IGCSE数学几何 Core Extended 圆定理 相似 作图

IGCSE数学几何 Core Extended 圆定理 相似 作图

Cambridge IGCSE 数学 0580 的几何模块叫 Geometry，Core 与 Extended 两层共用同一份术语表和同一套角度工具，却在圆定理、相似、对称三处拉出明显差距。先看清哪些条目属于哪一层，再决定练习时间投在哪里，比盲目刷整套历年卷省力得多。这份对照按考纲 2025 至 2027 版的条目顺序写，每一条都给出内容编号，方便你回官网核对。

Geometry in Cambridge IGCSE Mathematics 0580 shares one vocabulary list and one angle toolkit across both tiers, while circle theorems, similarity and symmetry pull the two tiers apart. Knowing which outcomes sit in Core and which ones appear only in Extended changes how a candidate plans practice. This comparison follows the outcome order of the 2025 to 2027 syllabus, with content codes attached to every point.

术语表的差别：plane 与 perpendicular bisector 只在 Extended 出现

0580 考纲 2025 至 2027 版的 C4.1 与 E4.1 都叫 Geometrical terms，两张词表并不相同。Extended 的 E4.1 在平面词汇里多出 plane 与 perpendicular bisector，Core 的 C4.1 只列 point、vertex、line、parallel、perpendicular、bearing 几项。立体图形一侧的差别更直接，E4.1 的词表里有 hemisphere 与 frustum，C4.1 明确写出 hemisphere 这个词不要求。圆的词汇同样分成两份，Extended 要求区分 major and minor arc，Core 只写到 arc 为止。很多学生以为术语题不单独考，实际卷面上每一个角度理由都要靠这些词作答。常见的错误是把 perpendicular bisector 当成 Core 内容背下来，白占了本可以给角度计算的复习额度。

Both tiers label the first outcome Geometrical terms, yet the two word lists differ. Extended adds plane and perpendicular bisector, plus hemisphere and frustum for solids, and it asks candidates to name major and minor arcs, while core stops at arc and states that hemisphere is not required. Many students treat vocabulary as a warm-up topic and then lose method marks because their reasons are written without the accepted terms. Check which list applies to your tier before memorising anything.

三角形、四边形、多边形与圆的术语怎么写对

C4.1 与 E4.1 都把图形名词列成清单，需要写对的是三角形四种、特殊四边形六种、多边形五种，再加上 regular 与 irregular 的区分。三角形要能写出 equilateral、isosceles、scalene、right-angled，四边形要能写出 square、rectangle、kite、rhombus、parallelogram、trapezium。多边形的名字从 pentagon 一直排到 hexagon、octagon、decagon，圆的词汇从 centre、radius、diameter、circumference 排到 semicircle、chord、tangent、arc、sector、segment。考纲在同一页写明考生不需要证明两个图形全等，这条说明正是 2025 版删掉 congruence criteria 之后留下的痕迹。阅卷时最常见的现象是学生认得出图形，答案里却写不出英文名，只剩一个含糊的说法。把全称抄在笔记本第一页，考试时直接调用。

The vocabulary lists cover four triangles, six special quadrilaterals and five named polygons, along with the regular and irregular distinction. Circle words run from centre, radius, diameter and circumference to semicircle, chord, tangent, arc, sector and segment. The syllabus notes that candidates are not expected to show that two shapes are congruent, which reflects the removal of the congruence criteria in the 2025 version. In practice, examiners see answers where a shape is correctly identified but the English name is missing, so learn the terms in English from the outset rather than translating them in the hall.

点、直线、三角形、四边形与平行线的角度计算

C4.6 与 E4.6 的第 1 条列出四类角度事实：同一点上的角之和为 360 度，直线上相邻两角之和为 180 度，对顶角相等，三角形内角和为 180 度，四边形内角和为 360 度。第 2 条把平行线里的三类关系写清楚：同位角相等，内错角相等，同旁内角之和为 180 度。考纲在同一条里要求会读 angle ABC 这类三字母记号，也要求答案里的理由使用正确的几何术语。很多学生以为算出数字就算完成，几何题的分值有一半压在理由上。常见的错误是把 alternate angles 与 corresponding angles 的理由写反，数字对而理由错，方法分照样拿不到。

The four basic angle facts are that angles at a point sum to 360 degrees, angles on a straight line sum to 180 degrees, vertically opposite angles are equal, and the angle sum is 180 degrees in a triangle and 360 degrees in a quadrilateral. Angles formed inside parallel lines follow three further rules, with corresponding angles equal, alternate angles equal and co-interior angles adding to 180 degrees. Reasons must be written with three-letter notation such as angle ABC. Many students produce the right number with the wrong reason, and the method mark goes with it.

正多边形与不规则多边形的内角外角

C4.6 第 3 条只要求已知和使用正多边形的角度性质，E4.6 第 3 条把范围扩到正多边形与不规则多边形。两条都点名要会用外角与内角，也要会用内角和。正 n 边形的每个外角等于 360 除以 n，内角等于 180 度减去外角，外角和固定为 360 度，内角和为 n 减 2 再乘 180 度。很多学生以为内角和公式要背很多条，其实一条就够全部命名多边形使用。答题时把 n 先写出来再代入，decagon 就不会被当成 octagon 处理。自检一下，如果算出的内角是 156 度，n 应当是 15，不是 10。

Core asks for the angle properties of regular polygons, while Extended extends the same outcome to irregular ones. For a regular polygon with n sides, each exterior angle is 360 divided by n, each interior angle is 180 minus that value, the exterior angles always total 360 degrees, and the angle sum is n minus 2 multiplied by 180 degrees. One formula covers every named polygon in the syllabus, so there is very little to memorise. Write n down before substituting, and an interior angle of 156 degrees points to 15 sides rather than 10.

圆的定理一：从两条到六条

这是几何模块里两层差距最大的一处。Core 的 C4.7 只列两条，半圆上的圆周角为 90 度，切线与半径的夹角为 90 度。Extended 的 E4.7 叫 Circle theorems I，在同一页列出六条，除前两条之外，还有圆心角等于同弧圆周角的两倍、同弧上的圆周角相等、圆内接四边形对角之和为 180 度、切线与弦的夹角等于对应角，最后一条就是 alternate segment theorem。考纲要求在答案里写出所用性质作为理由，不能只给数字。很多学生以为圆定理靠图形感觉，六条其实各自对应一个固定的角位置关系。常见的错误是把 angle at the centre 当成 angle at the circumference，答案直接少了一半。

This is where the two tiers separate most sharply. Core lists only two circle facts, namely that the angle in a semicircle is 90 degrees and the angle between a tangent and a radius is 90 degrees; Extended adds four more in Circle theorems I, covering the angle at the centre, angles in the same segment, opposite angles of a cyclic quadrilateral and the alternate segment theorem. Each rule describes a fixed positional relationship, so the diagram alone will not tell you which one applies. Mixing up the centre and the circumference is the most common slip in this topic.

圆的定理二：等弦、弦的中垂线与外点切线

E4.8 叫 Circle theorems II，内容不是新的角度计算，而是圆的三条对称性质。等长的弦到圆心距离相等，弦的垂直平分线一定经过圆心，从圆外一点引出的两条切线长度相等。这三条在 Core 内容里完全不出现，Core 考生不会在卷面上见到它们。Extended 考生要用它们解释某条线为什么是垂直平分线，或者为什么两段切线长相符。很多学生以为定理二只是定理一的补充，它的出题方式更像一个证明的第一步。写理由时把 equal chords 与 equidistant 两个词写出来，比补一条辅助线省时间。

Circle theorems II is not a further set of angle calculations; it collects three symmetry properties of a circle. Equal chords sit at equal distances from the centre, the perpendicular bisector of a chord passes through the centre, and the two tangents drawn from an external point are the same length. None of these appear in the core content, so core candidates will not meet them at all. Extended candidates use them to justify a perpendicular bisector or a pair of equal tangents, which usually opens a longer proof question.

相似：只算长度与长度面积体积三组比例

C4.4 只有一句话，计算相似图形的长度。E4.4 把同一主题拆成三条，除长度之外还要使用长度与面积之间的比例关系、长度与表面积体积之间的比例关系，并要求能给出简单的相似解释。三组比例可以记成一句话，长度比等于 k，面积比等于 k 的平方，体积比等于 k 的立方。Extended 还要会用几何理由说明两个三角形相似，这一步常常出现在证明题的第一行。很多学生以为相似就是把边长乘一个数，遇到面积比就忘了平方。常见的错误是把 3 比 4 的长度比直接当成体积比，正确答案应当是 27 比 64。

Core keeps similarity to a single line, which is calculating the lengths of similar shapes. Extended splits the same topic into three outcomes, adding the links between lengths and areas and between lengths, surface areas and volumes, plus simple explanations of why two triangles are similar. The ratios follow a fixed pattern, with lengths scaling by k, areas by k squared and volumes by k cubed. Length ratios are the ones students handle well, while volume ratios are where the marks go, since a 3 to 4 length ratio gives 27 to 64 for volume rather than 3 to 4.

尺规作图与弧线保留

C4.2 与 E4.2 的内容一致，要求测量和画线段与角，只用直尺和圆规按三条边的长度作三角形，会画也会用展开图。考纲在同一条里注明垂直平分线与角平分线的作图不要求，而作图弧线必须保留在卷面上。这两句话放在一起容易被读错，不要求作图并不等于不要求知道垂直平分线的性质。Extended 的 E4.1 词表里仍然保留 perpendicular bisector，圆定理二也用它来定位圆心。很多学生以为没画弧不影响评分，阅卷时留下的弧线正是作图分的依据。

Both tiers carry the same construction outcome: measure and draw lines and angles, build a triangle from three given side lengths using only a ruler and compasses, and draw and interpret nets. The syllabus adds that constructions of perpendicular bisectors and angle bisectors are not required, while construction arcs must be shown. These two sentences are easy to misread, because not constructing a bisector does not put the perpendicular bisector out of scope; Extended still lists it in the vocabulary and uses it in Circle theorems II. Arcs left on the page are exactly what an examiner looks for when awarding the construction mark.

比例尺与三向方位角

C4.3 与 E4.3 都要求画和读比例尺图，并使用三向方位角。考纲把方位角的定义写得很硬，从正北顺时针量起，取值从 000 度到 360 度，三点定位靠三位数字写出。同一页给的例子是已知 B 在 A 的方位角为 025 度，求 A 在 B 的方位角。做法是把 025 加 180 得 205 度，结果如果超过 360 就减去 360。画图时直尺必须用上，方向词 north、east、south、west 也要能读懂，考纲举的例子是 point D is due east of point C。常见的错误是把方位角从东边量起，或者忘了顺时针这个前提。

Scale drawings and three-figure bearings sit in the same outcome at both tiers. A bearing is measured clockwise from north and runs from 000 degrees to 360 degrees, which is the detail candidates most often forget. If the bearing of B from A is 025 degrees, the bearing of A from B is 205 degrees, found by adding 180 and subtracting 360 whenever the answer passes that value. Ruler work is compulsory, and compass words such as north, east, south and west are expected in answers like point D is due east of point C.

对称性：平面图形与立体图形

C4.5 与 E4.5 的第一条相同，要求识别平面图形的线对称与旋转对称的阶数，并把它和三角形、四边形、多边形的性质连起来。Extended 的 E4.5 多出第二条，要把对称性质用到柱体、圆柱、棱锥与圆锥这些立体图形上，例如找出对称面与对称轴。正方形有四条对称轴，旋转对称阶数为 4。等边三角形有三条对称轴，阶数为 3。平行四边形没有对称轴，旋转对称阶数为 2。很多学生以为平行四边形也有对称轴，这是这一节最容易出现的判断错误。

Both tiers start from plane figures, asking for lines of symmetry and the order of rotational symmetry, linked to the properties of triangles, quadrilaterals and polygons. Extended then applies the same idea to solids, identifying planes and axes of symmetry in prisms, cylinders, pyramids and cones. A square has four lines of symmetry and rotational symmetry of order 4, while an equilateral triangle has three lines and order 3. A parallelogram has rotational symmetry of order 2 but no line of symmetry, which is the judgement students most often get wrong.

两卷结构与公式表

2025 版考纲把四份卷子的分配写得很清楚。Core 考生考 Paper 1 非计算器卷，1 小时 30 分、80 分、占 50%，以及 Paper 3 计算器卷，同样 1 小时 30 分、80 分、占 50%。Extended 考生考 Paper 2 非计算器卷，2 小时、100 分、占 50%，以及 Paper 4 计算器卷，同样 2 小时、100 分、占 50%。考纲写明预计拿 D 或以下的考生应报考 Paper 1 与 Paper 3，目标是 A* 至 C 的学生读 Extended 内容。每份卷第二页提供 List of formulas，考纲同时说明并非所有需要的公式都会给出。角度性质与圆定理的理由从来不进公式表，只能靠平时记熟。考试安排在每年六月与十一月两个场次，印度另有三月场次。

The 2025 syllabus fixes the paper structure. Core candidates sit Paper 1, a non-calculator paper of 1 hour 30 minutes and 80 marks worth 50 per cent, and Paper 3, a calculator paper with the same duration and marks; extended candidates sit Paper 2 and Paper 4, each 2 hours and 100 marks and each worth 50 per cent. The syllabus states that candidates expected to achieve grade D or below should be entered for Paper 1 and Paper 3, while those targeting A* to C study the extended content. A List of formulas sits on page 2 of every paper, although the syllabus warns that not all required formulas are given, and geometric reasons never appear there at all.

几何模块的六周复习排期

六周时间够把几何模块走一遍，每周只抓一个抓手。前两周回到术语与角度，把三角形、四边形、多边形、圆的词表和角度理由各默写一遍。第三周集中处理圆定理，Extended 考生按六条加三条的顺序过，Core 考生只确认半圆角与切线半径这两条。第四周做相似与比例尺方位角，把长度比、面积比、体积比的关系抄成一张卡片。第五周处理尺规作图与对称，反复练三边作三角形并把弧线留在卷面上。第六周做整套计时卷，非计算器卷优先练，手算的准确率直接决定几何题能做完多少。很多学生以为几何靠灵感，排期表上的每一周都在减少失误率。

Six weeks is enough for the geometry module if each week carries one focus. Start with vocabulary and angle reasoning, writing the word lists and the accepted reasons out from memory. Move to circle theorems next, taking all six extended rules in sequence or the two core facts for the other tier. Similarity, scale drawings and bearings follow, with the length, area and volume ratios copied onto a single card, and construction plus symmetry practice comes before the first full timed paper under exam conditions. Give the non-calculator paper the earlier slot, since mental accuracy decides how much of the geometry section actually gets finished.

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