18. General Triangle Formulas
A general triangle is a triangle whose three sides may have different lengths and whose angles may also be different.
Let the three sides of a triangle be: a, b, c.
The corresponding angles can be represented by: A, B, C.
The general triangle formulas are very important for SSC, CGL, Banking, Railway, KPSC and other competitive examinations.
1. General Triangle Diagram
In the diagram, the height h is perpendicular to the chosen base.
Depending on the problem, any side can be selected as the base.
2. Semiperimeter of a Triangle
The semiperimeter is half of the perimeter of the triangle.
For sides a, b and c:
Here s represents the semiperimeter.
Example 1: Find the Semiperimeter
A triangle has sides 5 cm, 6 cm and 7 cm. Find its semiperimeter.
Formula:
Substitution:
Answer: 9 cm
3. Area of Triangle — Heron's Formula
When the three sides of a triangle are known, its area can be found using Heron's Formula.
First calculate the semiperimeter:
Then use:
This method is especially useful when the base and height are not directly given.
Example 2: Heron's Formula
Find the area of a triangle whose sides are 5 cm, 6 cm and 7 cm.
Step 1: Find the semiperimeter.
Step 2: Apply Heron's formula.
Answer: 6√6 cm²
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 directly.
Here:
- a = first side
- b = second side
- C = angle between a and b
Example 3: Using Two Sides and an Angle
Two sides of a triangle are 8 cm and 6 cm, and the included angle is 30°. Find the area.
Formula:
Substitute:
Since:
Answer: 12 cm²
5. Area Using Base and Height
The most basic area formula for a triangle is:
If the base is b and perpendicular height is h:
The height must be the perpendicular distance from the opposite vertex to the chosen base.
Example 4: Base and Height
A triangle has a base of 12 cm and a perpendicular height of 8 cm. Find its area.
Answer: 48 cm²
6. Perimeter of a General Triangle
The perimeter of a triangle is the sum of its three sides.
Example 5: Find the Perimeter
The sides of a triangle are 9 cm, 12 cm and 15 cm. Find its perimeter.
Answer: 36 cm
7. Inradius of a Triangle
The inradius is the radius of the circle inscribed inside the triangle.
The PDF gives the relation:
Therefore:
Where:
- r = inradius
- Area = area of triangle
- s = semiperimeter
Example 6: Find the Inradius
A triangle has area 30 cm² and semiperimeter 10 cm. Find its inradius.
Answer: 3 cm
8. General Triangle Formula Table
| Quantity | Formula | When to Use |
|---|---|---|
| Semiperimeter | s = (a+b+c)/2 | When three sides are known |
| Area — Heron's Formula | √[s(s−a)(s−b)(s−c)] | When all three sides are known |
| Area — Two Sides | ½ab sin C | When two sides and included angle are known |
| Area — Base & Height | ½ × Base × Height | When base and perpendicular height are known |
| Perimeter | a+b+c | When three sides are known |
| Inradius | Area/s | When area and semiperimeter are known |
9. Which Formula Should You Use?
Case 1 — Three sides are given
Use Heron's Formula.
Case 2 — Base and perpendicular height are given
Use:
Case 3 — Two sides and included angle are given
Use:
Case 4 — Area and semiperimeter are given
Use:
10. Mixed Solved Examples
Example 7: Three Sides Given
A triangle has sides 3 cm, 4 cm and 5 cm. Find its area.
First find the semiperimeter:
Using Heron's formula:
Answer: 6 cm²
Example 8: Find the Inradius
A triangle has area 84 cm² and sides 13 cm, 14 cm and 15 cm. Find its inradius.
First find the semiperimeter:
Now:
Answer: 4 cm
11. Practice Questions
Try to solve these questions before checking the answers.
1. Find the semiperimeter of a triangle whose sides are 7 cm, 8 cm and 9 cm.
2. Find the perimeter of a triangle whose sides are 11 cm, 13 cm and 15 cm.
3. Find the area of a triangle whose base is 16 cm and height is 9 cm.
4. Two sides of a triangle are 10 cm and 8 cm, and the included angle is 30°. Find its area.
5. Find the area of a triangle whose sides are 5 cm, 5 cm and 6 cm using Heron's formula.
6. A triangle has area 60 cm² and semiperimeter 12 cm. Find its inradius.
7. Find the area of a triangle whose sides are 3 cm, 4 cm and 5 cm.
12. Answers
| Question | Answer |
|---|---|
| 1 | 12 cm |
| 2 | 39 cm |
| 3 | 72 cm² |
| 4 | 20 cm² |
| 5 | 12 cm² |
| 6 | 5 cm |
| 7 | 6 cm² |
13. Important Exam Shortcuts
Shortcut 1: If all three sides are given, think immediately of Heron's Formula.
Shortcut 2: If base and perpendicular height are given, use ½ × Base × Height.
Shortcut 3: If two sides and the included angle are given, use ½ab sin C.
Shortcut 4: For perimeter, simply add the three sides.
Shortcut 5: For inradius, remember:
14. Common Mistakes to Avoid
1. In Heron's formula, calculate the semiperimeter first.
2. Do not use the full perimeter in place of the semiperimeter.
3. In ½ab sin C, the angle must be the included angle between the two given sides.
4. The height in ½ × Base × Height must be perpendicular to the selected base.
5. Area is expressed in square units such as cm² or m².
6. Perimeter and semiperimeter are expressed in linear units such as cm or m.
15. Quick Revision
Source: This section follows the General Triangle formulas in the provided Mensuration Formula PDF, including semiperimeter, Heron's formula, area using two sides and an included angle, area using base and height, perimeter, and inradius.
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