45-degree angle — Definition, Formula & Examples
A 45-degree angle is an angle that measures exactly 45°, which is precisely half of a right angle (90°). It is an acute angle and appears frequently in everyday shapes like the diagonal of a square.
An angle of measure 45° is formed when a ray rotates one-eighth of a full turn (360° ÷ 8 = 45°) from its initial position. Equivalently, it is the angle produced by bisecting a right angle into two congruent parts, each measuring 45°.
How It Works
You can recognize a 45-degree angle by comparing it to a right angle — it should look like exactly half of that familiar square corner. When you draw a diagonal across a square, it creates two 45-degree angles at each corner. A 45-45-90 triangle, one of the most common triangles in geometry, has two 45-degree angles and one 90-degree angle. To measure a 45-degree angle, place a protractor's center point on the vertex and read the scale at the 45° mark.
Worked Example
Problem: A right angle is divided into two equal parts by an angle bisector. What is the measure of each part?
Step 1: Write down the measure of a right angle.
Step 2: An angle bisector splits an angle into two equal halves. Divide by 2.
Step 3: Each part measures 45°, confirming both are 45-degree angles.
Answer: Each angle measures 45°.
Another Example
Problem: In a triangle, two angles each measure 45°. What is the measure of the third angle?
Step 1: The sum of all angles in any triangle is 180°.
Step 2: Add the two known angles.
Step 3: Subtract to find the missing angle.
Answer: The third angle is 90°, making this a 45-45-90 right triangle.
Why It Matters
The 45-degree angle is essential in elementary and middle-school geometry when studying triangles, symmetry, and angle relationships. The 45-45-90 triangle built from this angle appears constantly in construction, tiling, and design — carpenters use 45-degree miter cuts to join baseboards and picture frames at neat corners. Understanding this angle also prepares you for trigonometry, where sin 45° and cos 45° are among the first exact values you memorize.
Common Mistakes
Mistake: Confusing a 45-degree angle with a right angle because both look "sharp" or "square."
Correction: A right angle is 90° and forms a perfect square corner. A 45-degree angle is exactly half of that — noticeably more narrow. Use a protractor to check.
Mistake: Thinking the diagonal of any rectangle creates 45-degree angles.
Correction: A diagonal only makes 45-degree angles when the shape is a square (all sides equal). In a non-square rectangle, the diagonal creates unequal angles.