Inscribed Angle Theorem Calculator (Central Angle, Arc Ratio, Inscribed Quadrilateral)

Choose a mode and enter the angle (or the arc length ratio).

Mode

Inscribed angle (degrees)

Central angle (degrees)

Ratio for arc AB, m

Ratio for arc CD, n

Inscribed angle on arc AB (degrees)

One angle of the quadrilateral ∠A (degrees)

Enter numbers only. Decimals and fractions such as 250/7 work for angles and ratios.

Result and figure

Choose a mode on the left, enter the angle and press "Calculate". The result and a figure of the circle will appear here.

[ CLOSE ]

What you can do on this page
What is this calculation used for?
How to Use
Formulas and figures
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Using the inscribed angle theorem (an inscribed angle is half the central angle on the same arc), you can find the central angle from an inscribed angle (twice) or the inscribed angle from a central angle (half) right away. Angles that do not divide evenly, such as \(\dfrac{250}{7}^\circ\), are shown both exactly (as a fraction) and as a decimal
In the same circle, when arc AB : arc CD \(= m : n\), the inscribed angles are in the same ratio (inscribed angles are proportional to arc length). Use this to find one inscribed angle from the other
Opposite angles of a quadrilateral inscribed in a circle add up to \(180^\circ\). Use this to find the opposite angle from one angle
The result comes with a figure of the circle, so you can see where the arc, the central angle and the inscribed angle are, and check that inscribed angles on the same arc are equal wherever you look from (in the arc ratio mode, only when the two arcs are too large to fit in one circle, the figure is left out and just the numbers are shown)
The figures with the formulas explain why the theorems are true, such as "an angle inscribed in a semicircle is \(90^\circ\)" (Thales' theorem)

On this page, the inscribed angle and the central angle are the angles on one arc (the inscribed angle is more than 0° and less than 180°; the central angle is more than 0° and less than 360°).

What is this calculation used for?

Finding a ship's position (navigation and coastal surveying)

A ship near the coast can find its position by measuring the angle between two landmarks on shore (such as lighthouses). By the converse of the inscribed angle theorem, all the points that see two landmarks at the same angle lie on one circular arc. Measure the angles for two pairs of landmarks, and the ship is at the single point where the two arcs cross (a classic method called a horizontal sextant angle fix, plotted with a station pointer, or three-arm protractor).
It is still taught to ship officers as a backup for when GPS cannot be used.

Knowing where a shot is easiest (soccer and basketball)

In soccer, the angle between the two goalposts as seen by the shooter shows how wide the goal looks. By the inscribed angle theorem, all the spots with the same angle lie on one arc through the two posts. Moving sideways away from the center of the goal along this arc keeps the angle the same.
In sports analytics, this "shooting angle" is actually used as one of the main measures that explain shot success rates.

Finding the center of a round object (woodworking, crafts and machining)

The converse of Thales' theorem (if the inscribed angle is 90°, the chord is a diameter) makes it easy to find the center of a round board or a cylinder. Set the corner of a carpenter's square on the edge of the circle; the line joining the two points where its sides cross the edge is a diameter. Do this twice from different directions, and the center is where the two diameters cross.
A woodworker's center finder is this theorem turned into a tool.

Designing how the stage looks from the seats (theaters and stadiums)

The larger the angle between the two ends of the stage, the wider the stage looks. By the inscribed angle theorem, seats that see the stage at the same angle lie on the same arc, so arranging the seats in arcs keeps the view similar along each row.
From the semicircular theaters of ancient Greece to modern concert halls, seating that curves in arcs puts this property of geometry to work (real designs also take acoustics, exit routes and other factors into account).

Formulas and figures

Inscribed angle theorem (an inscribed angle is half the central angle)

Figure

Standard notation (the usual math form)

\(x\) \(=\) \(c\) \(\div\) \(2\)

In words (symbols replaced with words)

③ \(x\): inscribed angle on arc AB \(=\) ① \(c\): central angle on the same arc \(\div\) ② \(2\): takes half

The formula in words

① Take the \(c\): central angle on the same arc

② divide it by \(2\): takes half

③ and you get the \(x\): inscribed angle on arc AB

Quick example

When the central angle on arc AB is \(100^\circ\), the inscribed angle on the same arc is

inscribed angle on arc AB \(=\) central angle (\(100^\circ\)) \(\div\) \(2\)

\(100^\circ \div 2 = 50^\circ\)

Key idea

It also works the other way: "central angle \(c = \) inscribed angle \(x \times 2\)" (the "Inscribed angle → central angle" mode of this calculator). Why is it half? Call the center of the circle \(O\) and the point on the circle \(P\), and draw the line from \(P\) through \(O\). Since \(OA = OP\) and \(OB = OP\) (all radii), you get two isosceles triangles. The base angles of an isosceles triangle are equal, and an exterior angle of a triangle equals the sum of the two interior angles that are not next to it. So the central angle is exactly \(2\) times the inscribed angle. The theorem still holds when the central angle is over \(180^\circ\) (when the arc is longer than a semicircle). For example, the inscribed angle for a central angle of \(240^\circ\) is \(120^\circ\).

Inscribed angles on the same arc are equal

Figure

Standard notation (the usual math form)

In words (symbols replaced with words)

\(\angle APB\) \(=\) \(\angle AQB\)

① \(\angle APB\): inscribed angle seen from point \(P\) \(=\) ② \(\angle AQB\): inscribed angle seen from point \(Q\)

The formula in words

① As long as both look at the same arc AB, the \(\angle APB\): inscribed angle seen from point \(P\)

② is always equal to the \(\angle AQB\): inscribed angle seen from point \(Q\)

Quick example

If the inscribed angle on arc AB is \(50^\circ\) at point \(P\), then from any point on the circle

inscribed angle seen from \(P\) \(=\) inscribed angle seen from \(Q\) (\(50^\circ\))

\(\angle APB = \angle AQB = 50^\circ\)

Key idea

The reason is simple: both inscribed angles are half of the same central angle. There is only one central angle for arc AB, so the inscribed angle, which is half of it, is the same wherever you look from. But they are equal only when \(P\) and \(Q\) are on the same side of arc AB (on the circle but not on arc AB). If you put the point on the other side (on arc AB itself), the angle becomes \(180^\circ - x\). This leads to the last formula on this page: "opposite angles of a quadrilateral inscribed in a circle add up to \(180^\circ\)".

An angle inscribed in a semicircle is 90° (Thales' theorem)

Figure

Standard notation (the usual math form)

In words (symbols replaced with words)

\(\angle APB\) \(=\) \(90^\circ\)

① \(\angle APB\): inscribed angle on diameter AB \(=\) ② \(90^\circ\): right angle

The formula in words

① When chord AB is a diameter (the arc is a semicircle), the \(\angle APB\): inscribed angle on diameter AB

② is always a \(90^\circ\): right angle

Quick example

The central angle on a diameter is a straight angle, \(180^\circ\), so the inscribed angle is half of it

inscribed angle on a diameter \(=\) central angle (\(180^\circ\)) \(\div\) \(2\)

\(180^\circ \div 2 = 90^\circ\)

Key idea

This is a special case of the inscribed angle theorem called Thales' theorem. It is named after the ancient Greek mathematician Thales and is said to be one of the oldest recorded theorems. The converse is also true: if \(\angle APB = 90^\circ\) at a point \(P\) on the circle, chord AB is always a diameter. This converse helps you find the center of a round object. Put a right angle (such as a carpenter's square) against the circle to mark two points A and B; AB is then a diameter, so the center is where two such diameters cross.

Inscribed angles are proportional to arc length

Figure

Standard notation (the usual math form)

\(x_1 : x_2\) \(=\) \(m : n\)

In words (symbols replaced with words)

② \(x_1 : x_2\): ratio of the inscribed angles \(=\) ① \(m : n\): ratio of the arc lengths

The formula in words

① In the same circle (or circles with equal radii), given the \(m : n\): ratio of the arc lengths

② the \(x_1 : x_2\): ratio of the inscribed angles on each arc is the same ratio

Quick example

When arc AB : arc CD \(= 2 : 3\) and the inscribed angle on arc AB is \(30^\circ\), the inscribed angle on arc CD is

inscribed angle on arc CD \(=\) inscribed angle on arc AB (\(30^\circ\)) \(\times\) arc ratio factor \(\dfrac{3}{2}\)

\(30^\circ \times \dfrac{3}{2} = 45^\circ\)

Key idea

When an arc becomes 2 or 3 times longer, its central angle also becomes 2 or 3 times larger (the arc length of a sector is proportional to the central angle). The inscribed angle is half of that central angle, so inscribed angles are proportional to arc length too. Be careful: chord length is not proportional to the inscribed angle. For example, with inscribed angles of \(30^\circ\) and \(60^\circ\), the arc lengths are exactly \(1 : 2\), but the chord lengths are not \(1 : 2\) (a chord takes the straight shortcut, so it comes out shorter than proportional). Mixing up "arc" and "chord" is a classic trick on tests.

Opposite angles of a quadrilateral inscribed in a circle add up to 180°

Figure

Standard notation (the usual math form)

\(\angle A\) \(+\) \(\angle C\) \(=\) \(180^\circ\)

In words (symbols replaced with words)

① \(\angle A\): one angle \(+\) ② \(\angle C\): the opposite angle \(=\) ③ \(180^\circ\): straight angle

The formula in words

① In a quadrilateral inscribed in a circle, add \(\angle A\): one angle

② and \(\angle C\): the opposite angle

③ and you always get a \(180^\circ\): straight angle

Quick example

In a quadrilateral ABCD inscribed in a circle with \(\angle A = 110^\circ\), the opposite angle \(\angle C\) is

opposite angle \(\angle C\) \(=\) \(180^\circ\) \(-\) angle \(\angle A\) (\(110^\circ\))

\(180^\circ - 110^\circ = 70^\circ\)

Key idea

This also comes from the inscribed angle theorem. \(\angle A\) is the inscribed angle on arc BCD (the side that contains vertex C), and \(\angle C\) is the inscribed angle on arc DAB (the side that contains vertex A). Together the two arcs make exactly one full circle, so their central angles add up to \(360^\circ\). Each inscribed angle is half of its central angle, so the two add up to \(360^\circ \div 2 = 180^\circ\). From this you can also show that an exterior angle of an inscribed quadrilateral equals the interior angle opposite it. The exterior angle made by extending side BC is \(180^\circ - \angle C = \angle A\).

The inscribed angle theorem says "an inscribed angle is half the central angle on the same arc". From it comes a whole set of facts about angles in circles: "inscribed angles on the same arc are all equal", "an angle inscribed in a semicircle is \(90^\circ\)" (Thales' theorem), "inscribed angles are proportional to arc length" and "opposite angles of a quadrilateral inscribed in a circle add up to \(180^\circ\)".

Symbols and terms

Symbols

\(\angle\)	angle	The symbol for an angle. \(\angle APB\) is the angle with its vertex at point \(P\), formed by the two lines from \(P\) to \(A\) and \(B\). The middle letter is the vertex.
\(\overset{\frown}{AB}\)	arc A B	The symbol for arc AB, the part of the circle between two points \(A\) and \(B\) on it. The curved mark above the letters shows that it is the curved arc, not the straight segment AB.
\(x,\ c\)	ex, see	This page writes the inscribed angle as \(x\) and the central angle as \(c\). Using \(x\) for an unknown number is a familiar habit from textbooks, and \(c\) is the first letter of "center".
\(O\)	oh	The letter for the center of the circle. It is said to come from the first letter of "origin". The central angle is the angle with its vertex at \(O\).
\(P,\ Q\)	pee, cue	Letters for points on the circle, by the habit of using \(P\) for "point" and the next letter, \(Q\). The inscribed angle is the angle with its vertex at \(P\) or \(Q\).
\(m : n\)	m to n	The ratio of two amounts. On this page it is the ratio of the lengths of arc AB and arc CD. \(2 : 3\) tells you that arc CD is \(\dfrac{3}{2}\) times as long as arc AB.
\(^\circ\)	degrees	The symbol for the degree, a unit of angle. One full turn divided into 360 equal parts is \(1^\circ\). It is said that 360 was chosen for one turn because it has many factors and is easy to divide evenly.

Terms

inscribed angle	The angle formed by two chords drawn from one point on a circle to the ends \(A\) and \(B\) of an arc. It always goes with the arc it looks at (its intercepted arc), as in "the inscribed angle on arc AB". Wherever you move the vertex on the circle (off arc AB), its size stays the same.
central angle	The angle formed by the two radii drawn from the center \(O\) to the ends \(A\) and \(B\) of an arc. Each arc has only one central angle, and it is exactly twice the inscribed angle on the same arc.
inscribed angle theorem	The theorem that "an inscribed angle is half the central angle on the same arc" and "inscribed angles on the same arc are all equal". It is the central theorem of the circles unit in high school Geometry, and its proof uses isosceles triangles and the exterior angle of a triangle.
arc	The part of a circle between two points on it. Two points split the circle into two arcs; the shorter one is the minor arc and the longer one is the major arc. Plain "arc AB" usually refers to the shorter one (the minor arc).
major arc	When two points split a circle in two, the arc that is longer than a semicircle. Its central angle is more than \(180^\circ\). The inscribed angle theorem holds for either arc.
minor arc	When two points split a circle in two, the arc that is shorter than a semicircle. Its central angle is less than \(180^\circ\). The inscribed angle theorem holds for either arc.
chord	A straight line segment joining two points on a circle. An arc is the curved path along the circle, while a chord is the straight shortest path between the two points. Arc length is proportional to the inscribed angle, but chord length is not.
diameter	A chord through the center of a circle. It is the longest chord, twice as long as the radius. The arc on a diameter is exactly a semicircle, and the inscribed angle on it is always \(90^\circ\) (Thales' theorem).
radius	The distance (or the segment) from the center of a circle to a point on it. All radii of one circle are equal, and the isosceles triangles this creates are the key to proving the inscribed angle theorem.
Thales' theorem	The theorem that "an angle inscribed in a semicircle is \(90^\circ\)". It is a special case of the inscribed angle theorem (when the central angle is \(180^\circ\)). It is named after the ancient Greek mathematician Thales, and its converse (if the inscribed angle is \(90^\circ\), the chord is a diameter) is also true.
inscribed quadrilateral	A quadrilateral whose four vertices all lie on one circle (also called a cyclic quadrilateral). "Inscribed" tells you it fits exactly inside the circle. Its opposite angles have the special property of adding up to \(180^\circ\).
opposite angles	Angles across from each other in a quadrilateral. In quadrilateral ABCD, \(\angle A\) and \(\angle C\) are one pair of opposite angles, and \(\angle B\) and \(\angle D\) are the other.
isosceles triangle	A triangle with two sides of equal length. The two angles opposite the equal sides (the base angles) are equal. It shows up in the proof of the inscribed angle theorem because all radii are equal.
proportional	A relation where, when one amount becomes 2 or 3 times larger, the other also becomes 2 or 3 times larger. Arc length and central angle, and arc length and inscribed angle, are both proportional.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back to these topics is the fastest way forward.

Circle vocabulary (Grades 3–7)	Knowing the words center, radius, diameter and circumference, and that all radii of one circle are equal Knowing the difference between an arc (part of the circle) and a chord (a segment joining two points on the circle)
Angles and turns (Grade 4)	Knowing that a full turn is \(360^\circ\), a half turn (a straight angle) is \(180^\circ\) and a right angle is \(90^\circ\) Having a feel for the size of an angle, like one measured with a protractor
Isosceles triangles (high school Geometry)	Knowing that in a triangle with two equal sides, the base angles (the two angles opposite the equal sides) are equal It is used in the proof of the inscribed angle theorem as "radii are equal, so an isosceles triangle is formed"
Interior and exterior angles of a triangle (Grade 8)	Knowing that the angles of a triangle add up to \(180^\circ\), and that an exterior angle equals the sum of the two interior angles not next to it This exterior angle property is the heart of why the central angle is twice the inscribed angle
Ratios and proportional relationships (Grades 6–7)	Knowing what a ratio such as \(2 : 3\) is, and being able to scale by a ratio, as in \(30 \times \dfrac{3}{2}\) Having a feel for a proportional relationship, "when one doubles, so does the other"
Division and fractions (Grades 5–7)	Being able to do a division such as \(80 \div 2 = 40\) in your head or on paper Being able to keep an answer that does not divide evenly as an exact fraction, such as \(\dfrac{250}{7}\)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table for inscribed angle → central angle

Inscribed angle (degrees)	50
Central angle (degrees)	=B1*2

Table for central angle → inscribed angle

Central angle (degrees)	100
Inscribed angle (degrees)	=B1/2

Table for arc length ratio → inscribed angle

Ratio for arc AB, m	2
Ratio for arc CD, n	3
Inscribed angle on arc AB (degrees)	30
Inscribed angle on arc CD (degrees)	=B3*B2/B1

Table for opposite angles of an inscribed quadrilateral

Angle of the quadrilateral ∠A (degrees)	110
Opposite angle ∠C (degrees)	=180-B1

After pasting, the upper rows are your inputs and the bottom row is calculated automatically.
"*" is multiplication and "/" is division.
The first table finds the central angle from an inscribed angle of 50°; the answer is 100. The second goes the other way and finds the inscribed angle 50 from a central angle of 100°.
The third table is for arc AB : arc CD = 2 : 3 with an inscribed angle of 30° on arc AB. Inscribed angles are proportional to arc length, so the answer is 30 × 3 ÷ 2 = 45.
The fourth table is for a quadrilateral inscribed in a circle: enter 110 for ∠A and the opposite angle ∠C is 70.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table for inscribed angle → central angle

Inscribed angle (degrees)	50
Central angle (degrees)	=B1*2

Table for central angle → inscribed angle

Central angle (degrees)	100
Inscribed angle (degrees)	=B1/2

Table for arc length ratio → inscribed angle

Ratio for arc AB, m	2
Ratio for arc CD, n	3
Inscribed angle on arc AB (degrees)	30
Inscribed angle on arc CD (degrees)	=B3*B2/B1

Table for opposite angles of an inscribed quadrilateral

Angle of the quadrilateral ∠A (degrees)	110
Opposite angle ∠C (degrees)	=180-B1

The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the angles and ratios with your own numbers.

How to calculate it in Python

from fractions import Fraction

# Inscribed angle -> central angle (the central angle is twice the inscribed angle)
inscribed_angle = Fraction(250, 7)   # a fraction such as 250/7 degrees can be written as is
central_angle = inscribed_angle * 2
print(f"Inscribed angle {inscribed_angle}° -> central angle {central_angle}° ≈ {float(central_angle):.2f}°")

# Central angle -> inscribed angle (the inscribed angle is half the central angle)
central_angle = Fraction(100)
inscribed_angle = central_angle / 2
print(f"Central angle {central_angle}° -> inscribed angle {inscribed_angle}°")

# Arc length ratio -> inscribed angle (inscribed angles are proportional to arc length)
arc_ab, arc_cd = Fraction(2), Fraction(3)   # arc AB : arc CD = 2 : 3
angle_ab = Fraction(30)                     # inscribed angle on arc AB
angle_cd = angle_ab * arc_cd / arc_ab
print(f"Arc ratio {arc_ab}:{arc_cd}, inscribed angle {angle_ab}° -> other inscribed angle {angle_cd}°")

# Opposite angles of a quadrilateral inscribed in a circle (they add up to 180 degrees)
angle_a = Fraction(110)
angle_c = 180 - angle_a
print(f"∠A = {angle_a}° -> opposite angle ∠C = {angle_c}°")

With the fractions module from the standard library, angles that do not divide evenly, such as 250/7°, stay exact fractions with no rounding error. When you run it, you get the central angle "500/7° ≈ 71.43°" first, then the inscribed angle 50°, then the inscribed angle 45° on arc CD, and finally the opposite angle 70°. Change the numbers and run it.

How to write it in LaTeX and other math languages (copy and paste)

Inscribed angle theorem (an inscribed angle is half the central angle)

x = c / 2

x = \frac{c}{2}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>x</mi>
    <mo>=</mo>
    <mfrac>
      <mi>c</mi>
      <mn>2</mn>
    </mfrac>
  </mrow>
</math>

x = c/2

x = c/2

x := c/2;

x = c/2;

x = c/2

Inscribed angles on the same arc are equal

∠APB = ∠AQB

\angle APB = \angle AQB

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mo>&#x2220;</mo><mi>A</mi><mi>P</mi><mi>B</mi>
    <mo>=</mo>
    <mo>&#x2220;</mo><mi>A</mi><mi>Q</mi><mi>B</mi>
  </mrow>
</math>

/_ APB = /_ AQB

angleAPB == angleAQB

angleAPB = angleAQB;

angleAPB == angleAQB

∠APB = ∠AQB

An angle inscribed in a semicircle is 90° (Thales' theorem)

∠APB = 90°

\angle APB = 90^{\circ}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mo>&#x2220;</mo><mi>A</mi><mi>P</mi><mi>B</mi>
    <mo>=</mo>
    <msup><mn>90</mn><mo>&#x00B0;</mo></msup>
  </mrow>
</math>

/_ APB = 90^circ

angleAPB == 90 Degree

angleAPB = 90;

angleAPB = 90;

∠APB = 90°

Inscribed angles are proportional to arc length

x₁ : x₂ = m : n

x_1 : x_2 = m : n

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <msub><mi>x</mi><mn>1</mn></msub>
    <mo>:</mo>
    <msub><mi>x</mi><mn>2</mn></msub>
    <mo>=</mo>
    <mi>m</mi>
    <mo>:</mo>
    <mi>n</mi>
  </mrow>
</math>

x_1 : x_2 = m : n

x1/x2 == m/n

x1/x2 = m/n;

x1/x2 == m/n

x_1 : x_2 = m : n

Opposite angles of a quadrilateral inscribed in a circle add up to 180°

∠A + ∠C = 180°

\angle A + \angle C = 180^{\circ}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mo>&#x2220;</mo><mi>A</mi>
    <mo>+</mo>
    <mo>&#x2220;</mo><mi>C</mi>
    <mo>=</mo>
    <msup><mn>180</mn><mo>&#x00B0;</mo></msup>
  </mrow>
</math>

/_ A + /_ C = 180^circ

angleA + angleC == 180 Degree

angleA + angleC = 180;

angleA + angleC == 180

∠A + ∠C = 180°

How to have ChatGPT do the calculation

You are a calculation assistant for math (plane geometry and properties of circles). Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

Using the inscribed angle theorem (an inscribed angle is half the central angle on the same arc), show each of the following.
1. The central angle when the inscribed angle is 250/7° (both as a simplified fraction and as a decimal)
2. The inscribed angle when the central angle is 100°
3. In the same circle, with arc AB : arc CD = 2 : 3 and an inscribed angle of 30° on arc AB, the inscribed angle on arc CD (use the fact that inscribed angles are proportional to arc length)
4. In a quadrilateral inscribed in a circle with ∠A = 110°, the opposite angle ∠C (use the fact that opposite angles add up to 180°)

In Python, use the fractions module from the standard library to calculate exactly, and show the formulas you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula