45 Degree Angle: Construction and Examples

45-degree angle is a fundamental concept in geometry and trigonometry. An angle is a form of geometrical shape constructed by joining two rays to each other at their endpoints. The two lines joined together are called the arms of the angle.

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.

In this article, we will learn about, a 45-degree angle, how to draw a 45-degree angle using a compass and protractor, the properties of a 45-degree angle trigonometric values of a 45-degree angle and others in detail.

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.

We can construct a 45° angle,

• Using a Compass and Ruler
• Using Protractor

Construct a 45° Angle using a Compass and Ruler

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

Steps to Construct a 45°

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.

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:

Steps to Construct a 45°

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.

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 (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°) = √2 / 2 = 1/√2
• Cosine (cos): cos(45°) = √2 / 2 = 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.

Some Uses of a 45-degree angle are:

Geometry and Trigonometry

• Right-Angled Isosceles Triangle: A 45-degree angle is used to form right-angled isosceles triangles where the other two angles are also 45 degrees. This type of triangle has sides in a 1:1:√2​ ratio.

• Trigonometric Ratios: The sine and cosine of 45 degrees are both equal to √2 ​​, making it a crucial angle for solving trigonometric problems.

Engineering and Construction

• Structural Design: In civil engineering and architecture, 45-degree angles are used in designing trusses and frames due to their symmetrical properties.

• Staircases: 45-degree angles are often used in the design of staircase steps, making calculations straightforward.

Drafting and Design

• Technical Drawings: Drafters use 45-degree set squares for creating accurate angles and lines in technical drawings and blueprints.

• Graphic Design: 45-degree angles are used in various aspects of graphic design, such as creating grids, patterns, and isometric drawings.

Computer Graphics and Digital Imaging

• Pixel Art: In digital graphics, 45-degree angles are used to create diagonal lines and patterns, especially in pixel art and low-resolution graphics.

• 3D Modeling: In 3D computer graphics, 45-degree angles are used in constructing objects, especially in creating meshes and wireframes.

Articles Related to 45 degree Angle:

• Angles
• Types of Anglesa
• Acute Angle
• Right Angle
• Obtuse Angle

Example 1: Finding the missing side length in a 45-45-90 triangle. If one leg of a 45-45-90 triangle measures 6 cm, what is the length of the hypotenuse?

Solution:

Since the leg is 6 cm, the hypotenuse

= 6 cm × √2

= 8.49 cm (approximately)

Example 2: Finding the degree measure of an angle based on its position in a shape. All four angles in a rectangle measure 90 degrees. If we draw a diagonal line dividing the rectangle into two congruent right triangles, what is the measure of each of the angles where the diagonal line meets the rectangle’s sides?

Solution:

By dividing the rectangle diagonally, we create two 45-45-90 triangles.

Because each triangle has one right angle (90 degrees), the two remaining angles must add up to the remaining 90 degrees (since the angles in a triangle sum to 180 degrees).

So, each of the angles where the diagonal meets the sides of the rectangle measures

90 degrees / 2 = 45 degrees

Example 3: Using a 45-degree angle to solve for an unknown distance. You are standing next to a building. You look up and see the top of the building at a 45-degree angle. You are 10 meters away from the base of the building, and your eye level is 1.5 meters from the ground. How tall is the building?

Solution:

This situation creates a 45-45-90 triangle, where the distance from you to the building (10 meters) is one leg and the unknown building height is the other leg.

Distance from your eye to the top of the building (what we want to find) is the hypotenuse. Since the legs in a 45-45-90 triangle are congruent the building height is also 10 meters.

Then, to find the total height from the ground to the top of the building we add the height of your eye level:

10 meters (building height) + 1.5 meters (eye level)

= 11.5 meters

What does a 45-degree angle look like?

45-degree angle looks like a greater than (>) sign or lesser than sign (<). If we draw an angle bisector to a 90-degree angle the small angles thus formed will be 45-degree angles. It also looks like the open face of scissors.

What is 45 degree angle called?

A 45-degree angle is an acute angle. An acute angle lies between 0 degree and 90 degrees or in other words.An acute angle is one that is less than 90 degrees.

How much is Sin 45?

Sin 45 degrees is the value of sine trigonometric function for an angle equal to 45 degrees. The value of sin 45° is 1/√2 or 0.7071 (approx).

How much is Cos 45?

Cos 45 degrees is the value of cosine trigonometric function for an angle equal to 45 degrees. The value of cos 45° is 1/√2 or 0.7071 (approx).

How to calculate a 45-degree angle?

Create perpendicular lines to make a 90° angle. Divide the 90° angle in half to obtain a 45° angle.