Right-Angled Triangle Formulas

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12. Right-Angled Triangle Formulas

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 the right-angled triangle.

The other two sides are commonly called the perpendicular and base.

The important formulas in this section are:

  • Area
  • Perimeter
  • Altitude to the hypotenuse
  • Pythagoras theorem
  • Inradius

What Is a Right-Angled Triangle?

P B H M A B C M 90°

P = Perpendicular   |   B = Base   |   H = Hypotenuse   |   M = Altitude to hypotenuse

1. Important Properties of a Right-Angled Triangle

  • One angle is 90°.
  • The side opposite the 90° angle is the hypotenuse.
  • The hypotenuse is the longest side.
  • The other two sides are the base and perpendicular.
  • The altitude from the right-angle vertex to the hypotenuse divides the triangle into two smaller right-angled triangles.

2. Symbols Used in the Formulas

Symbol Meaning
P Perpendicular
B Base
H Hypotenuse
M Altitude to the hypotenuse

3. Area of a Right-Angled Triangle

The area of a right-angled triangle is:

Area = ½ × Base × Height

In a right-angled triangle, the perpendicular can be taken as the height when the base is the other perpendicular side.

Area = ½ × P × B

Example

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

Area = ½ × P × B

= ½ × 5 × 12

= 5 × 6

Area = 30 cm²

Answer: 30 cm²

4. Perimeter of a Right-Angled Triangle

The perimeter is the sum of all three sides.

According to the formula sheet:

Perimeter = P + B + H

Example

A right-angled triangle has perpendicular 5 cm, base 12 cm, and hypotenuse 13 cm. Find the perimeter.

Perimeter = P + B + H

= 5 + 12 + 13

Perimeter = 30 cm

Answer: 30 cm

5. Altitude to the Hypotenuse

When an altitude is drawn from the right angle to the hypotenuse, it forms a perpendicular line to the hypotenuse.

The formula given in the PDF is:

M = (P × B) / H

where:

  • M = altitude to the hypotenuse
  • P = perpendicular
  • B = base
  • H = hypotenuse

Example

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

= (6 × 8) / 10

= 48 / 10

M = 4.8 cm

Answer: 4.8 cm

6. Explanation of the Altitude Formula

The area of the same triangle can be calculated in two ways.

Using the perpendicular and base:

Area = ½ × P × B

Using the hypotenuse as the base and M as its corresponding height:

Area = ½ × H × M

Since both expressions represent the same area:

½PB = ½HM

Therefore:

M = PB / H

7. Pythagoras Theorem

For a right-angled triangle, the relationship between the three sides is given by the Pythagoras theorem.

The formula sheet represents the relation as:

H² = P² + B²

Therefore:

H = √(P² + B²)

This formula is useful when the two shorter sides are known and the hypotenuse has to be found.

Example 1: Find the Hypotenuse

A right-angled triangle has sides 6 cm and 8 cm. Find the hypotenuse.

H = √(P² + B²)

= √(6² + 8²)

= √(36 + 64)

= √100

H = 10 cm

Answer: 10 cm

8. Finding the Perpendicular

From Pythagoras theorem:

H² = P² + B²

Rearranging:

P = √(H² − B²)

Example

The hypotenuse of a right-angled triangle is 13 cm and the base is 12 cm. Find the perpendicular.

P = √(H² − B²)

= √(13² − 12²)

= √(169 − 144)

= √25

P = 5 cm

Answer: 5 cm

9. Finding the Base

Similarly:

B = √(H² − P²)

Example

The hypotenuse is 17 cm and the perpendicular is 8 cm. Find the base.

B = √(H² − P²)

= √(17² − 8²)

= √(289 − 64)

= √225

B = 15 cm

Answer: 15 cm

10. Inradius of a Right-Angled Triangle

The formula sheet gives the inradius as:

r = H / 2

where H is the hypotenuse.

Example

A right-angled triangle has a hypotenuse of 14 cm. Find the inradius using the formula given in the formula sheet.

r = H / 2

= 14 / 2

r = 7 cm

Answer: 7 cm

Important:

The provided PDF specifically shows the right-angled-triangle inradius formula as:

r = H / 2

For this website series, the formula presentation above follows the supplied PDF.

11. Complete Solved Example

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

  1. Hypotenuse
  2. Area
  3. Perimeter
  4. Altitude to the hypotenuse
  5. Inradius using the formula given in the PDF

Step 1: Find Hypotenuse

H = √(P² + B²)

= √(5² + 12²)

= √(25 + 144)

= √169

H = 13 cm

Step 2: Find Area

Area = ½ × P × B

= ½ × 5 × 12

Area = 30 cm²

Step 3: Find Perimeter

Perimeter = P + B + H

= 5 + 12 + 13

Perimeter = 30 cm

Step 4: Find Altitude to Hypotenuse

M = PB / H

= (5 × 12) / 13

M = 60/13 cm

Approximately:

M ≈ 4.62 cm

Step 5: Find Inradius

r = H / 2

= 13 / 2

r = 6.5 cm

Final Answers

  • Hypotenuse = 13 cm
  • Area = 30 cm²
  • Perimeter = 30 cm
  • Altitude to hypotenuse = 60/13 cm
  • Inradius = 6.5 cm

12. Which Formula Should You Use?

Given Information Formula
Base and perpendicular Area = ½PB
Three sides Perimeter = P+B+H
Two shorter sides H = √(P²+B²)
Hypotenuse and base P = √(H²−B²)
Hypotenuse and perpendicular B = √(H²−P²)
Base, perpendicular and hypotenuse M = PB/H
Hypotenuse r = H/2

13. Common Pythagorean Triples

Some commonly used right-angled triangle combinations are:

Perpendicular Base Hypotenuse
3 4 5
5 12 13
7 24 25
8 15 17
9 40 41
12 35 37

These combinations can save time in competitive-exam questions.

14. Common Mistakes

Mistake 1: Identifying the hypotenuse incorrectly

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

Mistake 2: Using the wrong Pythagoras arrangement

H² = P² + B²

The hypotenuse must be on the left side when writing the squared relationship.

Mistake 3: Forgetting the square root

If:

H² = 169

then:

H = √169 = 13

Mistake 4: Confusing altitude with perpendicular

The perpendicular P is one of the original sides of the right triangle, while M is the altitude drawn to the hypotenuse.

Mistake 5: Mixing units

Convert all measurements to the same unit before calculating area or perimeter.

15. Right-Angled Triangle Formula Chart

Quantity Formula
Area ½ × P × B
Perimeter P + B + H
Hypotenuse √(P² + B²)
Perpendicular √(H² − B²)
Base √(H² − P²)
Altitude to Hypotenuse PB / H
Inradius H / 2

16. Practice Questions

Question 1: Find the area of a right-angled triangle with base 15 cm and perpendicular 8 cm.

Question 2: Find the hypotenuse of a right-angled triangle whose sides are 9 cm and 12 cm.

Question 3: A right-angled triangle has hypotenuse 25 cm and base 7 cm. Find the perpendicular.

Question 4: A right-angled triangle has hypotenuse 17 cm and perpendicular 8 cm. Find the base.

Question 5: Find the perimeter of a right-angled triangle with sides 5 cm, 12 cm and 13 cm.

Question 6: A right-angled triangle has perpendicular 9 cm, base 12 cm and hypotenuse 15 cm. Find the altitude to the hypotenuse.

Question 7: Find the area of a right-angled triangle whose perpendicular is 10 cm and base is 24 cm.

Question 8: A right-angled triangle has hypotenuse 20 cm. Find its inradius using the formula given in the formula sheet.

17. Quick Revision

Area: ½PB
Perimeter: P+B+H
Pythagoras: H²=P²+B²
Hypotenuse: H=√(P²+B²)
Perpendicular: P=√(H²−B²)
Base: B=√(H²−P²)
Altitude to Hypotenuse: M=PB/H
Inradius: r=H/2

18. Competitive Exam Tip

When you see a right-angled triangle question, first identify the hypotenuse.

Then decide what the question is asking:

  • Area → use ½ × P × B
  • Hypotenuse → use H = √(P²+B²)
  • Missing side → use Pythagoras
  • Altitude to hypotenuse → use M = PB/H
  • Perimeter → add all three sides

Remember the common triples such as 3-4-5 and 5-12-13 to solve many questions quickly.

Formula Source:

The formulas in this section follow the RIGHT-ANGLED TRIANGLE section of the provided mensuration formula sheet. The source lists area, perimeter, altitude to hypotenuse, Pythagoras theorem and inradius.

Next Part:

Part 13 — Isosceles Triangle Formulas will cover area, perimeter and height, with diagrams, step-by-step explanations and solved examples based on the next triangle section of the provided formula sheet.