半円の面積計算機
半径または直径から半円の面積、円弧の長さ、底辺を含む全周囲長、重心の高さを瞬時に算出。
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.
半円の面積に関するよくある質問
円の面積の求め方、計算公式、単位換算に関する主な質問と回答
半円の面積の公式は何ですか？
半径を用いる場合は (π × r²) / 2、直径を用いる場合は (π × d²) / 8 です。
専門的な円の計算ツール
幾何学的パラメータに応じた最適な円の専門計算ツールを選択してください
Area of Circle Calculator
Find circle area from radius, diameter, or circumference with instant formulas.
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