45 Degree Angle: Definition & Construction

TL;DR

A 45 degree angle is an acute angle that measures exactly half of a right angle (90° ÷ 2 = 45°), and it is the angle each leg makes with the hypotenuse in an isosceles right triangle. This article covers the definition, how to construct one with a compass and by paper folding, its trigonometric values, where it shows up, and six worked examples.

What Is a 45 Degree Angle?

A 45 degree angle is an acute angle whose measure is 45°, which is exactly half of a right angle. An acute angle is any angle smaller than 90°, and 45° sits right at the midpoint between 0° (a flat ray) and 90° (a square corner).

Two equal 45° angles placed side by side rebuild a full right angle, since 45° + 45° = 90°. That is the cleanest way to hold the idea: a 45° angle is what you get when you fold a right-angled corner exactly onto itself.

The 45-45-90 Triangle

The most important place a 45° angle lives is inside the isosceles right triangle, often called the 45-45-90 triangle. One angle is the right angle (90°), and because a triangle's three angles sum to 180°, the remaining 90° splits evenly into two 45° angles.

Equal angles sit opposite equal sides, so the two 45° angles force the two legs to be equal in length: that is why the triangle is isosceles. If each leg is 1 unit, the Pythagorean theorem gives the hypotenuse as ( 1 + 1 = \sqrt{2} ). The fixed side ratio 1:1:2 is what makes this triangle worth memorising.

Trigonometric Values at 45°

Because the 45-45-90 triangle has known sides, the trig ratios at 45° come straight from 1:1:2. The sine is the opposite leg over the hypotenuse, and the cosine is the adjacent leg over the hypotenuse, so both equal the same fraction:

[ \sin 45° = \cos 45° = \frac{1}{\sqrt{2}} \approx 0.707. ]

The tangent is the opposite leg over the adjacent leg, and since those legs are equal:

[ \tan 45° = \frac{1}{1} = 1. ]

A tangent of exactly 1 is the signature of 45°: it is the only acute angle where rise equals run, so a line drawn at 45° goes up one unit for every one unit it moves across.

How to Construct a 45 Degree Angle

You do not need a protractor to draw a 45° angle. The most reliable method builds a right angle first, then bisects it, since 45° is half of 90°. Here is the compass-and-straightedge construction.

Draw a ray OA and construct a right angle at O, giving a second ray OB perpendicular to OA.
With the compass point on O, draw an arc that crosses both OA and OB, marking two points.
From each of those two points, draw equal arcs that cross each other inside the right angle.
Draw ray OC from O through that crossing point. OC bisects the right angle, so ( \angle AOC = 45° ).

How do you make a 45 degree angle without a protractor or compass?

Fold a square sheet of paper corner to corner. The diagonal crease bisects the paper's 90° corner, and each half of that corner is a perfect 45° angle.

Examples of the 45 Degree Angle

Example 1 - A right angle is bisected. What is the measure of each resulting angle?

Each angle measures 45°.

Example 2 - In a 45-45-90 triangle, one leg measures 5 cm. Find the correct leg for a second triangle where hypotenuse is 5 cm.

The leg is 3.54 cm, shorter than the 5 cm hypotenuse, as it must be.

Example 3 - The hour and minute hands of a clock form a 90° right angle at 3:00. The third hand bisects that angle, what angle does it make with the minute hand?

The third hand makes a 45° angle with the minute hand.

Example 4 - Two angles sit side by side on a straight line. The first is a 45° angle. What is the angle between the second angle's far arm and the line?

The partner is 135°.

Example 5 - A square has a diagonal drawn from one corner. What angle does the diagonal make with each side?

Diagonally, the angle is 45° with each side.

Example 6 - A ball is thrown so it lands as far as possible. Show why ( \theta = 45° ) gives the maximum range.

A 45° launch produces ( \sin(90°) = 1 ), the maximum possible range.

Where the 45 Degree Angle Shows Up

The reason a 45° angle is taught so early is that it is the angle of balance, the exact midpoint between horizontal and vertical.

Optimal projectile range. As Example 6 shows, a 45° launch maximises distance with no air resistance.
Carpentry and the mitre joint. Two boards cut at 45° meet to form a clean 90° corner.
Architecture and bracing. A 45° diagonal brace is the standard way to stop a rectangular frame from racking sideways.
The line y=x. The line through the origin at 45° to the x-axis is exactly y=x.

Key Takeaways

A 45 degree angle is an acute angle equal to half a right angle: 90°÷2=45°.
It is the base angle of the isosceles right (45-45-90) triangle, whose sides are in the ratio 1:1:2.
Its trig values are ( \sin 45° = \cos 45° = \frac{1}{\sqrt{2}} ) and ( \tan 45° = 1 ).
You can construct one by bisecting a right angle or folding a square sheet of paper along its diagonal.

Practice These Problems to Solidify Your Understanding

In a 45-45-90 triangle, each leg is 7 cm. Find the hypotenuse.
The hypotenuse of a 45-45-90 triangle is 10 cm. Find the length of each leg.
How many 45° angles fit inside a straight angle (180°)?

Answer to Question 1: hypotenuse = 7\sqrt{2} ≈ 9.9 cm. Answer to Question 2: each leg = ( \frac{10}{\sqrt{2}} ) ≈ 7.07 cm. Answer to Question 3: four.