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Area of Section of Lune Formula

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The Area of Section of Lune is the total quantity of plane occupied by the section formed between the larger and smaller Lunes in the Lune shape or the portion of intersection of circles in the Lune. Check FAQs

A Section = (π \cdot {r Smaller}^{2}) - ((2 \cdot A Triangle) + ({r Smaller}^{2} \cdot \arccos (\frac{{r Larger}^{2} - {r Smaller}^{2} - {d Centers}^{2}}{2 \cdot r Smaller \cdot d Centers})) - ({r Larger}^{2} \cdot \arccos (\frac{{r Larger}^{2} + {d Centers}^{2} - {r Smaller}^{2}}{2 \cdot r Larger \cdot d Centers})))

A_Section - Area of Section of Lune?r_Smaller - Radius of Smaller Circle of Lune?A_Triangle - Area of Triangle of Lune?r_Larger - Radius of Larger Circle of Lune?d_Centers - Distance of Centers of Circles of Lune?π - Archimedes' constant?

Area of Section of Lune Example

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Here is how the Area of Section of Lune equation looks like with Values.

Here is how the Area of Section of Lune equation looks like with Units.

Here is how the Area of Section of Lune equation looks like.

16.0283 = (3.1416 \cdot 5^{2}) - ((2 \cdot 20) + (5^{2} \cdot \arccos (\frac{8^{2} - 5^{2} - 10^{2}}{2 \cdot 5 \cdot 10})) - (8^{2} \cdot \arccos (\frac{8^{2} + 10^{2} - 5^{2}}{2 \cdot 8 \cdot 10})))

16.0283m² = (3.1416 \cdot {5m}^{2}) - ((2 \cdot 20m²) + ({5m}^{2} \cdot \arccos (\frac{{8m}^{2} - {5m}^{2} - {10m}^{2}}{2 \cdot 5m \cdot 10m})) - ({8m}^{2} \cdot \arccos (\frac{{8m}^{2} + {10m}^{2} - {5m}^{2}}{2 \cdot 8m \cdot 10m})))

16.0283 = (3.1416 \cdot 5^{2}) - ((2 \cdot 20) + (5^{2} \cdot \arccos (\frac{8^{2} - 5^{2} - 10^{2}}{2 \cdot 5 \cdot 10})) - (8^{2} \cdot \arccos (\frac{8^{2} + 10^{2} - 5^{2}}{2 \cdot 8 \cdot 10})))

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Area of Section of Lune Solution

Follow our step by step solution on how to calculate Area of Section of Lune?

FIRST Step Consider the formula

A Section = (π \cdot {r Smaller}^{2}) - ((2 \cdot A Triangle) + ({r Smaller}^{2} \cdot \arccos (\frac{{r Larger}^{2} - {r Smaller}^{2} - {d Centers}^{2}}{2 \cdot r Smaller \cdot d Centers})) - ({r Larger}^{2} \cdot \arccos (\frac{{r Larger}^{2} + {d Centers}^{2} - {r Smaller}^{2}}{2 \cdot r Larger \cdot d Centers})))

Next Step Substitute values of Variables

∴

A Section = (π \cdot {5m}^{2}) - ((2 \cdot 20m²) + ({5m}^{2} \cdot \arccos (\frac{{8m}^{2} - {5m}^{2} - {10m}^{2}}{2 \cdot 5m \cdot 10m})) - ({8m}^{2} \cdot \arccos (\frac{{8m}^{2} + {10m}^{2} - {5m}^{2}}{2 \cdot 8m \cdot 10m})))

Next Step Substitute values of Constants

∴

A Section = (3.1416 \cdot {5m}^{2}) - ((2 \cdot 20m²) + ({5m}^{2} \cdot \arccos (\frac{{8m}^{2} - {5m}^{2} - {10m}^{2}}{2 \cdot 5m \cdot 10m})) - ({8m}^{2} \cdot \arccos (\frac{{8m}^{2} + {10m}^{2} - {5m}^{2}}{2 \cdot 8m \cdot 10m})))

Next Step Prepare to Evaluate

∴

A Section = (3.1416 \cdot 5^{2}) - ((2 \cdot 20) + (5^{2} \cdot \arccos (\frac{8^{2} - 5^{2} - 10^{2}}{2 \cdot 5 \cdot 10})) - (8^{2} \cdot \arccos (\frac{8^{2} + 10^{2} - 5^{2}}{2 \cdot 8 \cdot 10})))

Next Step Evaluate

∴

A Section = 16.0283034451678m²

LAST Step Rounding Answer

∴

A Section = 16.0283m²

Area of Section of Lune Formula Elements

Variables

Constants

Functions

Area of Section of Lune

The Area of Section of Lune is the total quantity of plane occupied by the section formed between the larger and smaller Lunes in the Lune shape or the portion of intersection of circles in the Lune.

Symbol: A_Section

Measurement: AreaUnit: m²

Note: Value should be greater than 0.

Formulas to find Area of Section of Lune

Radius of Smaller Circle of Lune

The Radius of Smaller Circle of Lune is the radius of the circle with lesser size, out of the two circles using which the Lune is created.

Symbol: r_Smaller

Measurement: LengthUnit: m

Note: Value should be greater than 0.

Formulas to find Radius of Smaller Circle of Lune

Area of Triangle of Lune

The Area of Triangle of Lune is the total quantity of plane occupied by the triangle joining the centers of two circles of Lune and one of their intersecting point.

Symbol: A_Triangle

Measurement: AreaUnit: m²

Note: Value should be greater than 0.

Formulas to find Area of Triangle of Lune

Radius of Larger Circle of Lune

The Radius of Larger Circle of Lune is the radius of the circle with greater size, out of the two circles using which the Lune is created.

Symbol: r_Larger

Measurement: LengthUnit: m

Note: Value should be greater than 0.

Formulas to find Radius of Larger Circle of Lune

Distance of Centers of Circles of Lune

The Distance of Centers of Circles of Lune is the length of the line joining the centers of two circles with which the Lune is created.

Symbol: d_Centers

Measurement: LengthUnit: m

Note: Value should be greater than 0.

Formulas to find Distance of Centers of Circles of Lune

Archimedes' constant

Archimedes' constant is a mathematical constant that represents the ratio of the circumference of a circle to its diameter.

Symbol: π

Value: 3.14159265358979323846264338327950288

cos

Cosine of an angle is the ratio of the side adjacent to the angle to the hypotenuse of the triangle.

Syntax: cos(Angle)

arccos

Arccosine function, is the inverse function of the cosine function.It is the function that takes a ratio as an input and returns the angle whose cosine is equal to that ratio.

Syntax: arccos(Number)

Other formulas in Lune category

Go Area of Triangle of Lune

A Triangle = \frac{\sqrt{(r Smaller + r Larger + d Centers) \cdot (r Larger + d Centers - r Smaller) \cdot (d Centers + r Smaller - r Larger) \cdot (r Smaller + r Larger - d Centers)}}{4}

Go Area of Small Lune

A Small = (2 \cdot A Triangle) + ({r Smaller}^{2} \cdot \arccos (\frac{{r Larger}^{2} - {r Smaller}^{2} - {d Centers}^{2}}{2 \cdot r Smaller \cdot d Centers})) - ({r Larger}^{2} \cdot \arccos (\frac{{r Larger}^{2} + {d Centers}^{2} - {r Smaller}^{2}}{2 \cdot r Larger \cdot d Centers}))

Go Area of Large Lune

A Large = (π \cdot ({r Larger}^{2} - {r Smaller}^{2})) + (2 \cdot A Triangle) + ({r Smaller}^{2} \cdot \arccos (\frac{{r Larger}^{2} - {r Smaller}^{2} - {d Centers}^{2}}{2 \cdot r Smaller \cdot d Centers})) - ({r Larger}^{2} \cdot \arccos (\frac{{r Larger}^{2} + {d Centers}^{2} - {r Smaller}^{2}}{2 \cdot r Larger \cdot d Centers}))

How to Evaluate Area of Section of Lune?

Area of Section of Lune evaluator uses Area of Section of Lune = (pi*Radius of Smaller Circle of Lune^2)-((2*Area of Triangle of Lune)+(Radius of Smaller Circle of Lune^2*arccos((Radius of Larger Circle of Lune^2-Radius of Smaller Circle of Lune^2-Distance of Centers of Circles of Lune^2)/(2*Radius of Smaller Circle of Lune*Distance of Centers of Circles of Lune)))-(Radius of Larger Circle of Lune^2*arccos((Radius of Larger Circle of Lune^2+Distance of Centers of Circles of Lune^2-Radius of Smaller Circle of Lune^2)/(2*Radius of Larger Circle of Lune*Distance of Centers of Circles of Lune)))) to evaluate the Area of Section of Lune, The Area of Section of Lune formula is defined as the total quantity of plane bounded by the portion of intersection in the Lune shape. Area of Section of Lune is denoted by A_Section symbol.

How to evaluate Area of Section of Lune using this online evaluator? To use this online evaluator for Area of Section of Lune, enter Radius of Smaller Circle of Lune (r_Smaller), Area of Triangle of Lune (A_Triangle), Radius of Larger Circle of Lune (r_Larger) & Distance of Centers of Circles of Lune (d_Centers) and hit the calculate button.

FAQs on Area of Section of Lune

What is the formula to find Area of Section of Lune?

The formula of Area of Section of Lune is expressed as Area of Section of Lune = (pi*Radius of Smaller Circle of Lune^2)-((2*Area of Triangle of Lune)+(Radius of Smaller Circle of Lune^2*arccos((Radius of Larger Circle of Lune^2-Radius of Smaller Circle of Lune^2-Distance of Centers of Circles of Lune^2)/(2*Radius of Smaller Circle of Lune*Distance of Centers of Circles of Lune)))-(Radius of Larger Circle of Lune^2*arccos((Radius of Larger Circle of Lune^2+Distance of Centers of Circles of Lune^2-Radius of Smaller Circle of Lune^2)/(2*Radius of Larger Circle of Lune*Distance of Centers of Circles of Lune)))). Here is an example- 16.0283 = (pi*5^2)-((2*20)+(5^2*arccos((8^2-5^2-10^2)/(2*5*10)))-(8^2*arccos((8^2+10^2-5^2)/(2*8*10)))).

How to calculate Area of Section of Lune?

With Radius of Smaller Circle of Lune (r_Smaller), Area of Triangle of Lune (A_Triangle), Radius of Larger Circle of Lune (r_Larger) & Distance of Centers of Circles of Lune (d_Centers) we can find Area of Section of Lune using the formula - Area of Section of Lune = (pi*Radius of Smaller Circle of Lune^2)-((2*Area of Triangle of Lune)+(Radius of Smaller Circle of Lune^2*arccos((Radius of Larger Circle of Lune^2-Radius of Smaller Circle of Lune^2-Distance of Centers of Circles of Lune^2)/(2*Radius of Smaller Circle of Lune*Distance of Centers of Circles of Lune)))-(Radius of Larger Circle of Lune^2*arccos((Radius of Larger Circle of Lune^2+Distance of Centers of Circles of Lune^2-Radius of Smaller Circle of Lune^2)/(2*Radius of Larger Circle of Lune*Distance of Centers of Circles of Lune)))). This formula also uses Archimedes' constant and , Cosine (cos), Inverse Cosine (arccos) function(s).

Can the Area of Section of Lune be negative?

No, the Area of Section of Lune, measured in Area cannot be negative.

Which unit is used to measure Area of Section of Lune?

Area of Section of Lune is usually measured using the Square Meter[m²] for Area. Square Kilometer[m²], Square Centimeter[m²], Square Millimeter[m²] are the few other units in which Area of Section of Lune can be measured.

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Area of Section of Lune Formula

Area of Section of Lune Example

Area of Section of Lune Solution

Area of Section of Lune Formula Elements

Other formulas in Lune category

How to Evaluate Area of Section of Lune?

FAQs on Area of Section of Lune

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