What are the four fundamental formulas used in a Circle Calculator?

The four core circle formulas are: (1) Diameter d = 2r; (2) Circumference C = 2*pi*r = pi*d; (3) Circle Area A = pi*r^2 = (pi*d^2)/4 = C^2/(4*pi); and (4) Radius from Area r = sqrt(A/pi) or from Circumference r = C/(2*pi).

How do you calculate circle area if only the circumference is known?

Use the area of circumference formula: A = C^2 / (4*pi) ≈ C^2 / 12.56637. For example, if a circle has a circumference of 31.416 cm, its area is A = 31.416^2 / (4*pi) ≈ 78.54 cm^2.

How do you calculate circle sector area and arc length?

For a circle of radius r with central angle theta (in degrees): Arc Length s = (theta / 360°) * 2*pi*r = (pi*r*theta) / 180°; Sector Area A_sector = (theta / 360°) * pi*r^2 = 1/2 * r * s.

What is a circle segment and how is its area calculated?

A circular segment is the region bounded by a chord and the circular arc between the chord's endpoints. Its area is calculated as the sector area minus the triangle area formed by the radii: A_segment = A_sector - A_triangle = 1/2 * r^2 * ((pi*theta)/180° - sin(theta)).

How do you calculate chord length and sagitta (height)?

For radius r and central angle theta: Chord Length c = 2*r*sin(theta / 2); Sagitta (the perpendicular height from the chord's midpoint to the arc) h = r*(1 - cos(theta / 2)) = r - sqrt(r^2 - (c/2)^2).

How do you find the side length of an inscribed square inside a circle?

The diagonal of an inscribed square equals the circle's diameter (d = 2r). By the Pythagorean theorem, each side length is s_inscribed = r*sqrt(2) ≈ 1.4142 * r. The area of the inscribed square is A_square = 2*r^2.

What is the difference between a circumscribed square and an inscribed square?

An inscribed square lies entirely inside the circle with all 4 vertices touching the circumference (side length s = r*sqrt(2)). A circumscribed square contains the circle such that all 4 sides are tangent to the circle (side length s = 2r = d). The area of the circumscribed square is 4*r^2, exactly double the area of the inscribed square (2*r^2).

How does this Circle Calculator compute 3D cylinder volume and sphere properties?

Given a circle base with radius r, the calculator projects 3D dimensions: Cylinder Volume V_cylinder = pi*r^2*H (with height H), Cylinder Lateral Surface Area 2*pi*r*H, Sphere Volume V_sphere = (4/3)*pi*r^3, and Sphere Surface Area A_sphere = 4*pi*r^2.

How do you convert between metric and imperial circle dimensions?

Circle linear dimensions scale directly with standard length conversion factors (1 in = 2.54 cm, 1 ft = 0.3048 m), while area scales quadratically (1 in^2 = 6.4516 cm^2, 1 m^2 = 10.7639 ft^2).

How does this Circle Calculator work and what input modes does it support?

The calculator allows entering any single circle parameter (Radius, Diameter, Circumference, or Area) to instantly compute all remaining 2D dimensions, sector/segment geometry for any central angle theta, inscribed/circumscribed polygons, and 3D extruded volume with 1-click clipboard export.

100% Free • All-in-One 2D & 3D Circle Geometry Suite

Circle Calculator

Calculate circle radius ($r$), diameter ($d = 2r$), circumference ($C = 2\pi r$), and area ($A = \pi r^2$) from any single given measurement. Includes circular sector area, segment area, chord length, sagitta height, inscribed/circumscribed polygons, and 3D cylinder extrusion.

Circle Presets: Tap to load

Circle Dimension Solver

Select input parameter and specify circular measurements.

A = πr² • C = 2πr

Given Parameter

Circle Radius ($r$) r = d / 2

Central Sector & Segment Angle ($\theta$) Compute circular sector area, arc length, and chord metrics.

60.0° (1.047 rad)

Optional Cylinder Height ($H$) Extrude circular base into 3D cylinder volume.

$V = \pi r^2 H$

Key 2D Dimensions Glance

Radius ($r$) 5.000 m

Diameter ($d$) 10.000 m

Circumference ($C$) 31.416 m

Area ($A$) 78.540 m²

Calculated Circle Area ($A$)

78.540 m²

Radius: 5.000 m • Diameter: 10.000 m • Circumference: 31.416 m

Complete Circle Telemetry Matrix

⭕ Circle Radius ($r$)	5.000 m
📏 Circle Diameter ($d$)	10.000 m (2r)
🔄 Circumference ($C$)	31.416 m (2πr)
🍰 Circle Area ($A$)	78.540 m² (πr²)
🏹 Arc Length ($s$)	5.236 m (rθ)
🍕 Sector Area ($A_{\text{sector}}$)	13.090 m²
📐 Segment Area ($A_{\text{segment}}$)	2.265 m²
📏 Chord Length ($c$)	5.000 m (2r sin θ/2)
↕️ Sagitta Height ($h$)	0.670 m
⏹️ Inscribed Square Side	7.071 m (r√2)
🔲 Circumscribed Square	10.000 m (2r)
🥫 Cylinder Volume ($V_{\text{cyl}}$)	785.398 m³
🌐 Sphere Volume / Area	523.599 m³ • 314.159 m²

Why Use Our Circle Calculator? Real-World Problems It Solves

Circles are fundamental in manufacturing, architecture, sports equipment design, astronomy, and culinary economics. Our Circle Calculator provides an all-in-one geometric and trigonometric solver:

1. 4-Way Instant Parameter Isolation:

Input any single measurement—Radius ($r$), Diameter ($d$), Circumference ($C$), or Area ($A$)—and our engine immediately calculates the remaining three dimensions without manual algebraic manipulation.

2. Circular Sector & Segment Telemetry:

Specify any central angle ($\theta$) to instantly determine circular arc lengths ($s = r\theta$), pie sector areas, chord lengths ($c = 2r\sin(\theta/2)$), sagitta heights, and segment areas for civil arches and circular duct fabrication.

3. Inscribed & Circumscribed Structural Polygons:

Woodworkers and CNC machinists can instantly find the largest square timber beam side length ($s = r\sqrt{2}$) or equilateral triangle that can be milled from a circular tree trunk or cylindrical stock.

4. 3D Extrusion & Sphere Analytics:

Seamlessly project 2D circular cross-sections into 3D cylindrical volumes ($V = \pi r^2 H$) and spherical bodies ($V = \frac{4}{3}\pi r^3$) for tank sizing and hydraulic piping.

How to Use the Circle Calculator (Step-by-Step Guide)

Step 1: Choose Known Parameter

Select from Radius ($r$), Diameter ($d$), Circumference ($C$), or Area ($A$).

Step 2: Enter Value & Unit

Type your measurement value and select your preferred metric or imperial unit (m, cm, mm, in, ft, yd).

Step 3: Set Angle & Height

Adjust the central angle ($\theta$) for sector/segment geometry or enter a cylinder height ($H$).

Step 4: Copy Full Telemetry

Click Copy Circle Telemetry Card to copy a formatted report for engineering CAD, lab records, or homework.

All Circle Formulas & Mathematical Equations

1. 2D Circle Foundations

$$d = 2r, \quad C = 2\pi r = \pi d$$ $$A = \pi r^2 = \frac{\pi d^2}{4} = \frac{C^2}{4\pi}$$

2. Circular Sector & Arc Length

$$s = r\theta_{\text{rad}} = \frac{\pi r \theta^\circ}{180^\circ}$$ $$A_{\text{sector}} = \frac{1}{2}r^2\theta_{\text{rad}} = \frac{\theta^\circ}{360^\circ}\pi r^2$$

3. Chord Length, Sagitta & Segment Area

$$c = 2r\sin\left(\frac{\theta}{2}\right), \quad h = r\left(1 - \cos\left(\frac{\theta}{2}\right)\right)$$ $$A_{\text{segment}} = \frac{1}{2}r^2(\theta_{\text{rad}} - \sin\theta)$$

4. Inscribed & Circumscribed Polygons

$$s_{\text{inscribed square}} = r\sqrt{2}, \quad s_{\text{circumscribed square}} = 2r$$ $$s_{\text{inscribed triangle}} = r\sqrt{3}, \quad s_{\text{circumscribed triangle}} = 2r\sqrt{3}$$

Classroom Practice & Study Guide: Circle Geometry Calculations

Worked Circle Practice Problems with Solutions:

Problem 1: Pizza Surface Area Value Comparison

Which option provides more pizza: one 16-inch large pizza or two 10-inch medium pizzas?

Solution: Area of 16" pizza ($r=8\,\text{in}$): $A = \pi(8^2) \approx \mathbf{201.06\,\text{in}^2}$. Area of two 10" pizzas ($r=5\,\text{in}$): $2 \times \pi(5^2) = 2 \times 78.54 = \mathbf{157.08\,\text{in}^2}$. The single 16-inch pizza delivers $28\%$ more food!

Problem 2: Circular Sector & Arc Length

A circular sprinkler sprays water in a sector of radius $r = 8.0\,\text{m}$ with an angle of $\theta = 120^\circ$. Find the watered area and arc boundary.

Solution: $s = \frac{120^\circ}{360^\circ} \times 2\pi(8.0) = \frac{1}{3}(16\pi) \approx \mathbf{16.755\,\text{m}}$. $A_{\text{sector}} = \frac{120^\circ}{360^\circ}\pi(8.0^2) = \frac{64\pi}{3} \approx \mathbf{67.021\,\text{m}^2}$.

Problem 3: Milling Square Timber from a Round Log

A cylindrical tree trunk log has a diameter of $d = 40.0\,\text{cm}$ (radius $r = 20.0\,\text{cm}$). What is the largest square beam cross-section that can be milled?

Solution: $s_{\text{inscribed}} = r\sqrt{2} = 20.0 \times 1.4142 \approx \mathbf{28.284\,\text{cm}}$. Beam area is $s^2 = 800\,\text{cm}^2$.

Master Benchmark Circle Matrix Table

Exact geometric dimensions from unit circles to celestial bodies.

Circle Benchmarks

Object / Preset	Radius ($r$)	Diameter ($d$)	Circumference ($C$)	Circle Area ($A$)	Inscribed Square Side
Unit Circle	$1.000$	$2.000$	$2\pi \approx 6.283$	$\pi \approx 3.142$	$\sqrt{2} \approx 1.414$
Compact Disc (CD)	$60.00\,\text{mm}$	$120.00\,\text{mm}$	$376.99\,\text{mm}$	$113.10\,\text{cm}^2$	$84.85\,\text{mm}$
Pizza 12" Medium	$6.000\,\text{in}$	$12.000\,\text{in}$	$37.699\,\text{in}$	$113.10\,\text{in}^2$	$8.485\,\text{in}$
Pizza 16" Large	$8.000\,\text{in}$	$16.000\,\text{in}$	$50.265\,\text{in}$	$201.06\,\text{in}^2$	$11.314\,\text{in}$
Archery Target	$61.00\,\text{cm}$	$122.00\,\text{cm}$	$383.27\,\text{cm}$	$1.169\,\text{m}^2$	$86.27\,\text{cm}$
Basketball Rim	$9.000\,\text{in}$	$18.000\,\text{in}$	$56.549\,\text{in}$	$254.47\,\text{in}^2$	$12.728\,\text{in}$
Earth Equator	$6,378.1\,\text{km}$	$12,756.3\,\text{km}$	$40,075\,\text{km}$	$1.278 \times 10^8\,\text{km}^2$	$9,020\,\text{km}$
Moon Equator	$1,737.4\,\text{km}$	$3,474.8\,\text{km}$	$10,917\,\text{km}$	$9.483 \times 10^6\,\text{km}^2$	$2,457\,\text{km}$

Real-World Engineering, Optics & Manufacturing Applications

Everyday Engineering Implementations:

Civil Water Tanks & Silos: Computing cylinder base area and total capacity for municipal reservoirs and grain silos.
Optical Lens & Telescope Apertures: Light-gathering power scales directly with circular lens area ($A \propto d^2$), making a $200\,\text{mm}$ telescope collect $4\times$ more light than a $100\,\text{mm}$ lens.
HVAC Ductwork & Fluid Mechanics: Calculating hydraulic diameter and flow cross-sections for circular air ducts and plumbing pipes.
Timber Milling Yield: Maximizing square cross-section extraction from round logs using inscribed polygon ratios.

Frequently Asked Questions (FAQ)

Authoritative answers to common questions about circle formulas, radius, diameter, circumference, area, sector, and segment geometry.

Object / Preset	Radius (\(r\))	Diameter (\(d\))	Circumference (\(C\))	Circle Area (\(A\))	Inscribed Square Side
Unit Circle	\(1.000\)	\(2.000\)	\(2\pi \approx 6.283\)	\(\pi \approx 3.142\)	\(\sqrt{2} \approx 1.414\)
Compact Disc (CD)	\(60.00\,\text{mm}\)	\(120.00\,\text{mm}\)	\(376.99\,\text{mm}\)	\(113.10\,\text{cm}^2\)	\(84.85\,\text{mm}\)
Pizza 12" Medium	\(6.000\,\text{in}\)	\(12.000\,\text{in}\)	\(37.699\,\text{in}\)	\(113.10\,\text{in}^2\)	\(8.485\,\text{in}\)
Pizza 16" Large	\(8.000\,\text{in}\)	\(16.000\,\text{in}\)	\(50.265\,\text{in}\)	\(201.06\,\text{in}^2\)	\(11.314\,\text{in}\)
Archery Target	\(61.00\,\text{cm}\)	\(122.00\,\text{cm}\)	\(383.27\,\text{cm}\)	\(1.169\,\text{m}^2\)	\(86.27\,\text{cm}\)
Basketball Rim	\(9.000\,\text{in}\)	\(18.000\,\text{in}\)	\(56.549\,\text{in}\)	\(254.47\,\text{in}^2\)	\(12.728\,\text{in}\)
Earth Equator	\(6,378.1\,\text{km}\)	\(12,756.3\,\text{km}\)	\(40,075\,\text{km}\)	\(1.278 \times 10^8\,\text{km}^2\)	\(9,020\,\text{km}\)
Moon Equator	\(1,737.4\,\text{km}\)	\(3,474.8\,\text{km}\)	\(10,917\,\text{km}\)	\(9.483 \times 10^6\,\text{km}^2\)	\(2,457\,\text{km}\)

⭕ Circle Radius (\(r\))	5.000 m
📏 Circle Diameter (\(d\))	10.000 m (2r)
🔄 Circumference (\(C\))	31.416 m (2πr)
🍰 Circle Area (\(A\))	78.540 m² (πr²)
🏹 Arc Length (\(s\))	5.236 m (rθ)
🍕 Sector Area (\(A_{\text{sector}}\))	13.090 m²
📐 Segment Area (\(A_{\text{segment}}\))	2.265 m²
📏 Chord Length (\(c\))	5.000 m (2r sin θ/2)
↕️ Sagitta Height (\(h\))	0.670 m
⏹️ Inscribed Square Side	7.071 m (r√2)
🔲 Circumscribed Square	10.000 m (2r)
🥫 Cylinder Volume (\(V_{\text{cyl}}\))	785.398 m³
🌐 Sphere Volume / Area	523.599 m³ • 314.159 m²