Perimeter Formulas for Common Plane Figures | 常见平面图形周长公式汇总

📚 Perimeter Formulas for Common Plane Figures | 常见平面图形周长公式汇总

The perimeter of a plane figure is the total distance around its boundary. Mastering perimeter formulas is essential for solving geometry problems in A-Level mathematics and for applying them to real-world contexts such as fencing, framing, and construction.

平面图形的周长是指绕其边界一周的总长度。掌握周长公式对于解决A-Level数学中的几何问题以及在现实情境（如围栏、框架和建筑施工）中的应用至关重要。

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1. Square | 正方形

A square has four equal sides. If the side length is s, the perimeter is simply four times the side length.

正方形有四条相等的边。若边长为 s，则周长就是边长的四倍。

P = 4s

For example, a square with side length 5 cm has a perimeter of 4 × 5 = 20 cm.

例如，边长为5 cm的正方形，其周长为4 × 5 = 20 cm。

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2. Rectangle | 矩形（长方形）

A rectangle has two pairs of equal opposite sides. Let the length be l and the width be w. The perimeter is the sum of all four sides.

矩形有两条相等的长边和两条相等的宽边。设长为 l，宽为 w，周长等于四条边之和。

P = 2(l + w)

If l = 8 cm and w = 3 cm, then P = 2(8 + 3) = 22 cm.

若 l = 8 cm，w = 3 cm，则 P = 2(8 + 3) = 22 cm。

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3. Triangle | 三角形

The perimeter of any triangle is the sum of the lengths of its three sides. Let the sides be a, b, and c.

任意三角形的周长等于其三条边长度之和。设三边分别为 a、b、c。

P = a + b + c

For an equilateral triangle with side length a, this reduces to P = 3a. For an isosceles triangle with two equal sides a and base b, P = 2a + b.

对于边长为a的等边三角形，公式简化为P = 3a。对于两腰为a、底边为b的等腰三角形，P = 2a + b。

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4. Parallelogram | 平行四边形

A parallelogram has opposite sides equal in length. If the adjacent side lengths are a and b, the perimeter is twice the sum of the adjacent sides.

平行四边形的对边长度相等。若相邻两边长分别为 a 和 b，则周长等于相邻两边之和的两倍。

P = 2(a + b)

Note that the height of the parallelogram is not needed for the perimeter; only the side lengths matter.

注意：平行四边形的周长不需要用到高，只需要边长。

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5. Rhombus | 菱形

A rhombus is a parallelogram with all four sides equal. Therefore, its perimeter is the same as that of a square with the same side length.

菱形是四条边都相等的平行四边形。因此，其周长公式与同边长的正方形相同。

P = 4s

Although the diagonals of a rhombus are perpendicular, they do not appear directly in the perimeter formula.

虽然菱形的对角线互相垂直，但它们并不直接出现在周长公式中。

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6. Trapezoid (Trapezium) | 梯形

A trapezoid has one pair of parallel sides. The perimeter is simply the sum of all four side lengths. Let the parallel sides be a and b, and the non-parallel sides be c and d.

梯形有一对平行边。周长就是四条边长之和。设平行边为 a 和 b，非平行边为 c 和 d。

P = a + b + c + d

If the trapezoid is isosceles, then c = d, and the formula becomes P = a + b + 2c.

若梯形是等腰梯形，则 c = d，公式变为 P = a + b + 2c。

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7. Circle (Circumference) | 圆（周长）

The perimeter of a circle is called its circumference. It is proportional to the diameter or radius. Let the radius be r and the diameter be d = 2r.

圆的周长称为圆周。它与直径或半径成正比。设半径为 r，直径为 d = 2r。

C = 2πr = πd

For example, a circle with radius 7 cm has circumference C = 2 × π × 7 ≈ 43.98 cm.

例如，半径为7 cm的圆，其周长 C = 2 × π × 7 ≈ 43.98 cm。

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8. Sector of a Circle | 扇形

A sector is a part of a circle bounded by two radii and an arc. Its perimeter consists of the arc length plus the two radii. Let the radius be r and the central angle in radians be θ. The arc length is rθ (when θ is in radians).

扇形是由两条半径和一段圆弧围成的圆的一部分。其周长等于弧长加上两条半径。设半径为 r，圆心角（以弧度为单位）为 θ。弧长为 rθ（当θ用弧度表示时）。

P = rθ + 2r = r(θ + 2)

If the angle is given in degrees, convert it to radians first: θ(rad) = θ(deg) × π / 180°.

若角度以度为单位，需先转换为弧度：θ(弧度) = θ(度) × π / 180°。

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9. Regular Polygon | 正多边形

A regular polygon has all sides equal and all interior angles equal. If it has n sides and each side has length s, the perimeter is:

正多边形的所有边相等，所有内角相等。若它有 n 条边，每条边长为 s，则周长为：

P = n × s

For example, a regular hexagon with side length 4 cm has perimeter 6 × 4 = 24 cm.

例如，边长为4 cm的正六边形，其周长为6 × 4 = 24 cm。

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10. Ellipse (Approximate Perimeter) | 椭圆（近似周长）

The perimeter of an ellipse cannot be expressed with elementary functions. However, a good approximation uses the semi-major axis a and semi-minor axis b. One common approximation is Ramanujan’s formula:

椭圆的周长不能用初等函数精确表达。但可以使用半长轴 a 和半短轴 b 进行较好的近似。一种常用的近似是拉马努金公式：

P ≈ π [ 3(a + b) − √((3a + b)(a + 3b)) ]

Another simpler approximation is P ≈ π(a + b), which works well when the ellipse is nearly circular.

另一个更简单的近似为 P ≈ π(a + b)，当椭圆接近圆形时效果较好。

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11. Composite Figures | 组合图形

For composite figures, the perimeter is the sum of the lengths of all outer sides. Avoid counting internal shared edges. For example, if a semicircle is attached to a rectangle, the perimeter includes the straight side and the curved arc, but not the diameter inside the figure.

对于组合图形，周长等于所有外边界长度之和，内部共享边不要重复计算。例如，一个半圆附在矩形上，其周长包括直边和圆弧，但不包括图形内部的直径。

When calculating such perimeters, identify each boundary segment separately and sum them carefully.

计算此类周长时，应分别识别每一段边界并小心求和。

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12. Common Pitfalls and Exam Tips | 常见错误与备考提示

Always check the units. If the radius is given in cm, the perimeter is in cm. Remember to use radians for arc length formulas. Do not confuse diameter with radius; the circumference formula can be written as πd or 2πr.

始终检查单位。若半径以cm给出，周长单位就是cm。弧长公式中务必使用弧度。不要混淆直径与半径；圆周长公式可写为πd或2πr。

• For a rectangle, ensure you add the two different side lengths and multiply by 2 — not just add all four sides individually (though that also works).

  对于矩形，确保将两个不同边长相加后乘以2——当然逐个相加四条边也同样正确。

• For a sector, the perimeter includes both radii; many students forget to add 2r.

  对于扇形，周长包含两条半径；很多学生会忘记加上2r。

• When finding the perimeter of a composite shape, walk around the outline once, writing down every segment.

  求组合图形周长时，沿轮廓走一圈，写下每一段边界。

Practice with past exam questions to become fluent in recognizing which formula applies to each figure.

通过练习历年真题，熟练识别每种图形适用的公式。

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