A Circular Segment is the “crust” of a pizza slice. It is the region bounded by a chord and the arc subtended by the chord.
Specifically, a “Minor Segment” is the area less than a semicircle (Angle < 180°). If you cut a round loaf of bread with a straight knife, the smaller piece is the minor segment.
Calculator Features
1. Three Calculation Modes
Real-world problems rarely give you the central angle directly. This tool lets you solve the area starting from what you can measure:
Central Angle ($\theta$): If you know the theoretical geometry.
Chord Length (c): If you measure the straight-line width of the cut.
Height (h): Also called the “Sagitta”, this is the depth of the arc from the middle of the chord.
2. Full Property Derivation
Unlike basic tools, this calculator works backwards. Enter just the Height ($h$) and Radius ($r$), and it instantly derives the Angle, Chord Length, Arc Length, and Area.
3. Visual Verification
A live SVG diagram renders the specific segment you are calculating. This ensures you are solving for the intended shape (the “cap”) and not the main sector (the “slice”).
Formulas for Minor Segment
The area of a segment is found by subtracting the triangular part from the total sector area.
Let $r$ be the radius and $\theta$ be the central angle in radians.
Area Formula
Area $A = \frac{1}{2} r^2 (\theta – \sin \theta)$
Arc Length
Arc Length $s = r \theta$
Chord Length
Chord $c = 2r \sin(\theta/2)$
Height (Sagitta)
Height $h = r(1 – \cos(\theta/2))$
Where is this Used?
Architecture (Arches)
When designing a segmented arch or a window frame, architects calculate the area of the segment to determine glass sizes and material volumes.
Fluid Dynamics (Tanks)
In horizontal cylindrical tanks, the liquid often fills a circular segment cross-section. Calculating this area is the first step to determining the volume of liquid in the tank.
Carpentry
When cutting a round table to sit flush against a wall, the cutoff piece is a minor segment. Carpenters use the Chord/Height mode to mark the cut line accurately.
Tips for Accurate Input
Degrees vs Radians
Our calculator accepts degrees for simplicity (e.g., $90^\circ$), but converts them to radians internally for the math formulas. Make sure you don’t mix them up if checking manually.
Height Constraint
For a minor segment, the height ($h$) must always be less than the radius ($r$). If $h > r$, you are calculating a “Major Segment” (more than half the circle).
Frequently Asked Questions (FAQs)
1. What is the difference between a Sector and a Segment?
A **Sector** is the whole pizza slice (including the pointy center). A **Segment** is just the crust part (bounded by a straight line chord).
2. Can I calculate the Major Segment?
Yes. Calculate the total Circle Area ($\pi r^2$) and subtract the Minor Segment Area. Or use our dedicated Major Segment Calculator.
3. What is the “Sagitta”?
Sagitta means “arrow” in Latin. It refers to the height of the arc from the middle of the chord, looking like an arrow resting on a bow (the chord and arc).
Final Words
The Minor Segment Area Calculator is a bridge between linear measurements (chord, height) and circular geometry. By automating the trigonometric inversions, it saves you from tedious algebra and lets you solve practical problems in design and engineering instantly.
