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Area of Small Lune Formula

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The Area of Small Lune is the total quantity of plane occupied by the smaller lune portion in the Lune shape. Check FAQs

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}))

A_Small - Area of Small Lune?A_Triangle - Area of Triangle of Lune?r_Smaller - Radius of Smaller Circle of Lune?r_Larger - Radius of Larger Circle of Lune?d_Centers - Distance of Centers of Circles of Lune?

Area of Small Lune Example

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

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

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

62.5115 = (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}))

62.5115m² = (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}))

62.5115 = (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 Small Lune Solution

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

FIRST Step Consider the formula

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}))

Next Step Substitute values of Variables

∴

A Small = (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 Small = (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 Small = 62.5115128945771m²

LAST Step Rounding Answer

∴

A Small = 62.5115m²

Area of Small Lune Formula Elements

Variables

Functions

Area of Small Lune

The Area of Small Lune is the total quantity of plane occupied by the smaller lune portion in the Lune shape.

Symbol: A_Small

Measurement: AreaUnit: m²

Note: Value should be greater than 0.

Formulas to find Area of Small 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 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

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

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 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}))

Go Area of Section of Lune

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})))

How to Evaluate Area of Small Lune?

Area of Small Lune evaluator uses Area of Small Lune = (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 Small Lune, The Area of Small Lune formula is defined as total quantity of plane occupied by the smaller Lune portion of the Lune shape. Area of Small Lune is denoted by A_Small symbol.

How to evaluate Area of Small Lune using this online evaluator? To use this online evaluator for Area of Small Lune, enter Area of Triangle of Lune (A_Triangle), Radius of Smaller Circle of Lune (r_Smaller), 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 Small Lune

What is the formula to find Area of Small Lune?

The formula of Area of Small Lune is expressed as Area of Small Lune = (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- 62.51151 = (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 Small Lune?

With Area of Triangle of Lune (A_Triangle), Radius of Smaller Circle of Lune (r_Smaller), Radius of Larger Circle of Lune (r_Larger) & Distance of Centers of Circles of Lune (d_Centers) we can find Area of Small Lune using the formula - Area of Small Lune = (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 Cosine, Inverse Cosine function(s).

Can the Area of Small Lune be negative?

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

Which unit is used to measure Area of Small Lune?

Area of Small 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 Small Lune can be measured.

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Area of Small Lune Formula

Area of Small Lune Example

Area of Small Lune Solution

Area of Small Lune Formula Elements

Other formulas in Lune category

How to Evaluate Area of Small Lune?

FAQs on Area of Small Lune

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