Right-Angled Triangle & Isosceles Triangle

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19. Right-Angled Triangle & Isosceles Triangle Formulas

A triangle can be classified according to its angles and sides. Two important types covered in this section are the Right-Angled Triangle and the Isosceles Triangle.

This section explains the important formulas, diagrams, step-by-step examples, and exam shortcuts for both types.

1. Right-Angled Triangle

A right-angled triangle is a triangle in which one angle is exactly 90°.

The side opposite the 90° angle is called the hypotenuse. It is the longest side of a right-angled triangle.

Let:

  • P = Perpendicular
  • B = Base
  • H = Hypotenuse

2. Right-Angled Triangle Diagram

Perpendicular (P) Base (B) Hypotenuse (H) A B C

3. Area of a Right-Angled Triangle

The area of a right-angled triangle is calculated using its base and perpendicular.

Area = ½ × Base × Height

If the perpendicular is P and the base is B:

Area = ½ × B × P

Example 1: Find the Area

A right-angled triangle has a base of 12 cm and perpendicular of 5 cm. Find its area.

Area = ½ × B × P
= ½ × 12 × 5
= 6 × 5
Area = 30 cm²

Answer: 30 cm²

4. Perimeter of a Right-Angled Triangle

The perimeter is the sum of the perpendicular, base, and hypotenuse.

Perimeter = P + B + H

Example 2: Find the Perimeter

A right-angled triangle has sides 3 cm, 4 cm and 5 cm. Find its perimeter.

P = 3 + 4 + 5
P = 12 cm

Answer: 12 cm

5. Pythagoras Theorem

Pythagoras theorem is one of the most important formulas for a right-angled triangle.

It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

H² = P² + B²

Therefore:

H = √(P² + B²)

If the hypotenuse is known:

P = √(H² − B²)
B = √(H² − P²)
P B H

Example 3: Find the Hypotenuse

The perpendicular of a right-angled triangle is 6 cm and the base is 8 cm. Find the hypotenuse.

H² = P² + B²
H² = 6² + 8²
H² = 36 + 64
H² = 100
H = 10 cm

Answer: 10 cm

6. Altitude to the Hypotenuse

If a perpendicular is drawn from the right-angle vertex to the hypotenuse, it divides the original triangle into two smaller right-angled triangles.

The altitude to the hypotenuse is represented by M.

M = (P × B) / H
M B P H

Example 4: Find the Altitude to Hypotenuse

A right-angled triangle has perpendicular 6 cm, base 8 cm, and hypotenuse 10 cm. Find the altitude to the hypotenuse.

M = (P × B) / H
M = (6 × 8) / 10
M = 48 / 10
M = 4.8 cm

Answer: 4.8 cm

7. Inradius of a Right-Angled Triangle

For a right-angled triangle, the inradius can be calculated using the three sides.

r = (P + B − H) / 2

Where:

  • P = Perpendicular
  • B = Base
  • H = Hypotenuse

Example 5: Find the Inradius

Find the inradius of a right-angled triangle whose sides are 3 cm, 4 cm and 5 cm.

r = (P + B − H) / 2
r = (3 + 4 − 5) / 2
r = 2 / 2
r = 1 cm

Answer: 1 cm

Important Note About the Provided PDF

The provided PDF page places the expression (P + B − H) / 2 under the heading “Pythagoras Theorem”, while it shows H / 2 under “Inradius”.

These labels appear to be swapped/mislabeled in the source. For mathematical accuracy, this website section uses:

Pythagoras: H² = P² + B²
Inradius: r = (P + B − H) / 2

This note is included so that students are not confused by the source image.

8. Isosceles Triangle

An isosceles triangle is a triangle in which two sides are equal.

Let the two equal sides be a and the base be b.

The altitude from the vertex to the base divides the base into two equal parts.

BD = DC = b/2

9. Isosceles Triangle Diagram

a a b b/2 b/2 h α α A B C D

10. Area of an Isosceles Triangle

For an isosceles triangle with equal sides a and base b, the area can be calculated using:

Area = (b/4)√(4a² − b²)

The source also gives another form using the angle α:

Area = ½a² sin α

Example 6: Find the Area

An isosceles triangle has equal sides of 5 cm and base 6 cm. Find its area.

Given:

a = 5 cm,   b = 6 cm

Formula:

Area = (b/4)√(4a² − b²)

Substitution:

Area = (6/4)√(4×5² − 6²)
= 3/2 √(100 − 36)
= 3/2 √64
= 3/2 × 8
Area = 12 cm²

Answer: 12 cm²

11. Perimeter of an Isosceles Triangle

Since two sides are equal to a and the base is b:

Perimeter = a + a + b
Perimeter = 2a + b

Example 7: Find the Perimeter

The equal sides of an isosceles triangle are 10 cm each and the base is 12 cm. Find the perimeter.

P = 2a + b
P = 2(10) + 12
P = 20 + 12
P = 32 cm

Answer: 32 cm

12. Height of an Isosceles Triangle

When the equal sides are a and the base is b, the altitude divides the base into two equal parts.

Therefore:

BD = DC = b/2

Using Pythagoras theorem on one of the resulting right-angled triangles:

h = √(4a² − b²) / 2

Example 8: Find the Height

An isosceles triangle has equal sides of 5 cm and base 6 cm. Find its height.

h = √(4a² − b²) / 2
h = √(4×5² − 6²) / 2
h = √(100 − 36) / 2
h = √64 / 2
h = 4 cm

Answer: 4 cm

13. Area Using Base and Height

Once the height of an isosceles triangle is known, its area can also be found using the basic triangle formula.

Area = ½ × b × h

Example 9: Area Using Height

For an isosceles triangle with base 10 cm and height 12 cm, find the area.

Area = ½ × b × h
= ½ × 10 × 12
Area = 60 cm²

Answer: 60 cm²

14. Quick Formula Table

Triangle Type Quantity Formula
Right-Angled Triangle Area ½ × B × P
Perimeter P + B + H
Pythagoras H² = P² + B²
Altitude to Hypotenuse M = PB/H
Inradius r = (P+B−H)/2
Isosceles Triangle Area (b/4)√(4a²−b²)
Perimeter 2a+b
Height √(4a²−b²)/2

15. Which Formula Should You Use?

Right-Angled Triangle

Base and perpendicular given: Use the area formula.

Area = ½BP

Three sides given: Add them to find the perimeter.

P = P + B + H

Two sides given and hypotenuse is required: Use Pythagoras theorem.

H² = P² + B²

Isosceles Triangle

Equal sides and base given: Use the direct area formula.

Area = (b/4)√(4a²−b²)

Equal sides and base given: Use the height formula when height is required.

h = √(4a²−b²)/2

16. Practice Questions

1. Find the area of a right-angled triangle whose base is 15 cm and perpendicular is 8 cm.

2. Find the hypotenuse of a right-angled triangle whose perpendicular is 5 cm and base is 12 cm.

3. Find the perimeter of a right-angled triangle with sides 6 cm, 8 cm and 10 cm.

4. Find the altitude to the hypotenuse of a right-angled triangle with sides 6 cm, 8 cm and 10 cm.

5. Find the inradius of a 3-4-5 right-angled triangle.

6. Find the perimeter of an isosceles triangle whose equal sides are 8 cm each and base is 6 cm.

7. Find the height of an isosceles triangle whose equal sides are 5 cm each and base is 6 cm.

8. Find the area of an isosceles triangle whose equal sides are 5 cm each and base is 6 cm.

17. Answers

Question Answer
1 60 cm²
2 13 cm
3 24 cm
4 4.8 cm
5 1 cm
6 22 cm
7 4 cm
8 12 cm²

18. Important Exam Shortcuts

Shortcut 1: Remember the famous 3-4-5 triangle.

3² + 4² = 5²

Shortcut 2: In a right triangle, the hypotenuse is always the longest side.

Shortcut 3: The altitude to the hypotenuse is:

M = PB/H

Shortcut 4: For an isosceles triangle, the altitude from the vertex divides the base into two equal parts.

b/2 + b/2 = b

Shortcut 5: Isosceles triangle perimeter:

P = 2a + b

19. Common Mistakes to Avoid

1. Do not mistake the hypotenuse for the base.

2. The hypotenuse is always opposite the 90° angle.

3. In Pythagoras theorem, use the square of each side.

4. For an isosceles triangle, remember that the two equal sides are represented by a.

5. The base of an isosceles triangle is represented by b in these formulas.

6. Area must be written in square units such as cm² or m².

20. Quick Revision

Right-Angled Triangle

Area = ½BP
Perimeter = P+B+H
H² = P²+B²
M = PB/H
r = (P+B−H)/2

Isosceles Triangle

Area = (b/4)√(4a²−b²)
Area = ½a² sin α
Perimeter = 2a+b
Height = √(4a²−b²)/2

Source: Part 19 is based on page 9 of the provided Mensuration Formula PDF, which contains the Right-Angled Triangle and Isosceles Triangle sections. :contentReference[oaicite:1]{index=1}

Editorial note: The source image appears to have the Pythagoras and inradius expressions mislabeled. The standard mathematical forms have been used above so that the website content is suitable for students.