切口深一倍，
截面也大一倍吗？Double the height. Double the slice?
用一个平行于底面的平面去切圆锥，切口是一个圆。把切面从锥顶往下移，看看为什么它的面积比半径长得更快。
A plane parallel to a cone's base cuts a circle. Move it away from the apex to discover why its area grows faster than its radius.
01预测Predict
02比较两个切面Try two slices
比较 2 cm 和 4 cm 两个切面。上半截圆锥已经拿掉，方便你看金色的切口。转动视角看看：切口始终是圆，只是在屏幕上看起来像椭圆。
Compare 2 cm and 4 cm. The upper cone is removed so you can inspect the gold cut face. Turn the view to see that the slice is circular, even when it looks elliptical on screen.
转动视角只改变你看到的样子，不会改变图上的读数。Turning the view changes what you see, never the readouts on the figure.
03解释变化Explain what changed
看推理过程See the reasoning
由相似三角形得 r / R = h / H。这里 R = 4 cm、H = 8 cm，所以 r = h / 2。面积 A = πr²。h 加倍，r 也加倍；把加倍的半径平方，面积就变成 4 倍。
h = 2 cm 时，r = 1 cm，A = π cm²。h = 4 cm 时，r = 2 cm，A = 4π cm²。转动视角会改变你看到的形状，但不会改变这些数值。
Similar triangles give r / R = h / H. Here R = 4 cm and H = 8 cm, so r = h / 2. The area is A = πr². Doubling h doubles r, and squaring a doubled radius gives four times the area.
At h = 2 cm, r = 1 cm and A = π cm². At h = 4 cm, r = 2 cm and A = 4π cm². Turning the camera changes what you see, but never these values.
04迁移Transfer
这个模型的适用范围Where this model applies
这是一个直圆锥，用平行于底面的平面去切。h 从锥顶量起，不是从底面量起。h = 0 时截面缩成一个点，面积为零；h = 8 时截面就是底面。这里不涉及斜截面，也不计算体积。
A right circular cone, cut parallel to its base. h is measured from the apex, not from the base. At h = 0 the circle is a point with zero area; at h = 8 it equals the base. No oblique sections or volume calculation are implied.
图形是示意性的投影，计算由标注的尺寸驱动。
The picture is a schematic projection; the stated dimensions drive the calculations.
离线可用，不需要动画或网络。用 Tab 切换控件，方向键调节滑块。Works offline. No animation or network access is needed. Use Tab to reach controls and arrow keys on the sliders.
世界坐标在深度方向旋转后再投影。图形是示意，读数由标注的尺寸计算。World coordinates are rotated in depth and projected. The picture is schematic; the stated dimensions drive the calculations.