A 45-degree angle is a fundamental element in geometry, commonly used in various mathematical problems and practical applications. Constructing a 45-degree angle accurately is an essential skill for students and professionals alike. This article will guide you through the 45-degree angle construction, focusing on the steps of construction of a 45-degree angle using a compass and protector. Mastering this technique will enhance your understanding and ability to apply geometric principles effectively. Follow these detailed steps to confidently construct a 45-degree angle with precision.

A 45-degree angle is exactly half of a right angle, which measures 90 degrees. Two 45-degree angles placed together form a right angle. In degrees, a 45-degree angle is 45/360 of a complete circle since its measure is between 0 and 90 degrees, a 45-degree angle is classified as an acute angle.

What is a 45-degree Angle?

A 45-degree is a common angle measurement that divides a right angle (90 degrees) exactly in half. A 45-degree angle is a type of angle that measures 45°

Simple ways to understand what 45 degree Angle is are as follows:

• Visually: Imagine a square. The diagonal line from one corner to the opposite corner cuts the square exactly in half, creating two 45-degree angles.

• Units: Degrees are units used to measure angles. A circle can be divided into 360 degrees, so a 45-degree angle is 45/360 (= 1/4) of a complete circle.

Note: A 45-degree angle equals approximately π/4 radians.

45-Degree Angle Definition

A 45-degree angle is an acute angle that is exactly half of a right angle (90 degrees).

Dividing a Right Angle: An angle bisector of a 90-degree angle forms two equal angles, each measuring 45 degrees.

How to Draw 45 degrees Angle?

We can construct a 45° angle,

• Using a Compass and Ruler
• Using Protractor

Steps to Construct a 45° Angle using a Compass and Ruler

To construct a 45° angle using compass and ruler, we can use the following steps:

Step 1: Draw a line segment: Start by drawing a straight line segment on your paper. Label the endpoints A and B.

Step 2: Create a right angle: With the compass point placed at point A, set the compass to any convenient radius. Draw an arc that intersects line segment AB at point C.

Step 3: Mark the intersecting points: Without changing the compass setting, place the compass point on point C and draw another arc that intersects the first arc above line segment AB. Label this intersection point D.

Step 4: Draw the perpendicular line: Now, use a ruler (optional) to draw a vertical line (perpendicular to line segment AB) passing through point D. Label the point where this line intersects line segment AB as E.

By drawing the intersecting arcs in steps 2 and 3, you’ve created a right angle (angle AEC) because any angle formed by the radius of a circle is a right angle.

Drawing 45-Degree Angle

• Bisect Right Angle: With the compass point on point D (the intersection point of the arcs) and the same compass setting used previously, draw another arc that intersects both the first arc (above line AB) and the vertical line drawn in step 4. Label this new intersection point F.

• Draw 45-Degree Angle: Finally, draw a line segment from point A to point F. This line segment creates angle FAB which is the 45-degree angle you wanted to draw.

Steps to Construct a 45° Angle using Protractor

Below is the step-by-step process describing how to draw a 45-degree angle using a protractor:

To construct a 45° angle using protector, we can use the following steps:

Step 1: Draw a line segment: Start by drawing a straight line segment on your paper. Label one endpoint A.

Step 2: Place the protractor: Align the centre of the protractor’s base (the straight edge without markings) exactly at point A on the line segment.

Step 3: Locate the 45-degree mark: Look for the markings along the outer edge of the protractor, which is typically a circular scale. Find the degree marking labelled “45” (or somewhere between 40 and 50 degrees).

Step 4: Mark the angle: Make a small pencil mark on the line segment at the point that aligns with the 45-degree mark on the protractor’s outer scale. Label this point B.

Step 5: Draw the angle: Finally, use a ruler (optional) to draw a straight line segment from point A to the point you marked in step 4 (point B). This line segment creates angle CAB, which is your 45-degree angle.

Properties of 45-degree Angle

The most important property of a 45-degree angle is its measure about a right angle:

• Half of a right angle: A 45-degree angle is exactly half of a right angle which measures 90 degrees. This means two 45-degree angles placed together form a right angle.

Other Properties

• Special right triangle: When a right triangle has one angle measuring 90 degrees and the other two angles are congruent (equal), those other two angles each measure 45 degrees. This type of right triangle is called a 45-45-90 triangle.

• Side ratios in a 45-45-90 triangle: In a 45-45-90 triangle, the length of the hypotenuse (the side opposite the right angle) is 2​ times the length of either of the legs (the two sides forming the right angle).

• Unit circle: A 45-degree angle corresponds to a specific point on the unit circle in trigonometry. This point is where the x and y coordinates are each equal to 1/√2.
• Square diagonals: The diagonals of a square create two 45-degree angles with each of the square’s sides.

While not a property of the angle itself, a 45-degree angle is also:

• Classified as an acute angle because its measure is between 0 and 90 degrees.

Trigonometric Values of 45 Degree

Trigonometric values (sine, cosine, tangent, etc.) of a 45-degree angle are special because they have exact values that can be expressed without decimals.

Below are the main trigonometric values for a 45-degree angle:

• Sine (sin): sin(45°) = 1/√2
• Cosine (cos): cos(45°) = 1/√2
• Tangent (tan): tan(45°) = 1

There are other trigonometric functions like cotangent (cot), secant (sec), and cosecant (CSC), but these can be derived from the sine and cosine values using trigonometric identities.