Relationship Between Sides and Angles in a Triangle

How to relationship between sides and angles in a triangle, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

In any triangle, the lengths of the sides and the sizes of the angles are closely connected. Once you understand the relationship, the longest side is always opposite the largest angle, you can determine the relative ordering of sides just from angles, and vice versa. This is a key GED geometry concept.

What Is the Relationship Between Sides and Angles?

In any triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. This is sometimes called the Angle-Side Relationship or the Side-Angle Inequality. It works in both directions:

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• If you know which side is longest, the opposite angle is the largest.
• If you know which angle is largest, the opposite side is the longest.

Rules of the Relationship

Ordering sides from angles

Given angle measures, order the sides by matching each side to the angle opposite it.

• Largest angle → longest opposite side
• Smallest angle → shortest opposite side

Ordering angles from sides

Given side lengths, the angles follow the same ordering.

• Longest side → largest opposite angle
• Shortest side → smallest opposite angle

Computed example

A triangle has sides 3, 5, and 7. The angles opposite these sides are approximately 21.8°, 38.2°, and 120.0°, summing to 180°. The side 7 (longest) is opposite 120° (largest), and the side 3 (shortest) is opposite 21.8° (smallest). ✓

Step-by-Step Summary

1. Identify all three side lengths or all three angle measures.
2. Order the sides from shortest to longest (or angles from smallest to largest).
3. Match each side to its opposite angle: the nth-longest side is opposite the nth-largest angle.
4. Use this to answer questions like “Which side is longest?” or “Which angle is smallest?”

Watch: Triangle Inequality and Side-Angle Relationships (Video Lesson)

Khan Academy explains how the sizes of sides and angles relate to each other in a triangle:

Worked Examples

Example 1: A triangle has angles 30°, 60°, and 90°. Order the sides from shortest to longest.

30° < 60° < 90°, so the side opposite 30° is shortest, the side opposite 60° is next, and the side opposite 90° (the hypotenuse) is longest.

Example 2: A triangle has sides of length 4, 7, and 9. Which angle is the largest?

The longest side is 9, so the largest angle is opposite the side of length 9.

Example 3: In triangle ABC, angle (color{blue}{A = 45})°, angle (color{blue}{B = 80})°, angle (color{blue}{C = 55})°. Order the sides AC, BC, and AB from shortest to longest.

The side opposite angle A (= BC) has angle 45°, shortest side. The side opposite angle C (= AB) has angle 55°, middle. The side opposite angle B (= AC) has angle 80°, longest.
Order: BC < AB < AC.

Example 4: Sides are 3, 5, and 7. Without a calculator, which angle is obtuse?

The longest side is 7. By the angle-side relationship, the angle opposite side 7 is the largest. Since (color{blue}{3^{2} + 5^{2} = 9 + 25 = 34}) < (color{blue}{49 = 7^{2}}), the triangle is obtuse, and the obtuse angle is opposite the side of length 7.

More Practice: Math Antics, Triangles (Video)

Math Antics covers triangle classification and the connection between sides and angles:

Exercises

1. A triangle has angles 20°, 70°, and 90°. Which side is the longest?
2. A triangle has sides 8, 10, and 5. Which angle is the smallest?
3. In triangle PQR, (color{blue}{text{ PQ } = 12}), (color{blue}{text{ QR } = 7}), (color{blue}{text{ PR } = 9}). Order the angles from smallest to largest.
4. A triangle has angles 50°, 50°, and 80°. What can you say about two of its sides?
5. True or false: In a right triangle, the hypotenuse is always the longest side.
6. A triangle has sides 6, 6, and 6. What do you know about its angles?

Answers

1. The side opposite the 90° angle (the hypotenuse) is the longest.
2. The smallest angle is opposite the shortest side (5), so it faces the side of length 5.
3. The smallest angle is opposite the shortest side: (color{blue}{text{ QR } = 7}) → angle P is smallest; (color{blue}{text{ PR } = 9}) → angle Q is middle; (color{blue}{text{ PQ } = 12}) → angle R is largest.
4. The two 50° angles are equal, so the two sides opposite them are equal, the triangle is isosceles.
5. True, the right angle (90°) is the largest angle, so the hypotenuse is the longest side.
6. All sides are equal, so all angles are equal (each is 60°), equilateral triangle.

Frequently Asked Questions

What is the relationship between sides and angles in a triangle?

In every triangle, the longest side is opposite the largest angle and the shortest side is opposite the smallest angle. The rule works both ways, so you can order the sides from the angles or the angles from the sides. If a triangle has angles of 40°, 60° and 80°, the side across from the 80° angle is the longest and the side across from 40° is the shortest.

What is the relationship between the sides and the angles in special right triangles?

Special right triangles have fixed side ratios. In a 45°-45°-90° triangle the sides are in the ratio 1 : 1 : √2, so legs of 5 give a hypotenuse of 5√2. In a 30°-60°-90° triangle the ratio is 1 : √3 : 2, with the shortest side opposite 30° and the hypotenuse, twice the shortest side, opposite 90°. The largest angle still faces the longest side.

What is the relationship between sides and angles for each type of triangle?

In an equilateral triangle all three sides are equal and every angle is 60°. In an isosceles triangle the two equal sides sit opposite two equal angles. In a scalene triangle all sides and all angles are different. In a right triangle the hypotenuse is the longest side because it faces the 90° angle, and in an obtuse triangle the side across from the obtuse angle is the longest.

Is there a formula for the relation between sides and angles of a triangle?

Yes. The Law of Sines says a/sin A = b/sin B = c/sin C, and the Law of Cosines says c² = a² + b² − 2ab·cos C. For a triangle with sides 3, 5 and 7, the Law of Cosines gives cos C = (9 + 25 − 49) ÷ 30 = −0.5, so the angle opposite the 7 is 120°, the largest angle, just as the side-angle rule predicts.

What is the relationship between the sides of a triangle?

The sides of a triangle must satisfy the Triangle Inequality: any two sides added together must be longer than the third side. Lengths of 4, 6 and 9 work because 4 + 6 = 10 is more than 9, but 2, 3 and 7 cannot make a triangle because 2 + 3 = 5 is less than 7. The side lengths also decide the order of the angles opposite them.

Related Topics

• Triangles
• Triangle Inequality
• Area of a Triangle
• Polygons
• Area of a Trapezoid