How to find the volume of a cylinder

Although a cylinder is technically not a prism, it shares many of the properties of a prism. Like prisms, the volume is found by multiplying the area of one end of the cylinder (base) by its height.

Since the end (base) of a cylinder is a circle, the area of that circle is given by the formula :

Multiplying by the height h we get

where:π  is Pi, approximately 3.142r  is the radius of the circular end of the cylinderh  height of the cylinder

Volume of a sphereDefinition: The number of cubic units that will exactly fill a sphere .

Try this Drag the orange dot to adjust the radius of the sphere and note

how the volume changes.

The volume enclosed by a sphere is given by the formula

Where r is the radius of the sphere. In the figure above, drag the orange dot to change the radius of the sphere and note how the formula is used to calculate the volume. Since the 4, 3 and pi

are constants, this simplifies to approximately

This formula was discovered over two thousand years ago by the Greek philosopher Archimedes. He also realized that the volume of a sphere is exactly two thirds the volume of its circumscribed cylinder, which is the smallest cylinder that can contain the sphere.

Volume of a pyramidDefinition: The number of cubic units that will exactly fill a pyramid.

Try this Drag the orange dots to adjust the base size and height of the pyramid

and note how the volume changes.

The volume enclosed by a pyramid is one third of the base area times the

perpendicular height. As a formula: Where b is the area of the base of the pyramid and h is its height. The height must be measured as the vertical distance from the apex down to the base.

Volume of a coneDefinition: The number of cubic units that will exactly fill a cone .

Try this Drag the orange dots to adjust the radius and height of the cone and note how the volume changes.

The volume enclosed by a cone is given by the formula Where r is the radius of the circular base of the cone and h is its height. In the figure above, drag the orange dots to change the radius and height of the cone and note how the formula is used to calculate the volume .

Relation to a cylinder

Recall that the volume of a cylinder is

If you compare the two formulae, you will see one is exactly a third of the other. This means that the volume of a cone is exactly one third the volume of the cylinder with the same radius and height.

Such a cylinder is the "circumscribed cylinder" of the cone - the smallest cylinder that can contain the cone. In the

figure above, select "Show cylinder" to see the cone embedded in its circumscribed cylinder.

How to find the volume of a triangular prism

Recall that a prism has two congruent, parallel faces called the bases of the prism. The volume of any prism can be found by multiplying the area of one of the bases by its height. In the case of a triangular prism, each base is a triangle.

As a formula where a is the area of one triangular end face, and h is the height.

There are various ways to find the area of the triangle, use whichever method work with what you are given. In the above animation, the three sides are given, so here you would use Heron's Formula. But any method will do - below is a list of methods:

Methods for finding the triangle area

If you know: Use this

Base and altitude "Half base times height" method

All 3 sides Heron's Formula

Two sides and included angle Side-angle-side method

x,y coordinates of the vertices Area of a triangle- by formula (Coordinate Geometry) Area of a triangle - box method (Coordinate Geometry)

The triangle is equilateral Area of an equilateral triangle

Prism FormulaAn prism is a polyhedron with 2 polygonal bases parallel to each other. The two polygonal bases are joined by lateral faces.

The number of lateral faces are equal to the number of sides in the base. The lateral faces in a prism are perpendicular to the polygonal bases. Mostly the lateral faces are rectangle. In some cases the faces may be parallelogram.

The Prism Formula in general is given as,

urface area of a pyramidDefinition: The number of square units that will exactly cover the surface of a pyramid.

Try this Drag the orange dots to adjust the base size and height of the pyramid and note how the area changes.

The total surface area of any polyhedron, is sum of the surface areas of each face. In the case of a right pyramid, the side faces are all the same, so we can simply find the area of one and multiply by the number of faces. Once we add the area of the base, we have the total surface area.

The base

In the figure above, the base is a square. So to find its area we multiply the side length by itself. The base however can be any polygon. To find the area of a polygon see Area of a regular polygon.

In the figure above, click on 'reset'. The base side length is 10, so since the base is a square in this example, the base area is 102 or 100.

The sides

The sides of a pyramid are triangles. There are various ways to find the area of triangles (see Area of triangles.) We find the area of one face, then multiply by the number of faces.

In the figure above press 'reset'. We see from the front face that the base of the triangle is 10. We are also given the height* of the triangle - 11.    Recall that the area of a triangle is half the base times height, so each face has an area of 55. (half of 11 times 10). The total for the four faces is 220. (4 times 55).

*This is also called the "slant height" of the pyramid - to distinguish it from the perpendicular height.

Total AreaSo the total surface area of the above pyramid is

Area of the base 100

Area of the four faces = 4 times 55 220

TOTAL 320

As a formula

Since the base of a pyramid can be any polygon, and you may be given various different measurements, it's best to follow the method above to find the area. But in the particular case of a right square pyramid with the base side and

slant height given, the area is given by the formula Where b is the side length of the base, and h is the slant height.

By combining the 4 and the 2, this simplifies a little to

Derivation of the surface area of a cylinderSee also: Surface area of a cylinder

Try this Drag the orange dot to the left to "unroll" the cylinder.

The surface area of a cylinder can be found by breaking it down into three parts:

The two circles that make up the ends of the cylinder. The side of the cylinder, which when "unrolled" is a rectangle

In the figure above, drag the orange dot to the left as far as it will go. You can see that the cylinder is made up of two circular disks and a rectangle that is like the label unrolled off a soup can.

The area of each end disk can be found from the radius r of the circle. The area of a circle is πr2, so the combined area of the two disks is twice that, or2πr2. (See Area of a circle).

The area of the rectangle is the width times height. The width is the height h of the cylinder, and the length is the distance around the end circles. This is the circumference of the circle and is 2πr. Thus the rectangle's area is 2πr × h.

Combining these parts we get the final formula:where:π  is Pi, approximately 3.142r  is the radius of the cylinderh  height of the cylinder

Calculator

By factoring 2πr from each term we can simplify this to

Surface area of a right prism

b= area of a basep= perimeter of a baseh= height of the prism

Try this Change the height and dimensions of the triangular prism by dragging the orange

dots . Note how the surface area is calculated.

A right prism is composed of a set of flat surfaces .

The two base are congruent polygons. The lateral faces (or sides) are rectangles.

The total surface area is the sum of these.

Bases

Each base is a polygon. In the figure above it is a regular pentagon, but it can be any regular or irregular polygon. To find the area of the base polygons, see Area of a regular polygon and Area of an irregular polygon. Since there are two bases, this is doubled and accounts for the "2b" term in the equation above.

Lateral faces

Each lateral face (side) of a right prism is a rectangle. One side is the height of the

prism, the other the length of that side of the base .

Therefore, the front left face of the prism above is its height times width or

The total area of the faces is therefore If we

factor out the 'h' term from the expression we get

Note that the expression in the parentheses is the perimeter (p) of the base, hence we can write

the final area formula as

Regular prisms

If the prism is regular, the bases are regular polygons. and so the perimeter is 'ns' where s is the side length and n is the number of sides. In this case the surface area formula simplifies to

b= area of a base

n= number of sides of a base

s= length of sides of a base

h= height of the prism

The surface area of a cylinder can be found by breaking it down into three parts:

The two circles that make up the ends of the cylinder. The side of the cylinder, which when "unrolled" is a rectangle

In the figure above, drag the orange dot to the left as far as it will go. You can see that the cylinder is made up of two circular disks and a rectangle that is like the label unrolled off

a soup can.

The area of each end disk can be found from the radius r of the circle. The area of a circle is πr2, so the combined area of the two disks is twice that, or2πr2. (See Area of a circle).

The area of the rectangle is the width times height. The width is the height h of the cylinder, and the length is the distance around the end circles. This is the circumference of the circle and is 2πr. Thus the rectangle's area is 2πr × h.

Combining these parts we get the final formula:where:π is Pi, approximately 3.142r is the radius of the cylinderh height of the cylinder

Calculator

By factoring 2πr from each term we can simplify this to

Recall that a cone can be broken down into two parts - the top part with slanted sides, and the circular disc making the base. We can find the total surface area by adding these together.

The base is a circle of radius r. The area of as circle is given by For more, see Area of a circle.

The top section has an area given by where r is the radius at the base, and s is the slant height. See also Derivation of cone area.

The slant height is the distance along the cone surface from the top to the bottom rim. If you are given the perpendicular height, you can find the slant height using the Pythagorean Theorem. For more see Slant height of a cone.

By adding these together we get the final formula: This can be simplified by combining some terms, but we usually keep it this way because sometimes we want the area of each piece separately.

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