14. Isosceles Triangle Formulas
An isosceles triangle is a triangle in which two sides are equal.
In the formula sheet, the two equal sides are represented by a, while the base is represented by b.
Therefore:
The provided formula sheet gives formulas for:
- Area
- Perimeter
- Height
1. Diagram of an Isosceles Triangle
a = equal side | b = base | h = height | α = half of the vertex angle
2. Important Properties of an Isosceles Triangle
- Two sides are equal.
- The third side is called the base.
- The angles opposite the equal sides are equal.
- The height drawn from the vertex between the equal sides bisects the base.
- Therefore, the base is divided into two equal parts: b/2 and b/2.
- The height divides the triangle into two equal right-angled triangles.
3. Symbols Used in the Formulas
| Symbol | Meaning |
|---|---|
| a | Each of the two equal sides |
| b | Base of the triangle |
| h | Height / altitude |
| α | Half of the angle at the vertex |
4. Area of an Isosceles Triangle
The first area formula given in the PDF is:
Here:
- a = equal side
- b = base
This formula is useful when the equal side and base are known.
Example
An isosceles triangle has equal sides of 5 cm each and a base of 6 cm. Find its area.
Substitute:
a = 5 cm
b = 6 cm
Therefore:
= (6/4) × √(4 × 5² − 6²)
= 3/2 × √(4 × 25 − 36)
= 3/2 × √(100 − 36)
= 3/2 × √64
= 3/2 × 8
5. Area Using the Included Angle
The PDF also provides another area formula:
This form is useful when the equal side a and the angle α shown in the diagram are known.
Example
Suppose the equal sides of an isosceles triangle are 10 cm and the angle represented by α is 30°. Find the area using the formula from the PDF.
= ½ × 10² × sin 30°
= ½ × 100 × 1/2
= 50 × 1/2
6. Perimeter of an Isosceles Triangle
An isosceles triangle has two equal sides of length a and a base of length b.
Therefore, its perimeter is:
Example
An isosceles triangle has two equal sides of 8 cm each and a base of 10 cm. Find the perimeter.
= 2 × 8 + 10
= 16 + 10
Answer: 26 cm
7. Height of an Isosceles Triangle
The height formula given in the PDF is:
This formula is useful when the equal side a and base b are known.
Why Does This Formula Work?
The height divides the base into two equal parts.
The equal side a becomes the hypotenuse of a right-angled triangle.
Using Pythagoras theorem:
Therefore:
Taking the square root:
Example
An isosceles triangle has equal sides of 5 cm and base 6 cm. Find its height.
= √(4 × 5² − 6²) / 2
= √(100 − 36) / 2
= √64 / 2
= 8 / 2
Answer: 4 cm
8. Area Using Base and Height
The general triangle area formula can also be used:
For an isosceles triangle:
Using the height formula:
we get:
This gives the same area formula shown in the PDF.
9. Complete Solved Example
An isosceles triangle has two equal sides of 10 cm each and a base of 12 cm. Find:
- Height
- Area
- Perimeter
Step 1: Find Height
= √(4 × 10² − 12²) / 2
= √(400 − 144) / 2
= √256 / 2
= 16 / 2
Step 2: Find Area
= ½ × 12 × 8
= 6 × 8
Step 3: Find Perimeter
= 2 × 10 + 12
= 20 + 12
Final Answers
- Height = 8 cm
- Area = 48 cm²
- Perimeter = 32 cm
10. Relationship Between Area and Height
For every triangle:
For an isosceles triangle:
Since:
Substitution gives:
Therefore, if you know a and b, you can find the height first and then calculate the area.
11. Which Formula Should You Use?
| Given Information | Required Quantity | Formula |
|---|---|---|
| Equal side a and base b | Area | (b/4)√(4a²−b²) |
| Equal side a and angle α | Area | ½a² sin α |
| Equal side a and base b | Perimeter | 2a+b |
| Equal side a and base b | Height | √(4a²−b²)/2 |
| Base b and height h | Area | ½bh |
12. Quick Calculation Method
If the equal sides and base are given, follow these three steps:
Step 1
Find the height:
Step 2
Find the area:
Step 3
Find the perimeter:
13. Common Mistakes
Mistake 1: Forgetting the two equal sides
In an isosceles triangle:
Therefore:
Mistake 2: Using b instead of b/2
When the height is drawn, it divides the base into two equal parts:
Mistake 3: Forgetting the division by 2
The height formula is:
Do not forget the final /2.
Mistake 4: Confusing area and perimeter
Mistake 5: Using the wrong angle
In the PDF diagram, α is shown on each side of the altitude at the vertex. The area formula using this angle is:
14. Isosceles Triangle Formula Chart
| Quantity | Formula |
|---|---|
| Area | (b/4)√(4a²−b²) |
| Area using angle α | ½a² sin α |
| Perimeter | 2a+b |
| Height | √(4a²−b²)/2 |
| Area using height | ½bh |
15. Practice Questions
Question 1: Find the perimeter of an isosceles triangle whose equal sides are 8 cm each and base is 10 cm.
Question 2: Find the height of an isosceles triangle whose equal sides are 5 cm each and base is 6 cm.
Question 3: Find the area of an isosceles triangle whose equal sides are 5 cm each and base is 6 cm.
Question 4: An isosceles triangle has equal sides of 10 cm and a base of 12 cm. Find its height.
Question 5: An isosceles triangle has equal sides of 10 cm and base of 12 cm. Find its area.
Question 6: Find the perimeter of an isosceles triangle with equal sides 13 cm and base 10 cm.
Question 7: An isosceles triangle has equal sides of 15 cm and base 18 cm. Find its height.
Question 8: If the equal side is 10 cm and the angle α is 30°, find the area using: Area = ½a² sin α.
16. Quick Revision
17. Competitive Exam Tip
For an isosceles triangle, remember the relationship between the equal sides and the base:
If a and b are given, first find the height:
Then area becomes very easy:
For perimeter:
These three formulas are especially useful for competitive-exam mensuration questions.
The formulas in this section follow the ISOSCELES TRIANGLE section of the provided mensuration formula sheet. The source lists the area in two forms, perimeter, and height, along with the labelled triangle diagram.
Part 15 — Volume & Surface Area will continue with the next section available in the provided PDF.