📚 Constructing Triangles: SSS, SAS and ASA | 构造三角形：边边边、边角边和角边角
In KS3 Cambridge Mathematics, the ability to draw a triangle accurately from a set of given measurements is a core practical skill. It tests your understanding of geometric properties and your precision with instruments. This article focuses on the three standard construction cases — SSS (Side-Side-Side), SAS (Side-Angle-Side) and ASA (Angle-Side-Angle) — using only a ruler, a pair of compasses and a protractor. You will learn step-by-step methods, common pitfalls and how to check your work.
在剑桥初中数学中，根据给定测量数据精确画出一个三角形是一项核心实践技能。它既考查你对几何性质的理解，也考验你使用工具的精确度。本文重点介绍三种标准作图情形——边边边（SSS）、边角边（SAS）和角边角（ASA），仅需直尺、圆规和量角器。你将学会逐步操作的方法、常见错误以及如何检验你的作图结果。
1. Tools You Need | 你需要用到的工具
Before starting any construction, make sure your equipment is in good condition. A sharp pencil is essential for fine, accurate lines. Your ruler should have clear centimetre and millimetre markings. Compasses need to hold their radius firmly without slipping, and a protractor should be the standard 180° semicircular type with a clear baseline.
开始任何作图之前，确保你的工具状况良好。一支削尖的铅笔对于画出纤细精准的线条至关重要。直尺应具有清晰的厘米和毫米刻度。圆规需能牢固地保持半径不滑动，量角器应为标准的180°半圆型，带有清晰的基线。
- Ruler – for drawing and measuring straight line segments | 直尺 – 用于绘制和测量直线段
- Pair of compasses – for transferring lengths and drawing arcs | 圆规 – 用于转移长度和画弧线
- Protractor – for measuring and constructing angles | 量角器 – 用于测量和构造角度
- Sharp HB pencil – for crisp construction lines | 削尖的HB铅笔 – 用于画出清晰的作图线
- Eraser – only for correcting mistakes, not for heavy rubbing | 橡皮 – 仅用于纠正错误，不要用力摩擦
Keep your compass hinge tight so the radius does not change while you swing an arc.
保持圆规铰链拧紧，这样在画弧时半径不会改变。
2. SSS Construction: Given Three Sides | 边边边构造：已知三边
The SSS case is the most straightforward. You are given the lengths of all three sides, for example AB = 6 cm, BC = 5 cm and AC = 4 cm. No angle information is needed. You rely entirely on the compass to locate the third vertex by intersecting arcs.
边边边情况是最直接的。你已知三边的长度，例如AB = 6 cm，BC = 5 cm，AC = 4 cm。不需要任何角度信息。你完全依靠圆规通过弧线相交来定位第三个顶点。
Step 1: Draw the longest side as the base. Use your ruler to draw a line segment AB exactly 6 cm long. Label the endpoints A and B.
步骤1： 画出最长的一边作为底边。用直尺画出恰好6cm长的线段AB。标记端点A和B。
Step 2: Set your compasses to the length of the second side, 5 cm. Place the compass point on A and draw a large arc above (or below) the line AB. Make the arc long enough so it will clearly intersect with the next arc.
步骤2： 将圆规张开到第二条边的长度5cm。把圆规针尖放在A点上，在AB线段的上方（或下方）画一条大弧。弧线要足够长，确保与下一条弧线明显相交。
Step 3: Without changing the compass setting, set the compasses to the third side length, 4 cm. Place the point on B and draw a second arc that crosses the first one. The intersection point is vertex C.
步骤3： 改变圆规开度，设置为第三边长度4cm。将针尖放在B点，画第二条弧线与第一条相交。交点就是顶点C。
Step 4: Join C to A and C to B with straight lines using your ruler. You have now constructed triangle ABC.
步骤4： 用直尺分别连接C到A和C到B，画出直线。现在你已经构造出了三角形ABC。
Always check the lengths of AC and BC with your ruler to confirm they match 4 cm and 5 cm.
始终用直尺检查AC和BC的长度，确认它们分别为4 cm和5 cm。
3. SAS Construction: Given Two Sides and the Included Angle | 边角边构造：已知两边及其夹角
In the SAS case, you know the lengths of two sides and the angle between them. For instance, construct triangle PQR with PQ = 7 cm, PR = 5 cm and angle QPR = 50°. The ‘included’ angle is the one formed by the two given sides.
在边角边情形中，你知道两边的长度和它们之间的夹角。例如，构造三角形PQR，其中PQ = 7 cm，PR = 5 cm，∠QPR = 50°。这里的“夹角”指的是已知两边所夹的角。
Step 1: Draw the side that is adjacent to the given angle. It is often easiest to draw the longer one first. Use your ruler to draw PQ = 7 cm. Mark the vertex Q at one end and P at the other.
步骤1： 画出与已知角相邻的一边。通常先画较长的一边较为容易。用直尺画出PQ = 7 cm。在一端标记顶点Q，另一端为P。
Step 2: Place the centre of the protractor on point P, aligning the baseline exactly with line PQ. Starting from the zero on the PQ side, measure 50° and make a small mark. Remove the protractor and draw a long ray from P through this mark. This ray will contain the side PR.
步骤2： 将量角器的中心对准P点，基线精确对齐线段PQ。从PQ一侧的零刻度开始，量出50°并做一个轻小的标记。移开量角器，从P点出发经过该标记画出一条长射线。这条射线将包含边PR。
Step 3: Set your compasses to 5 cm. Place the point on P and draw an arc that cuts the ray drawn in Step 2. The intersection is the vertex R.
步骤3： 将圆规张开到5 cm。针尖放在P点，画弧与步骤2中的射线相交。交点即为顶点R。
Step 4: Join R to Q with a straight line to complete triangle PQR. Measure the length RQ and the remaining angles to verify your construction is consistent.
步骤4： 用直线连接R和Q，完成三角形PQR。测量RQ的长度和其余角度，以验证作图的准确性。
4. ASA Construction: Given Two Angles and the Included Side | 角边角构造：已知两角及其夹边
ASA construction requires you to know two angles and the side between them. For example, construct triangle XYZ with XY = 8 cm, ∠YXZ = 40° and ∠XYZ = 60°. Notice that the side XY is included between the two given angles.
角边角构造需要你知道两个角和它们的夹边。例如，构造三角形XYZ，已知XY = 8 cm，∠YXZ = 40°，∠XYZ = 60°。注意，边XY就是两个已知角所夹的边。
Step 1: Draw the given side XY = 8 cm. Label X and Y.
步骤1： 画出已知边XY = 8 cm。标记X和Y。
Step 2: At point X, use your protractor to construct an angle of 40° with XY as one arm. Draw a long ray from X into the interior region where the triangle will be.
步骤2： 在X点，用量角器以XY为一边构造一个40°的角。从X点向三角形将形成的内侧区域画一条长射线。
Step 3: At point Y, construct an angle of 60° on the same side of XY. Draw a ray from Y. The two rays will intersect at point Z.
步骤3： 在Y点，在XY的同侧构造一个60°的角。从Y点画出射线。两条射线将在Z点相交。
Step 4: The intersection Z is the third vertex. Thicken the triangle outline XYZ. To check accuracy, measure the third angle ∠XZY; it should be 80° because the angles in a triangle sum to 180°.
步骤4： 交点Z就是第三个顶点。加粗三角形XYZ的轮廓。为检查准确性，测量第三个角∠XZY，它应该是80°，因为三角形内角和为180°。
5. Working with Compasses and Arcs | 圆规与弧线的使用技巧
When constructing triangles, clean arcs are vital. Always hold the compasses by the top handle, not the legs, to avoid changing the radius. Draw arcs lightly so they can be erased later. The intersection of arcs must be clear; if they barely touch, make your arcs longer.
构造三角形时，清晰的弧线至关重要。始终握住圆规的顶部手柄，而不是两脚，以免改变半径。轻轻画弧，方便以后擦除。弧线的交点必须清晰；如果它们只是轻轻擦过，就把弧画得更长些。
If your compasses slip on the paper, place a small piece of masking tape at the pivot point or work on a slightly rough surface. Practice drawing arcs from both directions to confirm the radius is stable.
如果圆规在纸上打滑，可以在支点处贴一小片纸胶带，或在稍粗糙的表面上作图。练习从两个方向画弧，以确认半径稳定不变。
6. Common Mistakes and How to Avoid Them | 常见错误及如何避免
One frequent error is measuring the angle from the wrong end of the protractor. Always start from the zero mark along the line you are using, not the outer scale by default. A quick check: if the angle appears acute but the protractor reads 130°, you used the wrong scale.
一个常见的错误是从量角器错误的一端测量角度。务必从你所用边的零刻度开始，而不是默认使用外侧刻度。快速检查：如果角看起来是锐角但量角器显示130°，说明你用错了刻度。
Another mistake is drawing the initial side shorter than required. If you draw a base of 6 cm as exactly 6 cm, but your pencil line is thick, the effective length might be slightly off. Always use a sharp pencil and measure from the inner edge of the mark.
另一个错误是把起始边画得比要求短。如果你画一条6 cm的底边恰好为6 cm，但铅笔线很粗，有效长度可能稍有偏差。始终用尖铅笔并从标记内侧测量。
In SSS construction, students sometimes forget to check that the three sides can actually form a triangle (the triangle inequality: sum of any two sides > the third). For example, sides 3 cm, 4 cm and 8 cm will never meet because 3 + 4 < 8. Always check this before starting.
在边边边构造中，学生有时忘记检查三条边是否确实能构成三角形（三角形不等式：任意两边之和大于第三边）。例如，边长3 cm、4 cm和8 cm永远无法相交，因为3 + 4 < 8。在开始前务必检查这一点。
7. Constructing a Triangle Given RHS (Right Angle, Hypotenuse, Side) | 直角斜边边构造
A special case linked to SAS is the construction of a right-angled triangle when you know the hypotenuse and one other side. For instance, construct triangle ABC with right angle at B, hypotenuse AC = 10 cm, and AB = 6 cm. You can treat this as a SAS problem if you first construct the right angle.
与边角边相关的一个特例是，已知斜边和一条直角边构造直角三角形。例如，构造三角形ABC，∠B=90°，斜边AC=10 cm，AB=6 cm。如果先构造直角，就可以把它当作边角边问题来处理。
Step 1: Draw AB = 6 cm. At B, construct a 90° angle and draw a vertical ray upwards.
步骤1： 画出AB = 6 cm。在B点构造90°角，向上画一条垂直射线。
Step 2: Set compasses to 10 cm. With point on A, draw an arc cutting the vertical ray at C.
步骤2： 圆规张开到10 cm。以A为针尖画弧，与垂直射线交于C。
Step 3: Join C to A and C to B. The triangle is complete. Measure BC to verify.
步骤3： 连接C与A、C与B。三角形完成。测量BC以作验证。
8. Constructing Equilateral and Isosceles Triangles | 构造等边三角形和等腰三角形
An equilateral triangle is a special SSS construction where all sides are equal. If you need an equilateral triangle of side 5 cm, simply draw a base of 5 cm, then set compasses to 5 cm and draw two arcs from each endpoint. Their intersection gives the third vertex.
等边三角形是边边边的特例，所有边都相等。如果你需要画一个边长5 cm的等边三角形，只需画一条5 cm的底边，然后把圆规设为5 cm，从两个端点分别画弧，所得交点就是第三个顶点。
For an isosceles triangle, two sides are equal. You can use either SSS (if both equal sides and base are given) or SAS (if the vertex angle is given). Label the equal sides clearly and build arcs of equal radius from appropriate points.
对于等腰三角形，有两条边相等。你可以使用边边边法（如果已知两条等边和底边），或边角边法（如果已知顶角）。清楚地标记等边，并从相应点以相等半径画弧。
9. Using Construction to Solve Problems | 利用作图解决问题
Construction skills are not just for copying triangles. You can use them to find a missing distance if a diagram is drawn to scale. For instance, a treasure map problem may require you to locate a point given bearings and distances from two landmarks. Treat each bearing as an angle construction, and distances as radii on arcs.
作图技能不仅用于复制三角形。如果图画按比例绘制，你可以用它来找出缺失的距离。例如，藏宝图问题可能要求你根据两个地标的方向角和距离定位一个点。把每个方位角当作一个角的构造，把距离当作弧线的半径。
You can also construct a triangle to determine whether a set of measurements is ambiguous. In the SSA case (two sides and an angle not included), there may be two possible triangles. Sketching arcs will reveal if the second intersection exists.
你也可以通过构造三角形来判断一组测量数据是否有歧义。在SSA情况（两边及一个非夹角）中，可能存在两个符合的三角形。画弧线可以显示出第二个交点是否存在。
10. Accurate Labelling and Neatness | 准确的标注与整洁度
In any Cambridge KS3 assessment, marks are awarded for correct labelling of vertices A, B, C and side lengths. Use capital letters for vertices and lower-case letters for side lengths (often a opposite A, etc.), though exact convention may vary. Write the given measurements on your construction lines, and use a different colour or dashed lines for construction arcs so the final triangle stands out.
在任何剑桥初中评估中，正确标注顶点A、B、C和边长都能得分。通常顶点用大写字母，边长用小写字母（如a对应A的对边），不过具体惯例可能有所不同。把给定的测量数据写在作图纸上，并用不同颜色或虚线表示作图弧线，使最终三角形更加醒目。
A tidy construction is easier to mark and less prone to confusion. Keep your work area clear, erase superfluous arcs gently, and present the final triangle with solid boundaries.
整洁的作图更易于批改，也不容易混淆。保持作图区域整洁，轻轻擦掉多余的弧线，并用实线边界呈现最终的三角形。
11. Checking Your Triangle | 检验你的三角形
After completing the construction, always run a quick verification. Use your ruler to measure the sides that were not part of the original data; do they match expectations? Use your protractor to measure the remaining angles and check that the sum is as close to 180° as possible (within ±1° considering instrument accuracy).
完成作图后，一定要进行快速验证。用直尺测量那些原始数据中没有给出的边；它们是否符合预期？用量角器测量其余的角，并检查它们的和是否尽可能接近180°（考虑仪器精度，误差在±1°内）。
If the sum is off by more than 2°, re-examine your angle constructions. Most errors happen with the protractor alignment. Another test is to compare the triangle with a friend’s construction of the same data – they should be congruent.
如果角度和偏差超过2°，重新检查你构造的角。多数错误源于量角器的对齐。另一种检验方法是将你的三角形和同学用相同数据构造的三角形比较——它们应该全等。
12. Practice Problem and Summary | 练习题与总结
Try constructing triangle DEF with DE = 5.5 cm, EF = 6.5 cm and ∠DEF = 45°. Use SAS method. After completing, measure DF and angle DFE. Check that the three angles sum to 180°. This exercise consolidates all the skills discussed.
试着构造三角形DEF，DE = 5.5 cm，EF = 6.5 cm，∠DEF = 45°。使用边角边方法。完成后，测量DF和∠DFE。检查三个角之和是否为180°。这个练习可以巩固所讨论的全部技能。
To summarise, mastering triangle constructions builds a strong foundation for more advanced geometry. Remember: SSS uses arcs from both ends, SAS uses one angle with two adjacent sides, and ASA uses the included side with two base angles. Stay precise, check the triangle inequality, and always verify your final shape.
总结来说，掌握三角形构造为更高级的几何学奠定了坚实基础。记住：边边边从两端画弧，边角边使用一个角和两条相邻边，角边角使用夹边和两个底角。保持精确，检查三角形不等式，并且始终验证最终的图形。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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