Geometry

##### Related Lessons

- 3 4 5 Triangle
- 30 60 90 Triangle
- 45 45 90 Triangle
- 5 12 13 Triangle
- Arc Length Calculator
- Area of a Circle Calculator
- Area of a Hexagon
- Area of a Kite
- Area of a Parallelogram
- Area of a Pentagon
- Area of a Rectangle
- Area of a Rhombus
- Area of a Sector
- Area of a Semicircle
- Area of a Square
- Area of a Trapezoid
- Area of a Triangle
- Center of Mass Calculator
- Circumference Calculator
- Distance Formula
- Distance Formula Calculator
- Geometry Calculators
- How to Find the Height of a Triangle
- Isosceles Triangle Theorem
- Perimeter of a Circle
- Perimeter of a Pentagon
- Perimeter of a Rectangle
- Perimeter of a Rhombus
- Perimeter of a Semicircle
- Perimeter of a Square
- Perimeter of a Trapezoid
- Perimeter of a Triangle
- Properties of a Parallelogram
- Pythagorean Theorem Calculator
- Side Angle Side Theorem
- Side Splitter Theorem
- Similar Triangles
- Special Right Triangles
- Surface Area of a Cone
- Surface Area of a Cube
- Surface Area of a Cylinder
- Surface Area of a Hemisphere
- Surface Area of a Pyramid
- Surface Area of a Sphere
- Surface Area of a Triangular Prism
- Triangle Inequality Theorem
- Types of Triangles
- Vertical Angles
- Volume of a Cone
- Volume of a Cube
- Volume of a Hexagonal Prism
- Volume of a Pyramid
- Volume of a Sphere
- Volume of Hemisphere
- Volume of Parallelepiped

Tutors/teachers:

Nikkolas

Tutor and Aerospace Engineer

# 5 12 13 Triangle

#### Lesson Contents

## What is a 5 12 13 Triangle?

**The 5 12 13 triangle is an SSS special right triangle with the ratio between its side lengths as 5, 12, and 13.** It is a common Pythagorean triple that is worth memorizing to save time when dealing with right triangles. The other common SSS special right triangle is the 3 4 5 triangle.

### Using the 5 12 13 Ratio to our Advantage

A 5 12 13 triangle is considered a scalene triangle because all three of its sides have different lengths. If we come across a right triangle and two of the known sides are part of the 5 12 13 ratio, we can immediately determine that the third side will be the remaining number in the ratio. Note that the “13” side will always be the hypotenuse. **The 5 12 13 ratio is scalable and applies to any right triangle with sides that are any common multiple of the numbers 5, 12, and 13.**

### List of 5 12 13 Triangles

The following triangle side lengths are common multiples of the 5 12 13 ratio and are all considered 5 12 13 triangles. It is not necessary to memorize this list, but it is very useful to understand that 5 12 13 is a ratio that may manifest itself as smaller or larger numbers than the ratio itself.

– 2.5 6 6.5

– 5 12 13

– 7.5 18 19.5

– 15 36 39

### 5 12 13 Triangle Angles

A 5 12 13 triangle contains the following internal angles in degrees:**22.6°, 67.4°, 90°**.

And in radians:**0.39, 1.18, and 1.57**.

It is not expected to memorize these. We can use the soh cah toa rule and inverse trigonometric functions to solve for the angles since we know the side lengths. **See the image below for a visual of these angles.**

### What is the Area of a 5 12 13 Triangle?

Calculating the area of a 5 12 13 triangle, we get **A = ( ^{1}⁄_{2})(5)(12) = 30**. If the triangle conforms to the 5 12 13 ratio but is scaled, i.e. a 2.5 6 6.5 triangle, we still use the two shorter sides as the base and height. In this case, the area would be (

^{1}⁄

_{2})(2.5)(6) = 7.5.

## Example Problems

**Problem 1:**

Find the third side length of the 5 12 13 right triangle that has side lengths of 39 and 15.

Solution:

1.) Since we know this is a 5 12 13 triangle scaled from the ratio by an unknown factor, we must determine which two sides are given.

39/15 = 2.6

12/5 = 2.5

13/5 = 2.6

2.) Therefore, we have been given the “13” and “5” sides. We will now multiply by a ratio to get the “12” side.

3.) (^{12}⁄_{13})(39) = 36

4.) **The third side length is 36**.

**Problem 2:**

Which of the following sets of side lengths constitute a 5 12 13 triangle?

Triangle 1: 4, 9, 11.5

Triangle 2: 10, 24, 26

Triangle 3: 6, 13, 14

Triangle 4: 20, 48, 52

Solution:

The side lengths of **triangles 2 and 4 constitute a 5 12 13 triangle**.