University learnersSlice and relateExample · 1 question · 2 min

Slice Through a Surface

Inspect a fixed 3D paraboloid and its horizontal slice, then connect the visible circle to the equation x² + y² = 4.

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Your example

Horizontal slices of z = x² + y²

Plane	Slice	Radius
z = 1	x² + y² = 1	1
z = 9	x² + y² = 9	3
z = 4	?	?

The surface z = x² + y² is a bowl that opens upward. The table lists horizontal slices: at z = 1 the slice is x² + y² = 1 with radius 1; at z = 9 it is x² + y² = 9 with radius 3. The slice at z = 4 is still open.

Choose one answer

For z = x² + y², what shape appears when the horizontal plane z = 4 cuts the surface?

A circle around the z-axis: x² + y² = 4, with radius 2.A parabola: the same curve as the bowl's side view.

Show a hint

Hold the plane's height fixed, substitute it, and identify the two-variable equation that remains.

About this activity

What you practise: Move between a surface, a plane, an equation, and a family of level curves without treating them as the same object.

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For z = x² + y², what shape appears when the horizontal plane z = 4 cuts the surface? The surface z = x² + y² is a bowl that opens upward. The table lists horizontal slices: at z = 1 the slice is x² + y² = 1 with radius 1; at z = 9 it is x² + y² = 9 with radius 3. The slice at z = 4 is still open. A horizontal plane fixes z = 4 and leaves x and y free. Substituting gives x² + y² = 4: a circle around the origin with radius √4 = 2. A parabola appears only when you cut vertically, for example with the plane y = 0.