Arc Height Calculator (from Angle & Radius)

Geometric Calculators

This calculator determines the height of a circular arc, also known as the sagitta, based on the radius of the circle and the central angle subtended by the arc. Enter your values to get an instant result.

Visual Representation

A diagram showing the relationship between radius (r), angle (θ), and arc height (h).

What is the Height of an Arc?

The height of a circular arc, more formally known as the **sagitta**, is the distance from the center of the arc to the center of its chord. The term “sagitta” comes from the Latin word for “arrow,” as it resembles an arrow resting on a bow. This measurement is crucial in many fields, including architecture, engineering, optics, and carpentry, where creating or measuring curved surfaces is common. To calculate the height of an arc using the angle, you need two primary measurements: the **radius** of the circle from which the arc is taken and the **central angle** that the arc subtends.

Arc Height Formula and Explanation

The formula to calculate the height of an arc (sagitta) when you know the radius and the central angle is derived from basic trigonometry. It is given by:

h = r * (1 – cos(θ / 2))

This formula relates the arc’s height directly to the radius and half of the central angle. When using this formula, the angle measure must be consistent with the `cos` function’s expected input (typically radians in programming, though calculators can use degrees).

Variables for Arc Height Calculation

Variable	Meaning	Unit (Auto-Inferred)	Typical Range
h	Arc Height (Sagitta)	Length (e.g., meters, feet)	0 to 2r
r	Radius of the Circle	Length (e.g., meters, feet)	Any positive value
θ	Central Angle	Angular (degrees, radians)	0° to 360° (0 to 2π rad)
cos	Cosine Function	Unitless	-1 to 1

For more on calculating arc properties, see our arc length calculator.

Practical Examples

Example 1: Architectural Arch

An architect is designing a window with a rounded top. The circular arch must span a width that corresponds to a circle with a radius of 2 feet, and the central angle specified in the plans is 120 degrees.

• Inputs: Radius (r) = 2 ft, Angle (θ) = 120°
• Calculation:
  
  Half Angle = 120 / 2 = 60°
  
  cos(60°) = 0.5
  
  Height (h) = 2 * (1 – 0.5) = 1 ft
• Result: The height of the arch at its center will be 1 foot.

Example 2: Engineering a Curved Part

An engineer needs to mill a curved slot in a metal block. The curve is an arc from a circle with a radius of 50 millimeters and a central angle of 90 degrees.

• Inputs: Radius (r) = 50 mm, Angle (θ) = 90°
• Calculation:
  
  Half Angle = 90 / 2 = 45°
  
  cos(45°) ≈ 0.7071
  
  Height (h) = 50 * (1 – 0.7071) = 50 * 0.2929 ≈ 14.65 mm
• Result: The depth of the curved slot (its sagitta) is approximately 14.65 mm. You might also need the chord length formula for tooling paths.

How to Use This Arc Height Calculator

Follow these simple steps to find the height of any circular arc:

1. Enter the Radius: Input the radius of the circle that the arc belongs to. Select the appropriate unit of length (e.g., meters, feet, inches).
2. Enter the Central Angle: Input the angle created by the two lines from the center of the circle to the endpoints of the arc. Choose whether your angle is in degrees or radians.
3. View the Results: The calculator instantly provides the arc height (sagitta) in your selected unit. It also shows intermediate values like chord length and total arc length for your convenience.
4. Interpret the Visualization: The dynamic chart updates to give you a visual sense of the arc’s shape based on your inputs.

Key Factors That Affect Arc Height

Several factors influence the result when you calculate the height of an arc using the angle. Understanding them helps predict how changes will affect the arc’s shape.

• Radius (r): For the same central angle, a larger radius will result in a larger arc height. The height scales linearly with the radius.
• Central Angle (θ): For the same radius, a larger central angle will result in a much larger arc height. The relationship is not linear due to the cosine function. As the angle approaches 180°, the height approaches the radius.
• Units: Using consistent units is critical. Our calculator handles conversions, but when doing manual calculations, ensure the radius and height are in the same units.
• Measurement Accuracy: Small errors in measuring the radius or angle can lead to significant differences in the calculated height, especially for very large or very flat arcs.
• Half-Angle Calculation: The formula relies on the cosine of *half* the central angle. This is the most critical part of the calculation, as it defines the right-angled triangle used to find the height.
• Arc vs. Chord: Don’t confuse the arc length (the distance along the curve) with the chord length (the straight line distance between endpoints). Both are related to the height but are different measures. Check out a radius of an arc calculator if you have different starting values.

Frequently Asked Questions (FAQ)

What is a sagitta?

A sagitta is the formal geometric term for the height of an arc, measured from the midpoint of the chord to the arc itself.

What is the formula to calculate arc height from an angle?

The formula is h = r * (1 – cos(θ/2)), where ‘h’ is the height, ‘r’ is the radius, and ‘θ’ is the central angle.

What happens if the angle is 180 degrees?

If the angle is 180°, the arc is a semicircle. The height (sagitta) will be equal to the radius (r), as cos(180/2) = cos(90°) = 0, so h = r * (1 – 0) = r.

What if the angle is 360 degrees?

If the angle is 360°, the arc is a full circle. The height is ambiguous, but the formula gives h = r * (1 – cos(180°)) = r * (1 – (-1)) = 2r, which is the diameter of the circle.

Do I need to use radians or degrees?

Our calculator lets you choose. When calculating manually, ensure your calculator is set to the correct mode (degrees or radians) for the cosine function. Most mathematical formulas, including this one, are formally derived using radians.

Can I calculate the radius if I know the height and angle?

Yes, you can rearrange the formula to solve for the radius: r = h / (1 – cos(θ/2)). Our sagitta of an arc tool can help with that.

Does this work for any circle?

Yes, this formula applies to any perfect circular arc, regardless of the circle’s size.

How does arc height relate to circle segment area?

The arc height is a key dimension of a circle segment. Knowing the height helps in calculating the circle segment area, which is the area enclosed by the arc and its chord.