Discover
A loaf of bread shaped like a cylinder is sliced straight across. What shape is each slice?
- A circle
- A rectangle
Key idea
Solids from slices and from rotations.
Not to scale
A cross section is the shape where a plane cuts a solid. Slices parallel to the base of a prism or cylinder copy the base. Rotating a rectangle about a side gives a cylinder; a right triangle about a leg gives a cone; a semicircle about its diameter gives a sphere.
Worked example
A square pyramid.
Horizontal slice: a smaller square
Parallel to the base.
Vertical slice through the apex: a triangle
Through the apex.
Your turn, guided
Your turn: how many faces does a triangular prism have?
2 triangles + 3 rectangles.
Practice ★
- A cylinder of radius 4 is sliced by a plane parallel to its base. What shape is the cross section?
- A circle of radius 4
- A rectangle
- An ellipse
- A 9 by 6 rectangle is rotated about one of its sides. Which solid does it sweep out?
- A cylinder
- A cone
- A sphere
Watch out!
Practice ★★
- A cone of height 3 is sliced by a vertical plane through its apex. What shape is the cross section?
- An isosceles triangle
- A circle
- A rectangle
Exit ticket
Three questions to check you have got it.
- A cylinder of radius 4 is sliced by a plane parallel to its base. What shape is the cross section?
- A circle of radius 4
- A rectangle
- An ellipse
- A cylinder of radius 2 is sliced by a plane parallel to its base. What shape is the cross section?
- A circle of radius 2
- A rectangle
- An ellipse
- A cylinder of radius 5 is sliced by a plane parallel to its base. What shape is the cross section?
- A circle of radius 5
- A rectangle
- An ellipse
Summary card
Cross sections and rotations
Parallel slices copy the base; rectangle → cylinder, right triangle → cone, semicircle → sphere.
Watch out: the angle of the slice decides the shape.
End of the lesson
Next lesson: Volume of prisms and cylinders Back to the chapter