Volume and surface area of solids

Practice 5 items →

Prisms, cylinders, pyramids, cones, spheres — keep a table rather than memorizing in isolation. Students find volume and surface area, derive the slant height of a cone from r and h, and decide surface versus volume from the story.

Specification · for a parent or teacher

Part no.: volume-surface-3d
Course: Advanced Math 7 · Unit 13
Realm: Shapes
Difficulty: 4 of 4 — Grade 7 / Math 1 core. Structure, models, proofs on the grid.
Qty: 5 items

Part no. is the node's id; the realm is the kind of math; qty is how many practice items it carries.

Students can

use the table for prisms, cylinders, pyramids, cones, spheres; find slant height from $r$ and $h$ ; handle composite solids (cylinder + hemisphere) and “how much paint / how much water” stories that decide surface vs volume.

Ready when

they can find $V$ and total $S$ of a cone given $r$ and $h$ , producing $\ell$ first.

Formulas

Cylinder

V = \pi r^2 h,\quad S = 2\pi r^2 + 2\pi r h

Pyramid / cone

V = \tfrac{1}{3} B h

Cone surface

S = \pi r^2 + \pi r \ell,\quad \ell = \sqrt{r^2 + h^2}

Sphere

V = \tfrac{4}{3}\pi r^3,\quad S = 4\pi r^2

Watch for

pyramid $\frac12 Bh$ ; cone lateral area with $h$ instead of $\ell$ .

Before · requires

Volume and surface areavolume-surface-nets · Math 6
Perimeters and areas of plane figures (circles)circles-arcs-sectors · Math 7
Real numbersreals-radicals-sci-notation · Math 7

After · used by

— none

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