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A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently circular) to a point called the apex or vertex.

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A sphere is a 3-dimensional object shaped like a ball. Every point on the surface is the same distance from the center.

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A cylinder is a solid object with two identical flat ends that are circular and one curved side.

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Volume of a Cone - MathHelp.com - Math Help

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For a complete lesson on volume of a cone, go to http://www.MathHelp.com - 1000+ online math lessons featuring a personal math teacher inside every lesson!

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Soda can
Roll of toilet paper
Can of soup

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Ice cream cone
Party hat
Traffic cone

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Basketball
Globe
Orange

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Notice these interesting things:

The pointy end of a cone is called the apex

The flat part is the base

An object shaped like a cone is said to be conical

A Cone is a Rotated Triangle

A cone can be made by rotating a triangle!

The triangle is a right-angled triangle, and it gets rotated around one of its two short sides.

The side it rotates around is the axis of the cone.

Right vs Oblique Cone

When the apex is aligned on the center of the base it is a Right Cone otherwise it is an Oblique Cone:

Surface Area of a Cone

The Surface Area has two parts:

Which together makes:

Surface Area = π × r × (r + s)
Note: we can calculate
Example: h = 7 and r = 2

Total Surface Area ≈ 12.57 + 45.74 ≈ 58.31

Volume of a Cone

Play with it here. The formula also works when it "leans over" ( oblique) but remember that the height is always at right angles to the base:

Volume of a Cone vs Cylinder

The volume formulas for cones and cylinders are very similar:

So a cone's volume is exactly one third ( ) of a cylinder's volume.

You should order your ice creams in cylinders, not cones, you get 3 times as much!

Like a Pyramid

A cone is also like a pyramid with an infinite number of sides, see Pyramid vs Cone.

Different Shaped Cones

Construction Cone

This is almost a cone, but the top is chopped off (called a "truncated cone").

Also it has a wider base added so it doesn't fall over!

873,874,875,876, 1833, 1834, 3399, 3400, 3401, 3402

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Notice these interesting things: The pointy end of a cone is called the apex The flat part is the base An object shaped like a cone is said to be conical A cone can be made by rotating a triangle! The triangle is a right-angled triangle, and it gets rotated around one of its two short sides.

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Cylinder Facts
Notice these interesting things:

A wooden cylinder

An object shaped like a cylinder is said to be cylindrical.

Oblique Cylinder

When the two ends are directly aligned on each other it is a Right Cylinder otherwise it is an Oblique Cylinder:

Surface Area of a Cylinder

The Surface Area has these parts:

Which together make:

Surface Area = 2 × π × r × (r+h)

Example: h = 7 and r = 2

Try it yourself: cut some paper so it fits around a cylinder, then unwrap and measure it.

It will be h high and (the circumference of a circle) long:

Side area =
2 π rh

Don't forget the two end bits:

Total Surface Area
= 2(πr ²) + 2πrh
= 2πr(r+h)

Volume of a Cylinder

To calculate the volume we multiply the area of the base by the height of the cylinder:

And we get:

How to remember: Volume = pizza

Imagine you just cooked a pizza.

The radius is "z", and the thickness "a" is the same everywhere ... what is the volume?

(we would normally write "pi" as π, and z × z as , but you get the idea!)

Play with it here. The formula also works when it "leans over" ( oblique) but remember that the height is always at right angles to the base:

And this is why:

The stack leans over, but still has the same volume

Volume of a Tank

Learn how to find the volume of a partly filled horizontal cylinder like this one:

Volume of a Cone vs Cylinder

The volume formulas for cylinders and cones are very similar:

So a cone's volume is exactly one third ( ) of a cylinder's volume.

In future, order your ice creams in cylinders, not cones, you get 3 times as much!

Like a Prism

A cylinder is like a prism with an infinite number of sides, see Prism vs Cylinder.

It Doesn't Have to Be Circular

Usually when we say Cylinder we mean a Circular Cylinder, but you can also have Elliptical Cylinders, like this one:

And we can have stranger cylinders!

So long as the cross-section is curved and is the same from one end to the other, then it is a cylinder. But the area and volume calculations will be different than shown above.

More Cylinders

"We eat what we can
And what we can't we can"

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Cylinder Facts Notice these interesting things: A wooden cylinder An object shaped like a cylinder is said to be cylindrical. When the two ends are directly aligned on each other it is a Right Cylinder otherwise it is an Oblique Cylinder: The Surface Area has these parts: Which together make: Surface Area = 2 × π × r × (r+h) Try it yourself: cut some paper so it fits around a cylinder, then unwrap and measure it.

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Sphere Facts
Notice these interesting things:

It is perfectly symmetrical

All points on the surface are the
same distance"r" from the center

It has no edges or
vertices (corners)

It has one surface
(not a "face" as it isn't flat)

Glass Sphere.
Balls and marbles are shaped like spheres.

Largest Volume for Smallest Surface

Of all the shapes, a sphere has the smallest surface area for a volume. Or put another way it can contain the greatest volume for a fixed surface area.

Example: if you blow up a balloon it naturally forms a sphere because it is trying to hold as much air as possible with as small a surface as possible. Press the Play button to see.

In Nature

The sphere appears in nature when a surface wants to be as small as possible. Examples include bubbles and water drops. Can you think of more?

Volume and Surface Area

Example: r = 5

Try dragging point "A":

Also see Volume and Area of a Sphere Calculator

The Earth

The Planet Earth, our home, is nearly a sphere, except that it is squashed a little at the poles.

So it is technically a spheroid, but a very mild version of one.

Here is a more extreme spheroid:

Hemisphere

A hemisphere is an exact half of a sphere.

Other Cool Spheres

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Notice these interesting things: It is perfectly symmetrical All points on the surface are the same distance"r" from the center It has no edges or vertices (corners) It has one surface (not a "face" as it isn't flat) Glass Sphere. Balls and marbles are shaped like spheres.

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We get this amazing thing that the volume of a cone and sphere together make a cylinder (assuming they fit each other perfectly)

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Exactly half of a sphere

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Find the volume of the sphere, then divide it by two.