Formula for the diameter of the impact crater at the circumference

Center angle

Anyone interested in special angles on the circle will sooner or later encounter the central angle, the circumferential angle and the tendon tangent angle. In this chapter we take a closer look at the central angle.

definition

Given is a whole Circle.

In many tasks, however, it is not about a whole circle, but only about one part of which: Each section of the circular line is called a circular arc. An arc is limited by two points on the circle.

The angle,
whose Parting is at the center of the circle
and its leg intersect the boundary points of the circular arc,
called Center angle or Central angle.

For every arc there is exactly one Center angle.

Special case 1

The central angle across a Semicircular arc is a straight angle (\ (180 ^ \ circ \)).

Special case 2

The central angle over one Full circular arc is a full angle (\ (360 ^ \ circ \)).

Calculate center angle

Circumferential angle given

formula

According to the theorem of angles, the following applies:

\ begin {align *} \ alpha = 2 \ cdot \ beta \ end {align *}

manual

  1. Write down the formula
  2. Insert value for \ (\ beta \)
  3. Calculate result

example

example

Calculate the central angle \ (\ alpha \), which belongs to the circumferential angle \ (\ beta = 60 ^ \ circ \).

Write down the formula

\ begin {align *} \ alpha = 2 \ cdot \ beta \ end {align *}

Insert value for \ (\ beta \)

\ begin {align *} \ phantom {\ alpha} = 2 \ cdot 60 ^ \ circ \ end {align *}

Calculate result

\ begin {align *} \ phantom {\ alpha} = 120 ^ \ circ \ end {align *}

Arc and circumference given

formula

\ begin {align *} b = \ frac {\ alpha} {360 ^ \ circ} \ cdot u \ end {align *}

Convert formula to \ (\ alpha \)

\ begin {align *} b & = \ frac {\ alpha} {360 ^ \ circ} \ cdot u && {\ color {gray} | \ text {Swap sides}} \ [5px] \ frac {\ alpha} {360 ^ \ circ} \ cdot u & = b && {\ color {gray} |: u} \ [5px] \ frac {\ alpha} {360 ^ \ circ} & = \ frac {b} {u} && {\ color {gray} | \ cdot 360 ^ \ circ} \ [5px] \ alpha & = \ frac {b} {u} \ cdot 360 ^ \ circ \ end {align *}

manual

  1. Write down the formula
  2. Insert values ​​for \ (b \) and \ (u \)
  3. Calculate result

example

example

Calculate the center angle \ (\ alpha \), which leads to an arc of length \ (b = 1 {,} 5 ~ \ mathrm {km} \) and a circle with the circumference \ (u = 9 ~ \ mathrm {km} \) belongs.

Write down the formula

\ begin {align *} \ alpha = \ frac {b} {u} \ cdot 360 ^ \ circ \ end {align *}

Insert values ​​for \ (b \) and \ (u \)

\ begin {align *} \ phantom {\ alpha} = \ frac {1 {,} 5 ~ \ mathrm {km}} {9 ~ \ mathrm {km}} \ cdot 360 ^ \ circ \ end {align *}

Calculate result

\ begin {align *} \ phantom {\ alpha} = 60 \ ldots ^ \ circ \ end {align *}

Arc and radius given

formula

\ begin {align *} b = \ frac {\ alpha} {360 ^ \ circ} \ cdot 2 \ pi \ cdot r \ end {align *}

Convert formula to \ (\ alpha \)

\ begin {align *} b & = \ frac {\ alpha} {360 ^ \ circ} \ cdot 2 \ pi \ cdot r && {\ color {gray} | \ text {swap sides}} \ [5px] \ frac {\ alpha} {360 ^ \ circ} \ cdot 2 \ pi \ cdot r & = b && {\ color {gray} |: (2 \ pi \ cdot r)} \ [5px] \ frac {\ alpha } {360 ^ \ circ} & = \ frac {b} {2 \ pi \ cdot r} && {\ color {gray} | \ cdot 360 ^ \ circ} \ [5px] \ alpha & = \ frac {b } {2 \ pi \ cdot r} \ cdot 360 ^ \ circ \ end {align *}

manual

  1. Write down the formula
  2. Insert values ​​for \ (b \) and \ (r \)
  3. Calculate result

example

example

Calculate the center angle \ (\ alpha \), which belongs to an arc of length \ (b = 3 ~ \ mathrm {cm} \) and a circle with the radius \ (r = 5 ~ \ mathrm {cm} \). Round the result to one decimal place.

Write down the formula

\ begin {align *} \ alpha = \ frac {b} {2 \ pi \ cdot r} \ cdot 360 ^ \ circ \ end {align *}

Insert values ​​for \ (b \) and \ (r \)

\ begin {align *} \ phantom {\ alpha} = \ frac {3 ~ \ mathrm {cm}} {2 \ pi \ cdot 5 ~ \ mathrm {cm}} \ cdot 360 ^ \ circ \ end {align *}

Calculate result

\ begin {align *} \ phantom {\ alpha} & = 34 {,} 37 \ ldots ^ \ circ \ [5px] & \ approx 34 {,} 4 ^ \ circ \ end {align *}

Arc and diameter given

formula

\ begin {align *} b = \ frac {\ alpha} {360 ^ \ circ} \ cdot \ pi \ cdot d \ end {align *}

Convert formula to \ (\ alpha \)

\ begin {align *} b & = \ frac {\ alpha} {360 ^ \ circ} \ cdot \ pi \ cdot d && {\ color {gray} | \ text {swap sides}} \ [5px] \ frac {\ alpha} {360 ^ \ circ} \ cdot \ pi \ cdot d & = b && {\ color {gray} |: (\ pi \ cdot d)} \ [5px] \ frac {\ alpha} {360 ^ \ circ} & = \ frac {b} {\ pi \ cdot d} && {\ color {gray} | \ cdot 360 ^ \ circ} \ [5px] \ alpha & = \ frac {b} {\ pi \ cdot d} \ cdot 360 ^ \ circ \ end {align *}

manual

  1. Write down the formula
  2. Insert values ​​for \ (b \) and \ (d \)
  3. Calculate result

example

example

Calculate the center angle \ (\ alpha \), which belongs to an arc of length \ (b = 2 ~ \ mathrm {m} \) and a circle with the diameter \ (d = 4 ~ \ mathrm {m} \). Round the result to two decimal places.

Write down the formula

\ begin {align *} \ alpha = \ frac {b} {\ pi \ cdot d} \ cdot 360 ^ \ circ \ end {align *}

Insert values ​​for \ (b \) and \ (d \)

\ begin {align *} \ phantom {\ alpha} = \ frac {2 ~ \ mathrm {m}} {\ pi \ cdot 4 ~ \ mathrm {m}} \ cdot 360 ^ \ circ \ end {align *}

Calculate result

\ begin {align *} \ phantom {\ alpha} & = 57 {,} 295 \ ldots ^ \ circ \ [5px] & \ approx 57 {,} 30 ^ \ circ \ end {align *}

Circle section and area given

formula

\ begin {align *} A _ {\ text {circle section}} = \ frac {\ alpha} {360 ^ \ circ} \ cdot A _ {\ text {circle}} \ end {align *}

Convert formula to \ (\ alpha \)

\ begin {align *} A _ {\ text {circle section}} & = \ frac {\ alpha} {360 ^ \ circ} \ cdot A _ {\ text {circle}} && {\ color {gray} | \ text {pages swap}} \ [5px] \ frac {\ alpha} {360 ^ \ circ} \ cdot A _ {\ text {circle}} & = A _ {\ text {circle section}} && {\ color {gray} |: A_ {\ text {circle}}} \ [5px] \ frac {\ alpha} {360 ^ \ circ} & = \ frac {A _ {\ text {circle section}}} {A _ {\ text {circle}}} && {\ color {gray} | \ cdot 360 ^ \ circ} \ [5px] \ alpha & = \ frac {A _ {\ text {circle section}}} {A _ {\ text {circle}}} \ cdot 360 ^ \ circ \ end {align *}

manual

  1. Write down the formula
  2. Insert values ​​for \ (A _ {\ text {circle section}} \) and \ (A _ {\ text {circle}} \)
  3. Calculate result

example

example

Calculate the center angle \ (\ alpha \), which corresponds to a circle section of the area \ (A _ {\ text {circle section}} = 3 ~ \ mathrm {km} ^ 2 \) and a circle with the area \ (A _ {\ text {Circle}} = 9 ~ \ mathrm {km} ^ 2 \) belongs.

Write down the formula

\ begin {align *} \ alpha = \ frac {A _ {\ text {circle section}}} {A _ {\ text {circle}}} \ cdot 360 ^ \ circ \ end {align *}

Insert values ​​for \ (A _ {\ text {circle section}} \) and \ (A _ {\ text {circle}} \)

\ begin {align *} \ phantom {\ alpha} = \ frac {3 ~ \ mathrm {km} ^ 2} {9 ~ \ mathrm {km} ^ 2} \ cdot 360 ^ \ circ \ end {align *}

Calculate result

\ begin {align *} \ phantom {\ alpha} = 120 ^ \ circ \ end {align *}

Circle section and radius given

formula

\ begin {align *} A _ {\ text {circle section}} = \ frac {\ alpha} {360 ^ \ circ} \ cdot \ pi \ cdot r ^ 2 \ end {align *}

Convert formula to \ (\ alpha \)

\ begin {align *} A _ {\ text {circle section}} & = \ frac {\ alpha} {360 ^ \ circ} \ cdot \ pi \ cdot r ^ 2 && {\ color {gray} | \ text {Swap sides }} \ [5px] \ frac {\ alpha} {360 ^ \ circ} \ cdot \ pi \ cdot r ^ 2 & = A _ {\ text {circle section}} && {\ color {gray} |: (\ pi \ cdot r ^ 2)} \ [5px] \ frac {\ alpha} {360 ^ \ circ} & = \ frac {A _ {\ text {circle section}}} {\ pi \ cdot r ^ 2} && {\ color {gray} | \ cdot 360 ^ \ circ} \ [5px] \ alpha & = \ frac {A _ {\ text {section of a circle}}} {\ pi \ cdot r ^ 2} \ cdot 360 ^ \ circ \ end {align *}

manual

  1. Write down the formula
  2. Insert values ​​for \ (A _ {\ text {circle section}} \) and \ (r \)
  3. Calculate result

example

example

Calculate the center angle \ (\ alpha \), which corresponds to a circle section of the area \ (A _ {\ text {circle section}} = 3 ~ \ mathrm {cm} ^ 2 \) and a circle with the radius \ (r = 5 ~ \ mathrm {cm} \) heard. Round the result to one decimal place.

Write down the formula

\ begin {align *} \ alpha = \ frac {A _ {\ text {section of a circle}}} {\ pi \ cdot r ^ 2} \ cdot 360 ^ \ circ \ end {align *}

Insert values ​​for \ (A _ {\ text {circle section}} \) and \ (r \)

\ begin {align *} \ phantom {\ alpha} = \ frac {3 ~ \ mathrm {cm} ^ 2} {\ pi \ cdot (5 ~ \ mathrm {cm}) ^ 2} \ cdot 360 ^ \ circ \ end {align *}

Calculate result

\ begin {align *} \ phantom {\ alpha} & = 41 {,} 25 \ ldots ^ \ circ \ [5px] & \ approx 41 {,} 3 ^ \ circ \ end {align *}

Circle section and diameter given

formula

\ begin {align *} A _ {\ text {circle section}} = \ frac {\ alpha} {360 ^ \ circ} \ cdot \ frac {\ pi} {4} \ cdot d ^ 2 \ end {align *}

Convert formula to \ (\ alpha \)

\ begin {align *} A _ {\ text {circle section}} & = \ frac {\ alpha} {360 ^ \ circ} \ cdot \ frac {\ pi} {4} \ cdot d ^ 2 && {\ color {gray } | \ text {Swap sides}} \ [5px] \ frac {\ alpha} {360 ^ \ circ} \ cdot \ frac {\ pi} {4} \ cdot d ^ 2 & = A _ {\ text {Section of a circle }} && {\ color {gray} |: (\ tfrac {\ pi} {4} \ cdot d ^ 2)} \ [5px] \ frac {\ alpha} {360 ^ \ circ} & = \ frac { A _ {\ text {circle section}}} {\ frac {\ pi} {4} \ cdot d ^ 2} && {\ color {gray} | \ cdot 360 ^ \ circ} \ [5px] \ alpha & = \ frac {A _ {\ text {circle section}}} {\ frac {\ pi} {4} \ cdot d ^ 2} \ cdot 360 ^ \ circ \ end {align *}

manual

  1. Write down the formula
  2. Insert values ​​for \ (A _ {\ text {circle section}} \) and \ (d \)
  3. Calculate result

example

example

Calculate the center angle \ (\ alpha \), which corresponds to a circle section of the area \ (A _ {\ text {circle section}} = 10 ~ \ mathrm {m} ^ 2 \) and a circle with the diameter \ (d = 6 ~ \ mathrm {m} \) heard. Round the result to two decimal places.

Write down the formula

\ begin {align *} \ alpha = \ frac {A _ {\ text {circle section}}} {\ frac {\ pi} {4} \ cdot d ^ 2} \ cdot 360 ^ \ circ \ end {align *}

Insert values ​​for \ (A _ {\ text {circle section}} \) and \ (d \)

\ begin {align *} \ phantom {\ alpha} = \ frac {10 ~ \ mathrm {m} ^ 2} {\ frac {\ pi} {4} \ cdot (6 ~ \ mathrm {m}) ^ 2} \ cdot 360 ^ \ circ \ end {align *}

Calculate result

\ begin {align *} \ phantom {\ alpha} & = 127 {,} 323 \ ldots ^ \ circ \ [5px] & \ approx 127 {,} 32 ^ \ circ \ end {align *}