Area of a Spherical Triangle

The equation for the area of a spherical triangle can be proven by finding four areas.

Area of a spherical lune
Areas of three lunes that create the spherical triangle
Surface area of the sphere in terms of the areas of the lunes
Area of the spherical triangle

Area of a Spherical Lune

Let

L (x)

denote the area of a spherical lune where

x

is the angle between two great circles.

The surface area of a sphere is

4 π r^{2}

and its central angle measures

36 0^{\circ} .

Because the area of the spherical lune is proportional to the angle, the following proportion can be written.

\frac{L ( x )}{x} = \frac{4 π r ^{2}}{3 6 0 ^{\circ}}

The equation can now be solved for

L (x)

to find the area of the spherical lune.

\frac{L ( x )}{x} = \frac{4 π r ^{2}}{3 6 0 ^{\circ}}

▼

Solve for

L (x)

ReduceFrac

$\frac{a}{b} = \frac{a / 4}{b / 4}$

\frac{L ( x )}{x} = \frac{π r ^{2}}{9 0 ^{\circ}}

MultEqn

$LHS \cdot x = RHS \cdot x$

L (x) = \frac{π r ^{2}}{9 0 ^{\circ}} \cdot x

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

L (x) = \frac{x π r ^{2}}{9 0 ^{\circ}}

Area of the Spherical Lunes

Consider a two-dimensional view from the top of the pair of spherical lunes. Since the vertical angles are congruent by the Vertical Angles Theorem, a pair of opposite spherical lunes have the same area.

Each pair of great circles intersects at two opposite points. Consequently, the intersection of three great circles will create two spherical triangles that are opposite to each other. Based on the diagram,

x,

y,

and

z

are the measures of the interior angles of the spherical triangles.

Similar to

L (x),

areas

L (y)

and

L (z)

can be found.

L (x) = \frac{x π r ^{2}}{9 0 ^{\circ}} L (y) = \frac{y π r ^{2}}{9 0 ^{\circ}} L (z) = \frac{z π r ^{2}}{9 0 ^{\circ}}

Surface Area of the Sphere

To find the area

A

of the spherical triangle, the surface area of the sphere will be written in terms of the area of the spherical lunes and the area of the spherical triangle. It can be noted that all three pairs of lunes cover the entire sphere.

Now, add the surface areas of the spherical lunes. Since the lunes in a pair share the same surface area, each area will be doubled.

2 L (x) + 2 L (y) + 2 L (z)

Each pair of lunes intersects at the triangle and the opposite triangle, so the sum contains the area of the triangle six times. Consequently, four areas

A

of the triangle must be subtracted from the sum so that the area of each triangle is calculated only once. In this way, the surface area of the sphere,

4 π r^{2},

will be obtained.

2 L (x) + 2 L (y) + 2 L (z) - 4 A = 4 π r^{2}

Area of the Spherical Triangle

Finally, the expressions for the areas of the spherical lunes will be substituted into the equation to solve for

A .

2 L (x) + 2 L (y) + 2 L (z) - 4 A = 4 π r^{2}

SubstituteValues

Substitute values

2 (\frac{x π r ^{2}}{9 0 ^{\circ}}) + 2 (\frac{y π r ^{2}}{9 0 ^{\circ}}) + 2 (\frac{z π r ^{2}}{9 0 ^{\circ}}) - 4 A = 4 π r^{2}

▼

Solve for

A

Multiply

\frac{x π r ^{2}}{4 5 ^{\circ}} + \frac{y π r ^{2}}{4 5 ^{\circ}} + \frac{z π r ^{2}}{4 5 ^{\circ}} - 4 A = 4 π r^{2}

AddEqn

$LHS + 4 A = RHS + 4 A$

\frac{x π r ^{2}}{4 5 ^{\circ}} + \frac{y π r ^{2}}{4 5 ^{\circ}} + \frac{z π r ^{2}}{4 5 ^{\circ}} = 4 A + 4 π r^{2}

FactorOut

Factor out $4$

\frac{x π r ^{2}}{4 5 ^{\circ}} + \frac{y π r ^{2}}{4 5 ^{\circ}} + \frac{z π r ^{2}}{4 5 ^{\circ}} = 4 (A + π r^{2})

DivEqn

$LHS / 4 = RHS / 4$

\frac{x π r ^{2}}{1 8 0 ^{\circ}} + \frac{y π r ^{2}}{1 8 0 ^{\circ}} + \frac{z π r ^{2}}{1 8 0 ^{\circ}} = A + π r^{2}

SubEqn

$LHS - π r^{2} = RHS - π r^{2}$

\frac{x π r ^{2}}{1 8 0 ^{\circ}} + \frac{y π r ^{2}}{1 8 0 ^{\circ}} + \frac{z π r ^{2}}{1 8 0 ^{\circ}} - π r^{2} = A

RearrangeEqn

Rearrange equation

A = \frac{x π r ^{2}}{1 8 0 ^{\circ}} + \frac{y π r ^{2}}{1 8 0 ^{\circ}} + \frac{z π r ^{2}}{1 8 0 ^{\circ}} - π r^{2}

FactorOut

Factor out $π r^{2}$

A = (\frac{x}{1 8 0 ^{\circ}} + \frac{y}{1 8 0 ^{\circ}} + \frac{z}{1 8 0 ^{\circ}} - 1) π r^{2}

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Area of a Spherical Triangle

Proof

Area of a Spherical Lune

Area of the Spherical Lunes

Surface Area of the Sphere

Area of the Spherical Triangle

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