Grade 6 (age 11-12)

Cylinder, Cone, Sphere

A cylinder, a cone, and a sphere are round solids that show a curved surface wherever you look. You can spot them everywhere, like a tin can, a party hat, and a soccer ball. With the same base and height, a cone holds one third of a cylinder, while a sphere's volume depends only on its radius. Here you can spin the three shapes, unfold their nets, and compare their volumes side by side.

Exploring Round 3D Shapes

🥫 Round 3D Shapes Around Us

①Tin can → cylinder (circles top/bottom, rectangle side)

②Party hat → cone (round base, pointed tip)

③Soccer ball → sphere (round from any angle)

Radius r5

Height h8

Unfolding Nets

✂️ What If We Cut and Unfold?

①Cylinder: 2 circles + 1 rectangle

②Cone: 1 circle + 1 sector

③A sphere has no flat net (cannot be unfolded perfectly)

Comparing Volumes

💡 Cone is 1/3 of Cylinder

①Same base & height: 3 cones = 1 cylinder

②Pour water from cone to cylinder — exactly 3 fills!

③A sphere's volume depends only on its radius

Summary

Volume of a Cylinder

V = πr²h

Base area (πr²) × height (h)

Volume of a Cone

V = 13πr²h

1/3 of a cylinder (same base, same height)

Volume of a Sphere

V = 43πr³

Determined by radius alone (no height)

🎯 Key Points

①Cylinder: base (circle) × height → πr²h

②Cone: 1/3 of cylinder → (1/3)πr²h

③Sphere: proportional to radius³ → (4/3)πr³

④Nets: cylinder (circles + rectangle), cone (circle + sector)

⑤3 cones = 1 cylinder (proven by water experiment!)

Examples & Unit Review

Example 1

Find the volume of a cylinder with base radius 3 cm and height 10 cm. (Use π = 3.14)

Volume of a cylinder = base area × height = (π × radius × radius) × height.

(3.14 × 3 × 3) × 10 = 28.26 × 10 = 282.6 cm³.

▸ 282.6 cm³

The volume of a prism or cylinder is the base area times the height.

Example 2

When the side of a cylinder with base radius 5 cm is unrolled, what shape is it and what is its width? (Use π = 3.14)

The unrolled side is a rectangle whose width equals the circumference of the base.

Circumference = 2 × 3.14 × 5 = 31.4 cm, so the width is 31.4 cm.

▸ Rectangle, width 31.4 cm

The unrolled side rectangle has width = base circumference and height = cylinder height.

Unit Review

What is the volume of a cylinder with base radius 2 cm and height 5 cm? (Use π = 3.14)

①62.8 cm³

②31.4 cm³

③20 cm³

④125.6 cm³

▸ ① 62.8 cm³

Volume = base area × height = (3.14 × 2 × 2) × 5.

12.56 × 5 = 62.8, so 62.8 cm³.

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Area of a Circle

Space & 3D Figures