cm²
Approximately:
4. Area Using Two Sides and Included Angle
If two sides and the angle between those two sides are known, the area can be calculated using:
Here:
- a = first side
- b = second side
- C = angle between sides a and b
Example
Two sides of a triangle are 8 cm and 10 cm. The included angle is 30°. Find the area.
= ½ × 8 × 10 × sin 30°
Since:
Therefore:
A = ½ × 8 × 10 × ½
A = 20 cm²
Answer: 20 cm²
5. Important Sine Values for Triangle Questions
| Angle | sin θ |
|---|---|
| 0° | 0 |
| 30° | 1/2 |
| 45° | 1/√2 |
| 60° | √3/2 |
| 90° | 1 |
6. Area Using Base and Height
The basic area formula for any triangle is:
If the base is b and the corresponding height is h:
Example
The base of a triangle is 14 cm and its corresponding height is 9 cm. Find the area.
= ½ × 14 × 9
= 7 × 9
A = 63 cm²
Answer: 63 cm²
7. Finding Height from Area
From:
the height can be calculated as:
Example
The area of a triangle is 96 cm² and its base is 16 cm. Find the height.
= (2 × 96) / 16
= 192 / 16
h = 12 cm
Answer: 12 cm
8. Perimeter of a General Triangle
The perimeter is the sum of all three sides.
Example
The sides of a triangle are 9 cm, 12 cm and 15 cm. Find its perimeter.
= 9 + 12 + 15
P = 36 cm
Answer: 36 cm
9. Inradius of a Triangle
The inradius is the radius of the circle that lies inside the triangle and touches all three sides.
For a triangle:
where:
- r = inradius
- A = area
- s = semiperimeter
Example
The area of a triangle is 84 cm² and its semiperimeter is 14 cm. Find its inradius.
= 84 / 14
r = 6 cm
Answer: 6 cm
10. Inradius Using Heron's Formula
If the three sides are known, first find the semiperimeter and area. Then:
Since:
the inradius can also be expressed as:
Example
Find the inradius of a triangle with sides 5 cm, 6 cm and 7 cm.
We already know:
and:
Therefore:
= 6√6 / 9
r = 2√6 / 3 cm
Approximately:
11. Important Relation Between Area, Inradius and Semiperimeter
A very important mensuration relation is:
Therefore:
and:
12. Combined Exam Example
The sides of a triangle are 13 cm, 14 cm and 15 cm. Find:
- Perimeter
- Semiperimeter
- Area
- Inradius
Step 1: Perimeter
= 13 + 14 + 15
P = 42 cm
Step 2: Semiperimeter
= 42 / 2
s = 21 cm
Step 3: Area Using Heron's Formula
= √[21(21 − 13)(21 − 14)(21 − 15)]
= √(21 × 8 × 7 × 6)
= √7056
A = 84 cm²
Step 4: Inradius
= 84 / 21
r = 4 cm
Final Answers:
- Perimeter = 42 cm
- Semiperimeter = 21 cm
- Area = 84 cm²
- Inradius = 4 cm
13. Which Triangle Area Formula Should You Use?
| Given Information | Use This Formula |
|---|---|
| Base and corresponding height | A = ½bh |
| Three sides | A = √[s(s−a)(s−b)(s−c)] |
| Two sides and included angle | A = ½ab sin C |
| Area and semiperimeter | r = A/s |
| Three sides | s = (a+b+c)/2 |
14. Common Mistakes in General Triangle Questions
Mistake 1: Forgetting the ½
The basic area formula is:
Mistake 2: Using the wrong angle
For:
C must be the angle between sides a and b.
Mistake 3: Using Heron's formula without finding s
Always calculate:
before applying Heron's formula.
Mistake 4: Confusing perimeter and semiperimeter
Mistake 5: Mixing units
Make sure all sides and heights are converted to the same unit before calculating the area.
15. General Triangle Formula Chart
| Quantity | Formula |
|---|---|
| Perimeter | P = a+b+c |
| Semiperimeter | s = (a+b+c)/2 |
| Area — Base & Height | A = ½bh |
| Area — Heron's Formula | A = √[s(s−a)(s−b)(s−c)] |
| Area — Two Sides & Angle | A = ½ab sin C |
| Inradius | r = A/s |
| Area-Inradius Relation | A = rs |
16. Practice Questions
Question 1: Find the semiperimeter of a triangle whose sides are 8 cm, 10 cm and 12 cm.
Question 2: Find the perimeter of a triangle whose sides are 15 cm, 18 cm and 20 cm.
Question 3: Find the area of a triangle with base 16 cm and height 12 cm.
Question 4: Find the area of a triangle whose sides are 5 cm, 5 cm and 6 cm using Heron's formula.
Question 5: Two sides of a triangle are 12 cm and 10 cm and the included angle is 30°. Find the area.
Question 6: A triangle has area 120 cm² and semiperimeter 20 cm. Find its inradius.
Question 7: The area of a triangle is 150 cm² and its base is 15 cm. Find its height.
Question 8: The sides of a triangle are 13 cm, 14 cm and 15 cm. Find its area and inradius.
17. General Triangle Quick Revision
18. Mensuration Exam Tip
When you see a triangle question, first identify what information is given.
- Base + Height → use A = ½bh
- Three sides → use Heron's formula
- Two sides + included angle → use A = ½ab sin C
- Area + semiperimeter → use r = A/s
This simple identification step can save considerable time in competitive examinations.
The General Triangle formulas in this section are based on the provided mensuration formula sheet, which lists semiperimeter, Heron's formula, the two-side included-angle area formula, base-height area formula, perimeter and inradius.
Part 12 — Right-Angled Triangle Formulas will cover area, perimeter, Pythagoras theorem, altitude to the hypotenuse, important relationships and solved competitive-exam examples.