30-degree angle is an acute angle which is one-twelfth of a full circle (360 degrees). An angle formed when two rays meet at a point. A 30-degree angle is often used in architectural designs, such as the slope of roofs or ramps.
In this article, we will learn about 30-degree angles, how to measure 30-degree angles, properties of 30-degree angles along with a few examples based on it.
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An angle is a geometric figure formed by two rays, called the sides of the angle, sharing a common endpoint, known as the vertex.
What is 30-Degree Angle?
A 30-degree angle is an acute angle which is one-twelfth of a full circle (360 degrees). This means it is a relatively small angle.
Assume O to be the vertex—that is, the intersection of ray OA and ray OB.
A 30 degree angle is the one generated by these two rays if their angle measurement is 30 degrees. Ray OA and ray OB create an angle expressed as ∠AOB or ∠BOA.
∠AOB = ∠BOA = 30°
Radian Measure of 30-Degree
In radians, a 30-degree angle is equivalent to π/6 radians, since there are 2π radians in a full circle.
30-Degrees in Geometry
In geometry and trigonometry, it is among the often used angles. For example, in a 30–60–90 right triangle, the smallest angle is 30 degrees.
Trigonometric Values of 30-Degree
The trigonometric values of 30 degree angle are as follow:
- sin 30° = 1/2
- cos 30° = √3/2
- tan 30° = 1/√3 = √3/3
To visualize a 30-degree angle, imagine the angle formed by the hands of a clock at 1 o'clock. The angle between the hour and minute hands is 30 degrees.
Construction of 30-degree Angle Using Compass
the steps to construct a 30° angle using a compass are given below:
Step 1: Draw a straight line segment, say AB.
Step 2: Place a point, say O, on line AB. This will be the vertex of your angle.
Step 3: Draw an arc that intersects line AB at point C by placing the compass point on O. Any radius is can be taken for this arc.
Step 4: Position the compass point on C and draw a second arc that crosses the first one without altering the compass width. Name this point of intersection as D.
Step 5: Draw an arc above the line by placing the compass point on C.
Step 6: Position the compass point on D and draw an additional arc that crosses the one drawn in the previous step, all without adjusting the compass width. Put a point E on the junction here.
Step 7: Draw a straight line from O through E. This line OE forms a 30-degree angle with line OB.
Construction of 30-Degree Angle Using Protractor
To construct a 30° angle using a protector follow the steps given below:
Step 1: Sketch a segment of a straight line, such as AB.
Step 2: Mark point O on the line AB. This will be the vertex of your angle.
Step 3: Position the protractor so that its center point is exactly on point O and the baseline of the protractor is aligned with line AB.
Step 4: On the protractor, note the 30-degree mark. Depending on the direction you wish your angle to open, use the appropriate scale—inner or outer.
Step 5: On your paper, mark or place a tiny dot at the protractor's 30-degree point.
Step 6: Remove the protractor and draw a straight line passing through your mark at point O using a ruler. If necessary, note this new point C.
Step 7: ∠AOC is 30 degrees.
Properties of 30-Degree Angle
Some of the common properties and characteristics of 30° angle are:
- A 30-degree angle measures exactly 30 degrees, which is one-twelfth of a full circle (360 degrees).
- The complement of a 30-degree angle is 60 degrees because complementary angles add up to 90 degrees.
- The supplement of a 30-degree angle is 150 degrees because supplementary angles add up to 180 degrees.
- The angle bisector of a 30° angle divides it into two equal angles, each measuring 15°.
- In radians, a 30-degree angle is denoted as π\6 radians.
- In trigonometry, the sine of 30° is 1/2, the cosine of 30° is √3/2, and the tangent of 30° is 1/√3.
Real-Life Examples of 30 Degree Angle
Some real-life examples where 30 degree angle can be seen are:
- Staircases in buildings often have a slope with an angle of around 30 degrees. This angle provides a comfortable incline for walking up and down stairs.
- The pitch of many roofs is approximately 30 degrees. This angle allows for efficient water runoff and is common in residential construction.
- Some highway exit ramps and curved roads have banking angles of about 30 degrees. This slope helps vehicles navigate turns safely at higher speeds.
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30 Degree Angle - Solved Example
1. In a right triangle, one angle is 30 degrees, and the hypotenuse is 10 units long. Find the lengths of the other two sides.
Let the triangle be △ABC with ∠A=30∘, ∠B=60∘, and ∠C=90∘.
The hypotenuse AC is 10 units.
Using trigonometric ratios for a 30 degree angle:
Opposite side to ∠A:
BC=AC × sin (30∘)
=10 × 1/2
=5 units
Adjacent side to ∠A:
AB=AC × cos (30∘)
=10 × √3/2
=5√3 units.
2. Determine the slope of a road that makes a 30-degree angle with the horizontal.
The slope (m) is given by the tangent of the angle.
m=tan (30∘)
Hence, slope (m) = 1/√3 = √3/3.