G5 Geometry – Triangles and Quadrilaterals | G5几何 – 三角形与四边形

📚 G5 Geometry – Triangles and Quadrilaterals | G5几何 – 三角形与四边形

This revision article covers the core ideas of triangles and quadrilaterals, a vital part of the IGCSE Mathematics syllabus. You will learn their properties, classification, area formulas and common exam tips.

本篇复习文章覆盖IGCSE数学大纲中关于三角形与四边形的核心内容。你将学习它们的性质、分类、面积公式以及常见考试技巧。

1. Basics of Triangles | 三角形的基本性质

Every triangle has three sides and three interior angles. The sum of the three interior angles is always 180°.

每个三角形都有三条边和三个内角。三个内角的和始终为180°。

∠A + ∠B + ∠C = 180°

An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

三角形的一个外角等于与它不相邻的两个内角之和。

Also, the sum of any two sides must be greater than the third side. This is called the triangle inequality.

同时，任意两边之和必须大于第三边，这称为三角形不等式。

2. Classification of Triangles | 三角形的分类

Triangles can be classified by their sides or by their angles.

三角形可以根据边或角进行分类。

By sides: scalene (no equal sides), isosceles (two equal sides), equilateral (all three sides equal).

按边分类：不等边三角形（三边都不相等）、等腰三角形（两边相等）、等边三角形（三边都相等）。
By angles: acute (all angles less than 90°), right (one angle equals 90°), obtuse (one angle greater than 90°).

按角分类：锐角三角形（所有角小于90°）、直角三角形（一个角等于90°）、钝角三角形（一个角大于90°）。

3. Isosceles and Equilateral Triangles | 等腰三角形与等边三角形

In an isosceles triangle, the two equal sides are opposite the two equal angles. The two base angles are equal.

在等腰三角形中，两条相等的边对着两个相等的角，因此两个底角相等。

If AB = AC, then ∠B = ∠C

In an equilateral triangle, all three sides are equal and each angle is exactly 60°.

在等边三角形中，三条边都相等，并且每个角都是60°。

An isosceles triangle has one line of symmetry, while an equilateral triangle has three lines of symmetry.

等腰三角形有一条对称轴，等边三角形有三条对称轴。

4. Area and Perimeter of a Triangle | 三角形的面积与周长

The perimeter of a triangle is the sum of the lengths of its three sides.

三角形的周长等于三边长度之和。

Perimeter = a + b + c

The most common area formula is:

最常用的面积公式是：

A = ½ × base × height

Here the height is the perpendicular distance from the base to the opposite vertex.

这里的高是从底边到对边顶点的垂直距离。

For a triangle with sides a, b and c, we can also use Heron’s formula:

对于已知三边a、b、c的三角形，还可以使用海伦公式：

s = (a + b + c) ÷ 2, A = √[s(s − a)(s − b)(s − c)]

5. Basics of Quadrilaterals | 四边形的基本性质

A quadrilateral is a polygon with four sides, four vertices and four angles. The sum of its interior angles is 360°.

四边形是具有四条边、四个顶点和四个角的多边形，其内角和为360°。

Interior angle sum of any quadrilateral = 360°

The perimeter of a quadrilateral is the sum of its four side lengths.

四边形的周长等于四条边长之和。

Common quadrilaterals include parallelograms, rectangles, rhombuses, squares, trapeziums and kites.

常见的四边形包括平行四边形、矩形、菱形、正方形、梯形和风筝形。

6. Parallelograms, Rectangles, Rhombuses and Squares | 平行四边形、矩形、菱形与正方形

All of these shapes are special quadrilaterals with specific properties.

这些图形都是具有特殊性质的四边形。

Parallelogram: opposite sides are parallel and equal; opposite angles are equal; diagonals bisect each other.

平行四边形：对边平行且相等；对角相等；对角线互相平分。
Rectangle: a parallelogram with all angles equal to 90°, so opposite sides are equal and diagonals are equal.

矩形：所有角都是90°的平行四边形，因此对边相等且对角线相等。
Rhombus: a parallelogram with all four sides equal; diagonals bisect each other at right angles.

菱形：四条边都相等的平行四边形；对角线互相垂直平分。
Square: a parallelogram with all sides equal and all angles 90°. It is both a rectangle and a rhombus.

正方形：所有边相等且所有角为90°的平行四边形，它既是矩形又是菱形。

Area formulas: rectangle = l × w, square = s², parallelogram = b × h, rhombus = ½ × d₁ × d₂

7. Trapeziums and Kites | 梯形与风筝形

A trapezium has one pair of opposite sides parallel. In some countries the word ‘trapezoid’ is used.

梯形只有一组对边平行。有些国家也使用“trapezoid”来表示梯形。

Area of trapezium = ½ × (a + b) × h

where a and b are the parallel sides and h is the perpendicular height.

其中a和b是平行边的长度，h是垂直高。

A kite has two pairs of adjacent equal sides. Its diagonals cross at right angles, and one diagonal bisects the other.

风筝形有两组相邻的边相等，它的对角线互相垂直，且其中一条对角线平分另一条。

For a kite, the area is also ½ × (product of diagonals).

风筝形的面积也等于½乘以两条对角线乘积。

8. Polygon Angle Sums | 多边形的内角和与外角和

Triangles and quadrilaterals are polygons. A polygon with n sides has an interior angle sum of (n − 2) × 180°.

三角形和四边形都是多边形。有n条边的多边形的内角和为(n − 2) × 180°。

Sum of interior angles = (n − 2) × 180°

The sum of the exterior angles of any convex polygon is always 360°.

任何凸多边形的外角和始终为360°。

In a regular polygon, all interior angles are equal, so each interior angle equals (n − 2) × 180° ÷ n.

在正多边形中，所有内角都相等，因此每个内角等于(n − 2) × 180° ÷ n。

9. Congruence and Similarity | 全等与相似

Two triangles are congruent if they have exactly the same size and shape. We use tests SSS, SAS, ASA and RHS.

如果两个三角形大小和形状完全相同，则它们全等。我们使用SSS、SAS、ASA和RHS判定条件。

SSS: three corresponding sides are equal.

SSS：三条对应边都相等。
SAS: two sides and the included angle are equal.

SAS：两边及夹角相等。
ASA: two angles and the included side are equal.

ASA：两角及夹边相等。
RHS: right angle, hypotenuse and one other side are equal.

RHS：直角、斜边和一条直角边相等。

Similar triangles have the same shape but different sizes. Their corresponding angles are equal and corresponding sides are in the same ratio.

相似三角形形状相同但大小不同，它们的对应角相等，对应边成比例。

10. Common Exam Tricks and Errors | 常见考试技巧与易错点

Students often use the wrong base or height in the triangle area formula. Remember the height must be perpendicular to the base.

学生经常在三角形面积公式中选错底边或高，请记住高必须垂直于底边。

Another common error is thinking all rectangles are squares. A square is a rectangle, but a rectangle is not necessarily a square.

另一个常见错误是认为所有矩形都是正方形。正方形是矩形，但矩形不一定是正方形。

When calculating the interior angle of a regular polygon, use the formula correctly and do not forget the outer angles sum to 360°.

在计算正多边形的内角时，要正确使用公式，并且不要忘记外角和为360°。

Always write units (cm², m²) for area, and do not confuse perimeter units with area units.

面积一定要写单位（cm²、m²），并且不要把周长单位和面积单位混淆。

11. Worked Example | 例题精讲

Question: In a triangle, two angles are 40° and 70°. Find the third angle and state the type of triangle by its angles.

题目：在一个三角形中，两个角分别是40°和70°。求第三个角，并按角判断三角形的类型。

Third angle = 180° − 40° − 70° = 70°

Since all angles are less than 90°, the triangle is an acute triangle. Also, two angles are equal, so it is also an isosceles triangle.

由于所有角都小于90°，所以这是一个锐角三角形。同时有两个角相等，因此它也是等腰三角形。

Question: Find the area of a trapezium with parallel sides 8 cm and 12 cm, and height 5 cm.

题目：求一个梯形的面积，它的平行边分别为8 cm和12 cm，高为5 cm。

A = ½ × (8 + 12) × 5 = ½ × 20 × 5 = 50 cm²

12. Summary | 总结

Triangles and quadrilaterals appear in many IGCSE exam questions. Mastering their angle sums, area formulas and classification helps you solve problems quickly and accurately.

三角形与四边形在IGCSE考试中经常出现。掌握它们的内角和、面积公式以及分类，可以帮助你快速准确地解题。

Draw a clear diagram, write down known facts, and check that your units are correct.

画一个清晰的图形，写下已知条件，并检查单位是否正确。

Practice past paper questions until you feel confident with these geometrical shapes.

多练习历年真题，直到你对这些几何图形充满信心。

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G5 Geometry – Triangles and Quadrilaterals | G5几何 – 三角形与四边形