SOLID WORKSHOP

Unfold, slice, fill

Unfold solids into nets, slice them with a plane, pack them with cubes, and scale them up.

Faces

Highlight & count ↓

Edges

Highlight & count ↓

Vertices

Highlight & count ↓

F − E + V

Mode

Solid · Cube

Highlight & count

View

Euler table

Solid	F	E	V	F−E+V
Count a polyhedron's faces, edges and vertices; its row appears here.

Setting up the scene…

Concepts

Faces, edges, vertices

A face is a flat surface of the solid. An edge is the segment where two faces meet. A vertex is a point where three or more edges meet.

Turn the solid while you count: forgetting the hidden faces, or counting them twice, is the most common slip.

Euler's formula

For every polyhedron (without holes), faces minus edges plus vertices is always 2. Cube:

6 - 12 + 8 = 2

. Hexagonal prism:

8 - 18 + 12 = 2

. Know two of the numbers and you can find the third.

F - E + V = 2

Prism or pyramid?

Prism: two matching, parallel bases joined by rectangles. On an

n

-gon:

n + 2

faces,

3n

edges,

2n

vertices.

Pyramid: one base, triangles meeting at an apex.

n + 1

faces,

2n

edges,

n + 1

vertices. A regular tetrahedron is a triangular pyramid made of four equilateral triangles.

Round solids aren't polyhedra

A cylinder has 2 flat circular faces and 1 curved surface; a cone has 1 flat face, 1 curved surface and an apex; a sphere has a single curved surface. Their rims are curved, not straight edges.

Books count these differently: some call the rims curved edges (2 on a cylinder, 1 on a cone) and the cone's apex a vertex; others say these solids have no edges or vertices. This lab marks curved edges and the apex separately. Either way they are not polyhedra, so Euler's formula isn't used for them.

What is a net?

Cut a solid along some edges and lay it flat: the shape you get is a net. Fold it back and the solid returns.

A cube has 11 different nets. Rule of thumb: opposite faces never touch in the net; a row of four wraps around the cube and one lid goes on each side of the row.

Nets of cylinders and cones

Unroll a cylinder's side and you get a rectangle: one side is the height

h

, the other is the base circumference

2\pi r

(think of a can's label).

A cone's side unrolls into a sector: its radius is the slant height

\ell

and its arc is the base circumference

2\pi r

. The sector angle is

360^\circ \cdot \dfrac{r}{\ell}

2\pi r \times h \qquad \theta = 360^\circ\cdot\dfrac{r}{\ell}

Surface area = area of the net

A solid's surface area is the total area of all the pieces of its net. Grouping matching faces saves work: a rectangular prism has three pairs of equal rectangles. The unit is cm².

S = 2(lw + lh + wh)

Slant height

The height of a pyramid's triangles is not the pyramid's height

h

but the slant height

\ell

, from the apex down to the middle of a base edge. It is the hypotenuse of a right triangle. Cones work the same way:

\ell = \sqrt{h^2 + r^2}

\ell = \sqrt{h^2 + \left(\tfrac{a}{2}\right)^2}, \quad S = a^2 + 2a\ell

Surface area of round solids

Cylinder: two circles + one rectangle. Cone: one circle + one sector (

\pi r \ell

). Sphere: exactly the area of 4 great circles of the same radius; an orange's peel fills 4 circles.

S_{\text{cylinder}} = 2\pi r^2 + 2\pi r h

S_{\text{cone}} = \pi r^2 + \pi r\ell

S_{\text{sphere}} = 4\pi r^2

Volume = base area × height

Volume counts how many 1 cm³ unit cubes fit inside. Fill a prism layer by layer: each layer holds base area

B

cubes and there are

h

layers. The rule works for every prism and cylinder.

V = B \cdot h

Cone = one third of a cylinder

A cone of water fills the cylinder with the same base and height only one third of the way; it takes three cones. Pyramids and prisms work the same way. Pointed solids thin out quickly, so 'half' is too big a guess.

V_{\text{cone}} = \tfrac13 \pi r^2 h, \quad V_{\text{pyramid}} = \tfrac13 B h

Sphere = two thirds of its cylinder

The cylinder that just wraps a sphere has radius

r

and height

2r

. A sphere of water fills it ⅔ of the way (Archimedes was so proud of this he asked for it on his tomb).

V = \tfrac23 \cdot \pi r^2 (2r) = \tfrac43 \pi r^3

What is a cross-section?

Slice a solid with a flat plane and the new face you create is the cross-section. Each side of the section lies on one face of the solid, so the section has as many sides as faces the plane cuts. Cuts parallel to the base copy the base's shape.

Cube sections: 3 to 6 sides

A cube has 6 faces, so a section has at most 6 sides. A cut parallel to a face gives a square, a tilted cut parallel to an edge a rectangle, a tilted cut near a corner a triangle, and other tilted cuts quadrilaterals such as a trapezoid, or a pentagon. The cut through the centre perpendicular to a space diagonal is a regular hexagon.

Slicing a cone: circle, ellipse, parabola, hyperbola

A level cut gives a circle, a gentle tilt an ellipse. When the plane is parallel to the cone's side you get a parabola, steeper still a hyperbola. An upright cut through the apex is an isosceles triangle. These curves are called conic sections.

Scale: k, k², k³

Multiply every length of a solid by

k

: lengths grow

k

times, areas

k^2

times (

k \times k

squares) and volume

k^3

times (

k \times k \times k

cubes). Double a cube's edge and 8 old cubes fit inside, yet its surface is only 4 times larger.

L' = kL, \quad S' = k^2 S, \quad V' = k^3 V

cm, cm², cm³

Length is cm (a line), area cm² (1 cm × 1 cm squares), volume cm³ (1 cm × 1 cm × 1 cm cubes). The power tells you how many lengths are multiplied.

Skills and curriculum

Faces, edges and vertices

Grades 3–8

Identify prisms, pyramids, cylinders, cones and spheres; count faces, edges and vertices (Euler: F − E + V = 2).

Not started

MEB Grade 3 · Geometry of Objects (1) · MAT.3.3.1MEB Grade 8 · Geometric Solids · M.8.3.4.1, M.8.3.4.2, M.8.3.4.5, M.8.3.4.6CCSS 6.G.A.4

Nets

Grades 4–8

Match solids with their nets and tell which nets fold into which solid.

Not started

MEB Grade 4 · Geometric Solids and Shapes · M.4.2.1.4MEB Grade 7 · Geometric Quantities (1) · MAT.7.4.2MEB Grade 8 · Geometric Solids · M.8.3.4.1, M.8.3.4.2, M.8.3.4.5, M.8.3.4.6CCSS 6.G.A.4

Surface area

Grades 6–8

Compute surface area of prisms, pyramids and cylinders by adding the areas of a net's faces.

Not started

MEB Grade 7 · Geometric Quantities (1) · MAT.7.4.2, MAT.7.4.6MEB Grade 8 · Geometric Solids · M.8.3.4.3CCSS 6.G.A.4CCSS 7.G.B.6

Volume of prisms

Grades 5–7

Understand volume as packing unit cubes and use V = base area × height.

Not started

MEB Grade 7 · Geometric Quantities (1) · MAT.7.4.3, MAT.7.4.4, MAT.7.4.6CCSS 5.MD.C.5CCSS 6.G.A.2CCSS 7.G.B.6

Volume of cylinders, cones and spheres

Grades 8–12

Use the volume formulas for cylinders, cones and spheres and the cone = ⅓ cylinder relationship.

Not started

MEB Grade 8 · Geometric Solids · M.8.3.4.4MEB Grade 12 · Geometric Solids · MAT.12.3.2, MAT.12.3.3CCSS 8.G.C.9CCSS HSG-GMD.A.3

Cross-sections

Grades 7–10

Predict the shape of the cross-section when a plane slices a solid.

Not started

CCSS 7.G.A.3CCSS HSG-GMD.B.4

How scaling changes area and volume

Grades 7–10

Explain why scaling lengths by k scales area by k² and volume by k³.

Not started

MEB Grade 10 · Geometric Shapes · MAT.10.1.3CCSS 7.G.A.1

Mission 1/21Build0/21 ✓

Count the cube

Count the cube's faces, edges and vertices: use the buttons below to highlight each kind in turn and watch until the count finishes.

Try it in the simulation — the mission completes on its own when you reach the goal.

Unfold, slice, fill

Concepts

Skills and curriculum

Count the cube

All missions

How it works: Unfold, slice, fill

Keyboard and mouse