Grade 6 (age 11-12)

grade 6 Cylinder, Cone, Sphere

Cylinder, Cone, Sphere

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

↔️ Same base and height

The radius and height from the section above also drive this comparison. Same base, same height.

💡 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

④With radius 5 and height 8, cylinder πr²h ≈ 3.14×25×8 = 628. Cone is 1/3 ≈ 209. Sphere (4/3)πr³ ≈ 3.14×125×4/3 ≈ 523. Circumference 2πr was in the previous circle-area chapter.

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.

← Previous

Area of a Circle

Volume of Space Figures