अर्धवृत्त का क्षेत्रफल कैलकुलेटर
त्रिज्या या व्यास से किसी भी अर्धवृत्त का क्षेत्रफल, चाप की लंबाई, पूर्ण बंद परिमाप और गुरुत्व केंद्र निकालें।
Half Circle Area Calculator
What Is a Half Circle (Semicircle)? Anatomy & Core Properties
A semicircle is formed by bisecting a circle along its diameter (central angle θ = 180° or π radians). When enclosed by its flat diameter chord, it forms a planar 2D shape with fundamental architectural utility.
Unlike a full circle, a semicircle is not point-symmetric; it has a single vertical axis of reflectional symmetry. Because it features both a curved boundary and a flat baseline, calculating its physical boundaries requires separating curved arc from the flat diameter chord.
Understanding the distinction between curved arc (πr) and total closed perimeter (πr + 2r) is essential for window framing, garden bed curbing, and structural engineering.
Comprehensive Semicircle Geometric Relations
| Dimension | Formula (Radius r) | Formula (Diameter d) | Example (r = 10 cm) |
|---|---|---|---|
| Enclosed Area (A) | A = (πr²) / 2 | A = (πd²) / 8 | 157.08 cm² |
| Curved Arc Length (L) | L = π × r | L = (πd) / 2 | 31.42 cm |
| Straight Base Edge (b) | b = 2 × r | b = d | 20.00 cm |
| Total Closed Perimeter (P) | P = r(π + 2) ≈ 5.1416r | P = d(π/2 + 1) ≈ 2.5708d | 51.42 cm |
| Centroid Height (ȳ) | ȳ = (4r) / (3π) ≈ 0.4244r | ȳ = (2d) / (3π) ≈ 0.2122d | 4.24 cm |
| Base Moment of Inertia (I) | I_x = (πr⁴) / 8 | I_x = (πd⁴) / 128 | 3,926.99 cm⁴ |
4 Ways to Calculate Semicircle Geometry
Depending on whether you measure the center radius, outer diameter, or arch curve length, use the appropriate formula.
Radius Method: A = (πr²) / 2
The canonical textbook equation. Calculate full circle area (πr²) and divide by 2.
Diameter Method: A = (πd²) / 8
Directly use the flat bottom width (diameter d). Square diameter, multiply by π, and divide by 8.
Curved Arc Method: A = (L × r) / 2
Because a circular sector's area is half of arc length times radius, for a semicircle Area = (L × r) / 2.
Centroid & Pappus: V = 2π ȳ A
Revolving a semicircle of area A around its base generates a sphere of volume (4/3)πr³. Confirms ȳ = (4r) / (3π).
Algebraic Proof: Semicircle Area & Perimeter Formulas
Verifying the fundamental semicircle relationships from Euclidean principles:
Semicircle Dimensions Master Reference Table
Standard radius measurements from 1 to 20 units with arc length, perimeter, and area.
| Radius (r) | Diameter (d) | Arc Length (πr) | Perimeter [r(π+2)] | Centroid (ȳ) | Area [(πr²)/2] |
|---|---|---|---|---|---|
| 1 | 2 | 3.1416 | 5.1416 | 0.4244 | 1.5708 |
| 2 | 4 | 6.2832 | 10.2832 | 0.8488 | 6.2832 |
| 4 | 8 | 12.5664 | 20.5664 | 1.6977 | 25.1327 |
| 6 | 12 | 18.8496 | 30.8496 | 2.5465 | 56.5487 |
| 10 | 20 | 31.4159 | 51.4159 | 4.2441 | 157.0796 |
| 15 | 30 | 47.1239 | 77.1239 | 6.3662 | 353.4292 |
4 Detailed Solved Real-World Examples
Realistic architecture, landscaping, woodworking, and sports ramp calculations with exact steps.
Glass Area & Framing for 6-Foot Arched Top
An arched transom window sits atop a doorway with a 6-foot span (diameter):
Semicircular Patio Against a Garden House
A homeowner builds a stone patio extending 4 meters out from a straight backyard wall:
Half-Moon (Demi-Lune) Console Table
A solid oak entryway table has a flat rear edge of 120 cm (diameter):
Semicircular Halfpipe Section (10-Foot Radius)
Calculate cross-sectional airspace and curved riding surface for a ramp:
Common Mistakes to Avoid in Semicircle Calculations
Wrong: Area = (π × d²) / 4. Correct: Area = (π × d²) / 8. (πd²)/4 is the full circle's area; you must divide by 8 for a half circle.
The curved arc is just πr. If buying trim or fencing for a half circle, you must add the diameter baseline 2r: Total = πr + 2r.
The balance center of a flat semicircle is at 0.4244 × r from the straight base (roughly 42.4% of the radius upward).
Total closed perimeter is always 5.1416 × radius or 2.5708 × diameter. A fast shortcut for job site estimates.
Industry Applications of Semicircle Calculations
Architecture & Windows
Designing Roman arched transoms, Palladian windows, and masonry archway lintels.
Civil Bridge Engineering
Calculating waterway clearance area under semicircular stone and concrete barrel arches.
Custom Woodworking
Drafting drop-leaf tables, demi-lune entry consoles, and radial wooden marquetry veneers.
Skate Park Construction
Framing half-pipe vertical ramps, bowls, and transition curves with plywood sheet layouts.
Culvert Hydraulics
Computing half-full wetted perimeter and hydraulic radius (R_h = r / 2) for drainage culverts.
Landscape Hardscaping
Building circular patios against house foundations, fire pit seating rings, and planting retainers.
अर्धवृत्त के क्षेत्रफल से संबंधित प्रश्न
वृत्त के क्षेत्रफल, त्रिज्या, व्यास, परिधि और सूत्रों से संबंधित सभी प्रमुख प्रश्नों के उत्तर
अर्धवृत्त के क्षेत्रफल का सूत्र क्या है?
क्षेत्रफल = (π × r²) / 2 त्रिज्या से, अथवा (π × d²) / 8 व्यास से।
अर्धवृत्त का कुल परिमाप कैसे निकाला जाता है?
कुल परिमाप = चाप + व्यास = πr + 2r = r(π + 2) ≈ 5.1416 × r।
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Area of Circle Calculator
Find circle area from radius, diameter, or circumference with instant formulas.
व्यास से वृत्त का क्षेत्रफल कैलकुलेटर
Calculate circle area directly from diameter using A = (πd²)/4.
परिधि से वृत्त का क्षेत्रफल कैलकुलेटर
Determine circle area directly from perimeter using A = C²/(4π).
त्रिज्यखंड और वृत्तखंड क्षेत्रफल कैलकुलेटर
Circular sector and segment area from radius and central angle.
खोखले वृत्त का क्षेत्रफल कैलकुलेटर
Annulus, washer, and circular ring surface area between concentric circles.
वृत्ताकार अनुप्रस्थ काट का क्षेत्रफल कैलकुलेटर
Engineering cross-section for circular shafts, rebar, cables & tensile stress.
पाइप का क्षेत्रफल कैलकुलेटर
Internal fluid flow area vs solid pipe wall metal cross-sectional area.
समाकलन द्वारा वृत्त का क्षेत्रफल
Calculus derivation of circle area with polar, Cartesian, and shell integrals.