Essential 3D Shapes: Volume, Surface Area & Euler's Formula Explained

in this video i want to talk about some 3d figures that you need to be familiar with

and we're going to go over some equations so you know how to calculate the volume and the surface area of these

figures and we're going to talk about how to distinguish

or how to calculate the number of faces edges and vertices in some 3d shapes that we're going to

consider today and also in how it relates to euler's formula so let's go ahead and begin

the first shape that you need to be familiar with is the sphere it's a three-dimensional

object the circle is two-dimensional and so here's how we can draw the sphere it looks like a circle but

i'm going to draw these lines to indicate that this is a sphere now you need to know that the volume of

a sphere is four thirds pi r cube and the surface area is 4 pi

r squared where r is the radius of the sphere so make sure you write these things down

because you need to know them if you have a test on it now the next figure we need to talk

about is the cylinder and it looks like this this is just a rough sketch

so this is the radius of the cylinder and here we have the height

of the cylinder the volume of a cylinder it's the area of the base where the base

is basically a circle the area of a circle is pi r squared it's the area of the

base times the height so the base is pi r squared and the height is just h

so that's how you can calculate the volume of a cylinder

to calculate the total surface area of a cylinder it's the area of the base

plus the lateral surface area now the area of the base is pi r squared but notice that we have two bases one on

top and the one on the bottom so it's two pi r squared

the lateral area is the product of the circumference of the circle times the height

so this is the distance around the circle is the

circumference and the circumference of a circle is two pi r

so you could use this equation to calculate the total surface area of a cylinder

now the next figure that we need to discuss is the cone

so a cone has a circular base and this is the radius of the cone and here this is the height of the cone

so notice that we have a right triangle so this is r and this is h

now the hypotenuse of this right triangle is known as the slant height

which is l the volume of a cone is one third of the volume of a cylinder

so it's one third pi r squared times the height now the surface area

is the area of the base plus the lateral area just like before we know the area of the

base is the area of the circle and there's only one circle this time not two like

in the last problem so it's one pi r squared as opposed to two pi squared now the lateral area

is the area around the cone and it's one half

the perimeter times the slant height the perimeter of a circle

is two pi r keep in mind the perimeter is basically the circumference

of a circle and the slant and height that's just l so we can cancel one half and two half

of two is one so the surface area of a cone works out to be this equation it's pi r squared

times pi r l now how can we calculate the slant height

the slant height l you could find it based on the right triangle

you can use the pythagorean theorem to get it so for right triangle c squared is equal

to a squared plus b squared and c is the hypotenuse which correlates the slant height

so l squared is equal to r squared plus h squared so you can use this to calculate this

line height and once you have that you can calculate the surface area of

the cone and then you can easily calculate the volume of a cone using that formula

now let's move on to the next shape and this is the rectangular prism so let's draw this first

so let's call this the length the width and the height

the volume of a rectangular prism is simply the product of those three values is the length times the width

times the height now what about the surface area how can we calculate that

well first we need to add up the area of all six faces

so let's start with the face in the front and the one in the back notice that the area of each of those

faces is the width times the height where the height is h

and there's two of them so it's going to be 2wh so that's the area of the front and the

back now let's calculate the area of the sides

so let's say the front side and the back side or the side on the left

so notice that it has the length l and the height h

so the area of those two sides is going to be l h but times two or two lh

and then we need to calculate the area of the top face

and the bottom face so it's the width

times the length and then times two so that's going to be two lw

so this will give you the surface area of a rectangular prism now sometimes you may need to calculate

the length of the diagonal let's call it d if you ever need to do

that here's the formula you need d is equal to the square root of

l squared plus w squared plus h squared and so that's how you can calculate the length of the diagonal of a rectangular

prism now the next shape we need to talk about is the triangular prism

now there's multiple ways in which we can draw this figure so one way we can draw it

is like this the base could be a right triangle it can be an equilateral triangle

so it could be many things so in this example let's say that this is the base this is the height

and this is the height of the prism itself which i'm going to distinguish it from this h

by using capital h now we can also draw a triangle a triangular prism excuse me

like this so this would still be h in this case this is the base

and this would be the height of the triangle now the volume of a triangular prism

is the base times the height capital b is the area of the base and the area of the base can vary

in this case it's simply the area of a triangle for a right triangle

you can calculate the area using this formula it's one half base times height so this would equal capital b

now let's say if you have an equilateral triangle where all three sides are the same the

area is the square root of three over four times

s squared now let's say if you had a triangle where you have two sides an included

angle the area is one half a b

sine of angle c now if you have a triangle like this let's say if you know the base and the

height you could simply use a is equal to one half bh

now sometimes you may have a triangle where you're given all three sides and let's say all three sides are different

if you need to calculate the area of this triangle you need to calculate s first which is one half of the perimeter

of the triangle and then you can use huron's formula to calculate the area it's s times s

minus a times s minus b times s minus c and that result is all within the square root

so that's another way in which you can calculate the area of the triangle so once you find the area

you can replace b with it and then you can calculate the volume of a triangular prism

now sometimes you might see a triangular prism drawn this way and so you're still dealing with the

same type of shape this is the height of the prism this is the base

of the triangle and you could say this is the height of the triangle

now let's go back to this figure how can we calculate the surface area

of a triangular prism now keep in mind the formula can vary this is the base this is the height

and this is the height of the prism the general formula is this the surface area

is the area of the base plus the lateral area now the area of the

base is going to change depending on what type of triangle you have is it a right triangle do you have all three

sides are the three sides different the same and so this can vary but for this

particular triangle i'm just going to give an example the area of the base is going to be

the area of the triangle which is one half base times height now there's two of them we have

the triangle on the left and on the right so we've got to multiply that by two

so for this particular example capital b is two times one-half bh now the lateral area

is going to be the area of the other three sides or three faces rather

let's call the third side of the triangle l so since we have a right triangle

l squared is equal to b squared plus h squared so if we wish to calculate the area

of this face right here the top face it's going to be l times

the height of the prism so that's lh now let's say if we want to calculate

the area of the face in the back which is this face right here it has a height

h and another side length which is lowercase h

so it's going to be h times h and then if we want to calculate the area of the bottom face

it still has a height h but a width of

lowercase b so it's going to be bh so notice that

it's basically the perimeter of the triangle which is the sum of those three sides

h l and b times the height of the prism if you factor out capital h

you could see that p is equal to l plus h plus b so for this particular example the

lateral area is capital h times l plus

h plus b and i'm running out of space so thus we have the surface area is b times h

plus l plus h plus b times the height of the prism

so make sure you know this the lateral area is going to be the perimeter times the height

and so this is the surface area for this particular type of triangular prism

and you can use the general equation for all types of triangular prisms

now there's one more shape that i want to cover and it's the square pyramid

so it's a pyramid with the base of a square so i'm going to draw it like this

so let's call this b and this is going to be b as well that's the length of the base

and here we have the height of the pyramid which i'm going to call h

and this part here that's half of b if this whole thing is b

and this is the center then this part here is uh that's going to be b over 2.

and then this part is the slant height which we can call l

and this is the right triangle now the volume of a square based pyramid

is going to be one-third times the area of the base times the height of the pyramid

now the base is a square and the area of a square is length times width or b times b which is

b squared so capital b is equal to lowercase b squared that's

the area of the base or the area of the square so times the height

so therefore the volume of the square based pyramid is one-third b squared times h

where lowercase b is the length of the square on the bottom

now what about the surface area of a square based pyramid well we can use the general form for the

surface area of any 3d shape it's the area of the base plus the lateral area

now we know the area of the base is b squared but what's the lateral area well notice that there's four faces

of four triangular faces with one square face on the bottom

so i'm going to draw one of those four triangular faces the height of that triangular face is

the slant height which is l and it has a base b now we know that the area of a triangle

is one half base times height the base b is the same but the height is no longer h is the slant height l

now notice that we have four of those triangular faces so this is one of them

here's the second one that's the one in the front then we have the one on the left side

that's the third one and then there's the one in the back the fourth one

so the lateral area is going to be this times four so it's four times one half

bl and four times a half or four divided by two is 2.

so this is the surface area of a squared base pyramid it's b squared plus 2bl where l is the

slant height now how can we calculate the slant height

let's focus on this right triangle and so c squared is equal to a squared

plus b squared according to the pythagorean theorem but c is the slant height

a which we'll call b over two and let's square that and then b will be h

if you need to find a volume you could use that equation if you need to calculate the surface area

first you need to calculate the slant height using the height of the pyramid and the base

of the pyramid once you have the slant height you could plug it into this equation to calculate the surface area

so if you want to combine these two equations into one first we need to solve for l taking the

square root of both sides so then you'll get this equation l is the square root of this becomes b

squared over 4 once you square it plus h squared so i'm going to replace l with this

equation and first let's make some space so you could say the surface area

is b squared plus 2b times the square root

of b squared over 4 plus h squared so this will give you the surface area simply using

the base and the height of the square pyramid so here we have a rough sketch of a cube

and what do you think the volume and the surface area of the cube is going to be

well let's say the side length is x just like a rectangular prism the volume is going to be left times width times

height so it's x times x times x so the volume of a cube it's x cubed

where x is the side length now what about the surface area of a cube

well for rectangular prism we saw was 2l plus 2w plus 2lh

in this case it's going to be 2x squared or like 2 times x times x three times

so the surface area of a cube is six x squared let's make sure you know these formulas

now let's talk about the faces the edges and the vertices of a cube

this picture was a lot better than the last one let's call this a b

c and d and this is going to be e f

g and h so what are some of the faces of the cube how many faces does it have

so notice that we have the face in the front this is face abcd

another one is the one on the side dc

gh that's this one so a face is basically a flat surface

it's two dimensional now how many faces do we have in this cube

so we saw that a b c d was one face that was the one in the front we also have e f g h in the back

that's two on the right we have c d h g that's three

on the left a b f e that's four the bottom a d h e

that's five and the one on the top b c g f

that's six so let's say that f is equal to six now what about the number of edges

so what are some edges in this cube segment a b is an edge

so we can represent it like this bc is also an edge

and cd that's an edge as well so let's calculate the total number of

edges so a b is one b c is two

c d is three a d is four and then we have a e which is five

e h is six d h is seven g eight cg is nine

fg is ten bf is eleven and in the back we have ef which is uh twelve so a cube

has twelve edges now what about the vertices how many vertices does it have

so in a cube you can identify a vertex where three edges meet

so a is a vertex so it's one of the vertices so this is one b is another one that's two three four

five six seven eight

so there's eight vertices in a cube now euless formula tells us that the sum

of the faces and the vertices minus the number of edges is equal to two

so in this case we have six faces eight vertices and twelve edges six plus eight is

fourteen fourteen minus twelve does equal two

and so this is in harmony with euless formula for figures like these

that are just made up of faces now let's apply it for a triangular prism

so let's call this a b c

d e and f so what are some faces

in this uh problem so notice that abc is one face of the triangle

another face that we have is d e f and also we could say

b e f c is the face of this triangular prism

now let's calculate the total number of faces so the first one as you said was abc

the second one d e f the third

b e f c now a fourth one is the one on the bottom a d

f c and then the one on the back left a b e d

so that's the fifth one there's a total of five faces this figure

now what about edges let's name some edges so a b is an edge

we could say that b e is an edge and d f is edge so we can write them

this way now let's calculate the total number of edges so this is one

two let me write them so this is a b is one b c is two a c is

three d e is four e f is five

d f is six b e is seven a d is eight c f is nine

so i calculated a total of nine edges now the vertices we just gotta count the letters

because that's where three edges will meet so this is one

two three four

five six so i counted six vertices

now let's use euler's formula to confirm that this is equal to two so the faces plus the vertices

minus the edge lengths has to equal two so we have five faces six vertices and nine edges

five plus six is eleven eleven minus nine is 2. so this is in harmony with euler's

formula now let's go over one more example and so that is

the square based pyramid so let's call this a b

c d and e so let's calculate the number of faces

so we have face abcd that's the base of the pyramid another face is aeb

that's 2 and then ebc is another one so that's stream

and then e c d that's four and then the one in back e a d

that's five so there are five faces in this figure now the number of

vertices are the letters a b c d e

so there's five vertices now how many edges are there so we have edge a b

that's one a e that's another one

b e that's number three let's use a different color bc is four e c is five

d c is six a d is seven and then e d is eight

so i got a total of eight edges so the faces

and the vertices minus the edges has to equal two so f is five

v is five and e is eight five plus five is ten ten minus eight is two

so this is another figure that works in harmony with you less formula and so that's it for this video

hopefully gave you a good introduction into 3d shapes such as spheres cylinders cones

rectangular prisms square based pyramids and things like that so thanks for watching

you