Геометрия и величины
AQA · GCSE · Mathematics Тема 4 5:27 Английское озвучивание · Английский + китайские субтитры (встроенные)
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Five families carry this unit.
本单元有五类要点。
A polygon's interior angles add to n minus 2 times 180, and its exterior angles always add to 360.
多边形内角和是 (n − 2) × 180°，外角和总是 360°。
A sector is theta over 360 of the circle.
扇形是圆的 θ/360。
Cones and spheres have their own volumes, and these are not on the formulae sheet.
圆锥和球有各自的体积公式，而且公式表上没有。
Right triangles use Pythagoras and SOH CAH TOA.
直角三角形用勾股定理和 SOH CAH TOA。
And a length scale factor k gives areas k squared and volumes k cubed.
长度比例因子为 k 时，面积乘 k²，体积乘 k³。
Here is a regular octagon.
这是一个正八边形。
Extend one side: the turn to the next side is the exterior angle, 45 degrees.
延长一条边：转向下一条边的角就是外角，45°。
Walking round, the turns add to one full turn, 360, so 360 divided by 45 gives 8 sides.
绕一圈，所有转角加起来是一整圈 360°，所以 360 ÷ 45 = 8 条边。
The interior angle is 180 minus 45, which is 135.
内角是 180 − 45 = 135°。
For a hexagon the same steps give 60 and 120.
正六边形同样得到 60° 和 120°。
Construct with ruler and compasses and keep the arcs as evidence.
用直尺和圆规作图，并保留作图弧线作为证据。
Points equidistant from two points lie on the perpendicular bisector; a fixed distance from a point is a circle.
到两点距离相等的点在垂直平分线上；到一点距离固定的点构成一个圆。
An enlargement sends a point P to C plus k times P minus C.
放大把点 P 变到 C 加 k 乘 (P 减 C)。
With centre 1, 1, the point 3, 2 and scale factor 2 goes to 5, 3.
以 (1, 1) 为中心，点 (3, 2) 在比例因子 2 下变到 (5, 3)。
At Higher, a scale factor of minus 2 sends it through the centre to minus 3, minus 1, on the opposite side.
高阶内容中，比例因子为负 2 时，它穿过中心变到 (负 3, 负 1)，在另一侧。
Take a circle of radius 5 and a chord 3 from the centre.
取半径为 5 的圆，一条弦到圆心距离为 3。
The perpendicular from the centre bisects the chord, so a right triangle appears: radius 5 as the hypotenuse, 3 as one leg.
从圆心作的垂线平分这条弦，于是出现一个直角三角形：半径 5 是斜边，3 是一条直角边。
The half-chord is root of 25 minus 9, which is 4.
半弦长是 √(25 − 9) = 4。
So the chord is 8 centimetres.
所以弦长 8 厘米。
Two tangents from P touch the circle at A and B, and angle APB is 24 degrees.
从 P 作的两条切线分别切圆于 A 和 B，∠APB = 24°。
Draw the radii OA and OB: a tangent meets a radius at 90 degrees.
连半径 OA 和 OB：切线与半径成 90°。
So the quadrilateral OAPB has angles 90, 90 and 24, and the angle at the centre is 156.
所以四边形 OAPB 有 90°、90° 和 24° 三个角，圆心角是 156°。
The angle at the circumference on the same arc is half of it: 78 degrees.
同一段弧上的圆周角是它的一半：78°。
A sector of radius 6 with angle 120.
半径 6、圆心角 120° 的扇形。
120 over 360 is a third of the circle.
120/360 是圆的三分之一。
The whole circumference is 12 pi, so the arc is 4 pi.
整圆周长是 12π，所以弧长是 4π。
The whole area is 36 pi, so the sector is 12 pi.
整圆面积是 36π，所以扇形面积是 12π。
Working backwards, a sector of area 9 pi is a quarter of 36 pi, so its angle is 90 degrees.
反过来，面积为 9π 的扇形是 36π 的四分之一，所以它的角是 90°。
Here is a cone with radius 3 and height 4.
这是一个半径 3、高 4 的圆锥。
Its slant is 5, from Pythagoras.
由勾股定理，母线长 5。
The volume is a third of pi r squared h: a third of pi times 9 times 4, which is 12 pi.
体积是 ⅓πr²h：⅓ × π × 9 × 4 = 12π。
The curved surface is pi r l: pi times 3 times 5, so 15 pi.
侧面积是 πrl：π × 3 × 5 = 15π。
Add the base, 9 pi, for a total of 24 pi.
再加上底面 9π，总表面积 24π。
A cuboid is 4 by 3 by 12.
长方体尺寸 4 × 3 × 12。
Lay the base flat: its diagonal is root of 9 plus 16, so base diagonal 5.
先看底面：对角线是 √(9 + 16)，底面对角线为 5。
That diagonal and the 12-cm edge make a triangle that stands upright inside the box.
这条对角线和 12 厘米的棱构成一个竖立在盒子里的直角三角形。
Its hypotenuse is the space diagonal, root of 25 plus 144, which is 13.
它的斜边就是体对角线，√(25 + 144) = 13。
And tan alpha is 12 over 5, so alpha is about 67.4 degrees.
tan α = 12/5，所以 α 约 67.4°。
Cut an equilateral triangle of side 2 in half: half an equilateral triangle has sides 1, root 3 and 2.
把边长为 2 的等边三角形对半切开：半个等边三角形的三边是 1、√3 和 2。
So sin 30 is a half, cos 30 is root 3 over 2 and tan 60 is root 3.
所以 sin 30° = ½，cos 30° = √3/2，tan 60° = √3。
A right isosceles triangle with legs 1 has hypotenuse root 2, so sin 45 and cos 45 are both root 2 over 2.
两直角边为 1 的等腰直角三角形斜边是 √2，所以 sin 45° 和 cos 45° 都是 √2/2。
Sides 6 and 8 enclose 60 degrees.
两边 6 和 8 夹 60° 角。
The cosine rule gives c squared as 36 plus 64 minus 96 times a half: 52.
余弦定理：c² = 36 + 64 − 96 × ½ = 52。
So c is root 52, which is 2 root 13.
所以 c = √52 = 2√13。
The area is a half times 6 times 8 times sine 60, which is 12 root 3.
面积是 ½ × 6 × 8 × sin 60° = 12√3。
When a sine rule gives an angle, an obtuse supplement may fit too.
用正弦定理求角时，它的钝角补角也可能成立。
A column vector's top entry is horizontal and its bottom entry vertical.
列向量上面一项是水平分量，下面一项是竖直分量。
With a equals 4, 1 and b equals 1, 3, adding tip to tail gives a plus b equals 5, 4.
a 等于 (4, 1)，b 等于 (1, 3)，首尾相接相加得 a 加 b 等于 (5, 4)。
From A at 1, 2 to B at 5, 5 the vector is 4, 3, and its length is 5 by Pythagoras.
从 A(1, 2) 到 B(5, 5) 的向量是 (4, 3)，由勾股定理，它的长度是 5。
To prove three points lie on a straight line, show that two vectors are multiples of each other and share a point.
要证明三点共线，就证明两个向量互为倍数并且有一个公共点。
Check yourself.
自我检测。
An exterior angle of 40 means 360 over 40: 9 sides.
外角 40° 意味着 360 ÷ 40 = 9 条边。
The reverse of a bearing of 070 is 070 plus 180: 250.
方位角 070° 的反方位是 070 + 180 = 250°。
A radius-4 circle has area 16 pi.
半径 4 的圆面积是 16π。
Cos 60 is a half.
cos 60° = ½。
And doubling every length multiplies the volume by 2 cubed, which is 8.
所有长度加倍，体积乘以 2³ = 8。