Area Of A Chord Of A Circle

Understanding the geometry of a circle often leads students and professionals to a common point of confusion: the terminology surrounding the "area of a chord.Worth adding: " Strictly speaking, a chord is a line segment connecting two points on a circle's circumference. As a one-dimensional object, a chord possesses length, not area. When mathematicians, engineers, or students search for this phrase, they are almost always looking for the area of a circular segment—the region bounded by a chord and the arc it subtends Small thing, real impact. Still holds up..

This thorough look will clarify the terminology, derive the necessary formulas, and walk you through step-by-step calculations for finding the area of a segment, a sector, and the associated triangle. Whether you are solving homework problems, designing a curved architectural element, or calculating material volumes for a cylindrical tank, mastering these concepts is essential.

The Critical Distinction: Chord vs. Segment vs. Sector

Before diving into calculations, we must visualize the components of a circle divided by a straight line.

• The Chord: The straight line segment connecting Point A and Point B on the circle. It has length ($c$), not area.
• The Arc: The curved portion of the circumference between Point A and Point B.
• The Circular Segment: The region bounded by the chord and the arc. This is the "area" you are likely looking for. It looks like a cap or a slice of an orange with the bottom cut straight across.
• The Circular Sector: The "pizza slice" region bounded by two radii (to Point A and Point B) and the arc.
• The Central Angle ($\theta$): The angle formed at the circle's center by the two radii connecting to the chord's endpoints.

The Fundamental Relationship:

Area of Segment = Area of Sector – Area of Isosceles Triangle

This relationship is the key to solving almost every problem involving the area bounded by a chord.

Core Formulas You Need

All calculations stem from the radius ($r$) and the central angle ($\theta$). The angle must be in radians for the standard formulas below to work directly. If your angle is in degrees, convert it first: $ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} $

1. Area of the Sector ($A_{\text{sector}}$)

$ A_{\text{sector}} = \frac{1}{2} r^2 \theta $

2. Area of the Isosceles Triangle ($A_{\text{triangle}}$)

The triangle formed by the two radii and the chord has sides $r, r, c$. Its area can be found using the trigonometric formula for a triangle with two sides and an included angle: $ A_{\text{triangle}} = \frac{1}{2} r^2 \sin(\theta) $

3. Area of the Segment ($A_{\text{segment}}$) — The Target Formula

Subtracting the triangle from the sector gives the standard formula for the area bounded by the chord (the segment): $ A_{\text{segment}} = \frac{1}{2} r^2 (\theta - \sin \theta) $ (Angle $\theta$ in radians)

If $\theta$ is in degrees: $ A_{\text{segment}} = \frac{1}{2} r^2 \left( \frac{\pi \theta}{180} - \sin \theta \right) $

4. Chord Length ($c$)

While not an area, the chord length is often a given variable or a required intermediate step: $ c = 2r \sin\left(\frac{\theta}{2}\right) $

5. Sagitta (Height of the Segment, $h$)

The distance from the midpoint of the chord to the midpoint of the arc (the "height" of the cap): $ h = r - r \cos\left(\frac{\theta}{2}\right) = r \left(1 - \cos\left(\frac{\theta}{2}\right)\right) $ Alternatively, using Radius and Chord Length (Pythagorean theorem): $ h = r - \sqrt{r^2 - \left(\frac{c}{2}\right)^2} $

Step-by-Step Calculation Scenarios

Real-world problems rarely hand you the radius and the angle in radians directly. Here are the three most common scenarios and how to solve them Worth knowing..

Scenario A: Given Radius ($r$) and Central Angle ($\theta$)

This is the most direct application.

Example: Find the area of the segment for a circle with radius 10 cm and a central angle of 120° Less friction, more output..

1. Convert Angle to Radians: $\theta = 120 \times \frac{\pi}{180} = \frac{2\pi}{3} \approx 2.094 \text{ radians}$.
2. Calculate $\sin(\theta)$: $\sin(120°) = \sin(60°) = \frac{\sqrt{3}}{2} \approx 0.866$.
3. Apply Segment Formula: $A = \frac{1}{2} (10)^2 \left( \frac{2\pi}{3} - \frac{\sqrt{3}}{2} \right)$ $A = 50 \left( 2.094 - 0.866 \right)$ $A = 50 (1.228)$ $A \approx 61.4 \text{ cm}^2$

Scenario B: Given Radius ($r$) and Chord Length ($c$)

You have the "width" of the segment but not the angle.

Example: A circular window has a radius of 5 ft. A horizontal bar (chord) crosses it 6 ft wide. Find the area of the smaller segment (the top cap).

1. Find Half-Chord: $c/2 = 3 \text{ ft}$.
2. Find Half-Angle ($\theta/2$) using Sine: $\sin(\theta/2) = \frac{c/2}{r} = \frac{3}{5} = 0.6$. $\theta/2 = \arcsin(0.6) \approx 0.6435 \text{ radians}$.
3. Find Full Angle ($\theta$): $\theta \approx 1.287 \text{ radians}$.
4. Find $\sin(\theta)$: Use double angle identity: $\sin(\theta) = 2 \sin(\theta/2) \cos(\theta/2)$. $\cos(\theta/2) = \sqrt{1 - 0.6^2} = 0.8$. $\sin(\theta) = 2(0.6)(0.8) = 0.96$.
5. Calculate Area: $A = \frac{1}{2} (5)^2 (1.287 - 0.96)$ $A = 12.5 (0.327)$ $A \approx 4.09 \text{ ft}^2$

Scenario C: Given Chord Length ($c$) and Segment Height / Sagitta ($h$)

Common in construction (archways, tank volume) where the radius is unknown.

Example: An arched doorway has a width (chord) of 4 meters and a height (sagitta) of 1 meter.