Remote Exterior Angle Calculator
Calculate the Exterior Angle
Enter the measures of the two remote interior angles of a triangle to find the measure of the exterior angle.
| Remote Angle 1 (α) | Remote Angle 2 (β) | Exterior Angle (γ) | Third Interior Angle |
|---|---|---|---|
| 30° | 60° | 90° | 90° |
| 45° | 45° | 90° | 90° |
| 70° | 50° | 120° | 60° |
| 20° | 100° | 120° | 60° |
What is a Remote Exterior Angle?
An exterior angle of a triangle is an angle formed by one side of the triangle and the extension of an adjacent side. Every triangle has six exterior angles (two at each vertex, which are vertically opposite and thus equal). When we talk about “the” exterior angle at a vertex, we usually mean one of these.
For any exterior angle of a triangle, the two remote interior angles are the interior angles of the triangle that are not adjacent to that exterior angle. Our Remote Exterior Angle Calculator helps you find the measure of an exterior angle if you know the measures of these two remote interior angles.
This concept is fundamental in Euclidean geometry and is often used to solve problems involving angles and triangles. Anyone studying basic geometry, trigonometry, or even more advanced mathematics will find this calculator useful. Common misconceptions include confusing the exterior angle with the adjacent interior angle or forgetting which angles are the “remote” ones.
Remote Exterior Angle Formula and Mathematical Explanation
The relationship between an exterior angle and its remote interior angles is very straightforward:
Exterior Angle = Sum of the two Remote Interior Angles
If we denote the two remote interior angles as α (alpha) and β (beta), and the exterior angle as γ (gamma), the formula is:
γ = α + β
Derivation:
- The sum of the interior angles of any triangle is always 180°. Let the three interior angles be α, β, and θ (theta), where θ is the interior angle adjacent to the exterior angle γ. So, α + β + θ = 180°.
- An exterior angle (γ) and its adjacent interior angle (θ) form a linear pair, meaning they lie on a straight line. Therefore, their sum is 180°. So, γ + θ = 180°.
- From step 1, we have θ = 180° – (α + β).
- Substitute this into step 2: γ + (180° – (α + β)) = 180°.
- Simplifying, we get γ – α – β = 0, which means γ = α + β.
This proves that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. Our Remote Exterior Angle Calculator uses this exact formula.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| α (alpha) | Measure of the first remote interior angle | Degrees (°) | 0° < α < 180° |
| β (beta) | Measure of the second remote interior angle | Degrees (°) | 0° < β < 180° |
| γ (gamma) | Measure of the exterior angle | Degrees (°) | 0° < γ < 180° (as it relates to α+β < 180) |
| α + β | Sum of remote interior angles | Degrees (°) | 0° < α + β < 180° |
| θ (theta) | Interior angle adjacent to γ | Degrees (°) | 0° < θ < 180° |
Practical Examples (Real-World Use Cases)
Example 1: Designing a Roof Truss
Imagine an engineer is designing a simple triangular roof truss. They know two interior angles at the base are 35° and 45°. They need to find the exterior angle formed by extending one of the base sides to plan for an adjoining structure.
- Remote Interior Angle 1 (α) = 35°
- Remote Interior Angle 2 (β) = 45°
- Using the Remote Exterior Angle Calculator or formula: Exterior Angle (γ) = 35° + 45° = 80°.
The exterior angle is 80°.
Example 2: Navigation
A navigator is plotting a course that involves three points, forming a triangle. They measure two interior angles of the triangular path as 60° and 80°. To determine the turning angle at one vertex if they were to continue straight, they need the exterior angle.
- Remote Interior Angle 1 (α) = 60°
- Remote Interior Angle 2 (β) = 80°
- Using the Remote Exterior Angle Calculator or formula: Exterior Angle (γ) = 60° + 80° = 140°.
The exterior turning angle would be 140°.
How to Use This Remote Exterior Angle Calculator
- Enter Remote Interior Angle 1 (α): Input the measure of one of the remote interior angles in degrees into the first input field.
- Enter Remote Interior Angle 2 (β): Input the measure of the other remote interior angle in degrees into the second input field.
- View Results: The calculator will instantly display the measure of the remote exterior angle (γ), along with the sum and the third interior angle of the triangle. The diagram will also update.
- Reset: Click the “Reset” button to clear the inputs and results and return to the default values.
- Copy: Click “Copy Results” to copy the main result and intermediate values to your clipboard.
Ensure that the sum of the two interior angles is less than 180°, as they are part of a triangle. The calculator will show an error if the sum is 180° or more, or if individual angles are not positive.
Key Factors That Affect Remote Exterior Angle Results
The measure of the remote exterior angle is solely and directly determined by the measures of the two remote interior angles.
- Measure of Remote Interior Angle 1 (α): As this angle increases, the exterior angle increases proportionally.
- Measure of Remote Interior Angle 2 (β): Similarly, as this angle increases, the exterior angle increases proportionally.
- Sum of α and β: The exterior angle is exactly the sum of α and β. If their sum increases, the exterior angle increases. The sum must be less than 180°.
- The Third Interior Angle (θ): While not directly used to calculate γ using α and β, it’s related: θ = 180° – (α + β), and γ = 180° – θ. If α or β change, θ changes, and so does γ.
- Type of Triangle: Whether the triangle is acute, obtuse, or right-angled will influence the possible values of α and β, and thus γ. For instance, in an obtuse triangle, one of α or β (or the angle adjacent to γ) will be greater than 90°.
- Geometric Constraints: The fact that these angles form a triangle imposes the constraint that α > 0, β > 0, and α + β < 180°.
Frequently Asked Questions (FAQ)
- What is an exterior angle of a triangle?
- An exterior angle is formed by one side of the triangle and the extension of an adjacent side. It is supplementary to the adjacent interior angle.
- What are remote interior angles?
- The remote interior angles are the two interior angles of the triangle that are not adjacent to the exterior angle in question.
- What is the formula for the remote exterior angle?
- The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles (γ = α + β).
- Can a remote interior angle be 0 or 180 degrees?
- No, for a valid triangle, each interior angle must be greater than 0 and less than 180 degrees. Their sum (α + β) must also be less than 180 degrees.
- Why is the exterior angle equal to the sum of the remote interior angles?
- Because the sum of all three interior angles is 180°, and the exterior angle and its adjacent interior angle also sum to 180°. This leads to the equality.
- How many exterior angles does a triangle have?
- A triangle has six exterior angles, two at each vertex (forming vertically opposite pairs, so three distinct measures, each supplementary to an interior angle).
- Can I use this Remote Exterior Angle Calculator for any triangle?
- Yes, this principle applies to all types of triangles (scalene, isosceles, equilateral, right, acute, obtuse).
- What if I know the exterior angle and one remote interior angle?
- You can rearrange the formula: Remote Interior Angle 2 = Exterior Angle – Remote Interior Angle 1.
Related Tools and Internal Resources
- Triangle Angle Calculator: Calculate missing angles in a triangle given other angles or sides.
- Triangle Area Calculator: Find the area of a triangle using various formulas.
- Sum of Interior Angles of a Polygon Calculator: Calculate the sum of interior angles for any polygon.
- Geometry Calculators: A collection of calculators for various geometric shapes and problems.
- Math Calculators: Explore a wide range of mathematical calculators.
- Straight Line Angle Calculator: Understand angles on a straight line.