Cross-Sections and Solids of Revolution
Aligned to G-GMD.B.4 — Common Core State Standards for Mathematics.
What this lesson teaches
Slicing a 3D solid with a plane produces a 2D cross-section whose shape depends on the slice's angle. Rotating a 2D shape around an axis sweeps out a 3D solid: a rectangle rotated about a side gives a cylinder, a right triangle gives a cone.
Worked example
What solid results from rotating a right triangle about one leg? Stand a right triangle with legs vertical and horizontal, and spin it about the vertical leg. The horizontal leg sweeps a circular base and the hypotenuse sweeps the slanted surface, producing a cone. Its radius equals the horizontal leg and its height equals the vertical leg.
Practice questions
- What 2D shape is the horizontal cross-section of a cylinder?
- Rotate a rectangle about one of its sides. What solid do you get?
- A cube is sliced by a plane parallel to one face. Name the shape of the cross-section.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Secciones transversales y sólidos de revolución
Cortar un sólido 3D con un plano produce una sección transversal 2D cuya forma depende del ángulo del corte. Rotar una forma 2D alrededor de un eje genera un sólido 3D: un rectángulo rotado sobre un lado da un cilindro, un triángulo rectángulo da un cono.
Ejemplo: ¿Qué sólido resulta de rotar un triángulo rectángulo sobre un cateto? Coloca un triángulo rectángulo con catetos vertical y horizontal, y gíralo sobre el cateto vertical. El cateto horizontal barre una base circular y la hipotenusa barre la superficie inclinada, produciendo un cono. Su radio es el cateto horizontal y su altura es el cateto vertical.
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