Cylinders, cones and spheres are solids with curved surfaces.
A tin can is a cylinder, a party hat is a cone and a football is a sphere, so they are easy to find around you.
A cylinder and a cone both have circular bases, and with the same base and height a cone holds one third as much as a cylinder; a sphere has no height, and its size is set by its radius alone.
Change the radius and the height to compare, in turn, the three shapes, the nets of the cylinder and the cone, and the three volumes.
Round solids
🥫 Round solids around you
①A tin can is a cylinder: its top and bottom are equal circles and its side is a curved surface
②A party hat is a cone: it has one circular base, and its side narrows to a point called the apex
③A football is a sphere: it looks like a circle from every direction, and every point on its surface is one radius from the center
④All three can be made by spinning a flat shape around one of its sides: a rectangle gives a cylinder, a right triangle a cone and a semicircle a sphere
5
8
seegongsik.com
A cylinder, a cone and a sphere drawn with radius r and height h
Unfolding the nets
✂️ Cut open and laid flat
①A cylinder unfolds into 2 circles for the bases and 1 rectangle for the side
②The rectangle is as wide as the circumference of the base, 2πr, and as tall as the cylinder, so making the radius bigger in the picture stretches the rectangle sideways
③A cone unfolds into 1 circle and 1 sector, and the arc of the sector is as long as the circumference of the base
④A sphere cannot be laid flat however you cut it, so it has no net
seegongsik.com
The nets of the cylinder and the cone
Comparing volumes
↔️ Same radius, same height
The radius and height you chose above are used in the volume picture below too. The cylinder and the cone share the same base and height, and the sphere has the same radius.
seegongsik.com
Volumes of a cylinder, a cone and a sphere with the same radius and height
💡 A cone is one third of a cylinder
①With the same base and height, you have to pour a full cone into the cylinder three times to fill it, so the cone's volume is one third of the cylinder's
②With the starting radius 5 and height 8, the cylinder is 3.14 × 5 × 5 × 8 = 628 and the cone is one third of that, about 209.3
③A sphere's volume depends only on its radius; with the same radius 5 it is about 523.3
④Change the height and only the cylinder and cone bars change, while the sphere's bar stays put
Summary
Volume of a cylinder
V = πr²h
Area of the base (πr²) × height (h)
Volume of a cone
V = 13πr²h
One third of a cylinder with the same base and height
Volume of a sphere
V = 43πr³
Set by the radius alone; a sphere has no separate height
🎯 At a glance
①Cylinder: two equal circular bases, volume πr²h
②Cone: one circular base and an apex, volume one third of the cylinder's
③Sphere: every point on the surface is the same distance from the center, and the volume depends only on the radius
④Nets: a cylinder is 2 circles and a rectangle, a cone is 1 circle and a sector
⑤Three cones of water fill a cylinder with the same base and height
Examples & Unit Review
Example 1
Find the volume of a cylinder with base radius 3 cm and height 10 cm. (Use 3.14 for π.)
1
Volume of a cylinder = area of the base × height = (3.14 × radius × radius) × height.
2
(3.14 × 3 × 3) × 10 = 28.26 × 10 = 282.6, so the volume is 282.6 cm³.
▸ 282.6 cm³
As with a rectangular prism, the volume of a shape with the same cross-section all the way up is the base area times the height.
Example 2
If you unroll the side of a cylinder with base radius 5 cm, what shape do you get and how wide is it? (Use 3.14 for π.)
1
The side unrolls into a rectangle whose width equals the circumference of the base.
2
Circumference = π × diameter = 3.14 × 10 = 31.4, so the rectangle is 31.4 cm wide.
▸ Rectangle, width 31.4 cm
The second picture starts with radius 5, so the rectangle in the cylinder net has exactly this width.
Unit Review
What is the volume of a cylinder with base radius 2 cm and height 5 cm? (Use 3.14 for π.)
①62.8 cm³
②31.4 cm³
③20 cm³
④125.6 cm³
▸ ① 62.8 cm³
1
Volume = area of the base × height = (3.14 × 2 × 2) × 5.