Defining a Circle Segment
Segment is a geometric figure that is obtained by cutting off part of a circle with a chord.
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This figure is located between the chord and the arc of the circle.
ChordThis is a segment lying inside a circle and connecting two arbitrarily chosen points on it.
When cutting off part of a circle with a chord, you can consider two figures: this is our segment and an isosceles triangle, the sides of which are the radii of the circle.
The area of a segment can be found as the difference between the areas of a sector of a circle and this isosceles triangle.
The area of a segment can be found in several ways. Let's look at them in more detail.
Formula for the area of a circle segment using the radius and arc length of the circle, the height and base of the triangle
S = 1 2 ⋅ R ⋅ s − 1 2 ⋅ h ⋅ a S=\frac(1)(2)\cdot R\cdot s-\frac(1)(2)\cdot h\cdot aS=2 1 ⋅ R⋅s −2 1 ⋅ h⋅a
R R R- radius of the circle;
s s s- arc length;
h h h- height of an isosceles triangle;
a a a- the length of the base of this triangle.
Given a circle, its radius is numerically equal to 5 (cm), the height, which is drawn to the base of the triangle, is equal to 2 (cm), the length of the arc is 10 (cm). Find the area of a circle segment.
Solution
R=5 R=5 R=5
h = 2 h=2 h =2
s = 10 s=10 s =1
0
To calculate the area, we only need the base of the triangle. Let's find it using the formula:
A = 2 ⋅ h ⋅ (2 ⋅ R − h) = 2 ⋅ 2 ⋅ (2 ⋅ 5 − 2) = 8 a=2\cdot\sqrt(h\cdot(2\cdot R-h))=2\cdot\ sqrt(2\cdot(2\cdot 5-2))=8a =2 ⋅ h ⋅ (2 ⋅ R − h ) = 2 ⋅ 2 ⋅ (2 ⋅ 5 − 2 ) = 8
Now you can calculate the area of the segment:
S = 1 2 ⋅ R ⋅ s − 1 2 ⋅ h ⋅ a = 1 2 ⋅ 5 ⋅ 10 − 1 2 ⋅ 2 ⋅ 8 = 17 S=\frac(1)(2)\cdot R\cdot s-\frac (1)(2)\cdot h\cdot a=\frac(1)(2)\cdot 5\cdot 10-\frac(1)(2)\cdot 2\cdot 8=17S=2 1 ⋅ R⋅s −2 1 ⋅ h⋅a =2 1 ⋅ 5 ⋅ 1 0 − 2 1 ⋅ 2 ⋅ 8 = 1 7 (see sq.)
Answer: 17 cm sq.
Formula for the area of a circle segment given the radius of the circle and the central angle
S = R 2 2 ⋅ (α − sin (α)) S=\frac(R^2)(2)\cdot(\alpha-\sin(\alpha))S=2 R 2 ⋅ (α − sin(α))
R R R- radius of the circle;
α\alpha α
- the central angle between two radii subtending the chord, measured in radians.
Find the area of a circle segment if the radius of the circle is 7 (cm) and the central angle is 30 degrees.
Solution
R=7 R=7 R=7
α = 3 0 ∘ \alpha=30^(\circ)α
=
3
0
∘
Let's first convert the angle in degrees to radians. Because the π\pi π
A radian is equal to 180 degrees, then:
3 0 ∘ = 3 0 ∘ ⋅ π 18 0 ∘ = π 6 30^(\circ)=30^(\circ)\cdot\frac(\pi)(180^(\circ))=\frac(\pi )(6)3
0
∘
=
3
0
∘
⋅
1
8
0
∘
π
=
6
π
radian. Then the area of the segment is:
S = R 2 2 ⋅ (α − sin (α)) = 49 2 ⋅ (π 6 − sin (π 6)) ≈ 0.57 S=\frac(R^2)(2)\cdot(\alpha- \sin(\alpha))=\frac(49)(2)\cdot\Big(\frac(\pi)(6)-\sin\Big(\frac(\pi)(6)\Big)\Big )\approx0.57S=2 R 2 ⋅ (α − sin(α)) =2 4 9 ⋅ ( 6 π − sin ( 6 π ) ) ≈ 0 . 5 7 (see sq.)
Answer: 0.57 cm sq.
The area of a circular segment is equal to the difference between the area of the corresponding circular sector and the area of the triangle formed by the radii of the sector corresponding to the segment and the chord limiting the segment.
Example 1
The length of the chord subtending the circle is equal to the value a. The degree measure of the arc corresponding to the chord is 60°. Find the area of the circular segment.
Solution
A triangle formed by two radii and a chord is isosceles, so the altitude drawn from the vertex of the central angle to the side of the triangle formed by the chord will also be the bisector of the central angle, dividing it in half, and the median, dividing the chord in half. Knowing that the sine of the angle is equal to the ratio of the opposite leg to the hypotenuse, we can calculate the radius:
Sin 30°= a/2:R = 1/2;
Sc = πR²/360°*60° = πa²/6
S▲=1/2*ah, where h is the height drawn from the vertex of the central angle to the chord. According to the Pythagorean theorem h=√(R²-a²/4)= √3*a/2.
Accordingly, S▲=√3/4*a².
The area of the segment, calculated as Sreg = Sc - S▲, is equal to:
Sreg = πa²/6 - √3/4*a²
Substituting numeric value Instead of the value a, you can easily calculate the numerical value of the segment area.
Example 2
The radius of the circle is equal to a. The degree measure of the arc corresponding to the segment is 60°. Find the area of the circular segment.
Solution:
The area of the sector corresponding to a given angle can be calculated using the following formula:
Sc = πа²/360°*60° = πa²/6,
The area of the triangle corresponding to the sector is calculated as follows:
S▲=1/2*ah, where h is the height drawn from the vertex of the central angle to the chord. According to the Pythagorean theorem h=√(a²-a²/4)= √3*a/2.
Accordingly, S▲=√3/4*a².
And finally, the area of the segment, calculated as Sreg = Sc - S▲, is equal to:
Sreg = πa²/6 - √3/4*a².
The solutions in both cases are almost identical. Thus, we can conclude that to calculate the area of a segment in the simplest case, it is enough to know the value of the angle corresponding to the arc of the segment and one of two parameters - either the radius of the circle or the length of the chord subtending the arc of the circle forming the segment.
The mathematical value of area has been known since ancient Greece. Even in those distant times, the Greeks found out that an area is a continuous part of a surface, which is limited on all sides by a closed contour. This is a numerical value that is measured in square units. Area is a numerical characteristic of both flat geometric figures (planimetric) and the surfaces of bodies in space (volumetric).
Currently, it is found not only in the school curriculum in geometry and mathematics lessons, but also in astronomy, everyday life, construction, design development, manufacturing and many other human subjects. Very often we resort to calculating the areas of segments on a personal plot when designing a landscape area or during renovation work on an ultra-modern room design. Therefore, knowledge of methods for calculating various areas will be useful always and everywhere.
To calculate the area of a circular segment and a sphere segment, you need to understand the geometric terms that will be needed during the computational process.
First of all, a segment of a circle is a fragment of a flat figure of a circle, which is located between the arc of a circle and the chord cutting it off. This concept should not be confused with the sector figure. These are completely different things.
A chord is a segment that connects two points lying on a circle.
The central angle is formed between two segments - radii. It is measured in degrees by the arc on which it rests.
A segment of a sphere is formed when a part is cut off by some plane. In this case, the base of the spherical segment is a circle, and the height is the perpendicular emanating from the center of the circle to the intersection with the surface of the sphere. This intersection point is called the vertex of the ball segment.
In order to determine the area of a sphere segment, you need to know the cut-off circle and the height of the spherical segment. The product of these two components will be the area of the sphere segment: S=2πRh, where h is the height of the segment, 2πR is the circumference, and R is the radius of the great circle.
In order to calculate the area of a circle segment, you can resort to the following formulas:
1. To find the area of a segment by the most in a simple way, it is necessary to calculate the difference between the area of the sector in which the segment is inscribed, and whose base is the chord of the segment: S1=S2-S3, where S1 is the area of the segment, S2 is the area of the sector and S3 is the area of the triangle.
You can use an approximate formula for calculating the area of a circular segment: S=2/3*(a*h), where a is the base of the triangle or h is the height of the segment, which is the result of the difference between the radius of the circle and
2. The area of a segment different from a semicircle is calculated as follows: S = (π R2:360)*α ± S3, where π R2 is the area of the circle, α is the degree measure of the central angle, which contains the arc of the circle segment, S3 is the area of the triangle that was formed between the two radii of the circle and the chord, which has an angle at the central point of the circle and two vertices at the points of contact of the radii with circle.
If angle α< 180 градусов, используется знак минус, если α >180 degrees, plus sign applied.
3. You can calculate the area of a segment using other methods using trigonometry. As a rule, a triangle is taken as a basis. If the central angle is measured in degrees, then the following formula is acceptable: S= R2 * (π*(α/180) - sin α)/2, where R2 is the square of the radius of the circle, α is the degree measure of the central angle.
4. To calculate the area of a segment using trigonometric functions, you can use another formula, provided that the central angle is measured in radians: S= R2 * (α - sin α)/2, where R2 is the square of the radius of the circle, α is the degree measure of the central angle.
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