how do u find an area and a volume of a pentagon?

Question

i know u dont like to answer these questions but i m stuck on my math project. i know area is width times length but on pentagon, how does it work? and volume is height times length times width but how is it on pentagon?

Answer 1

A pentagon is a 2D shape and so has 0 volume.

To work out the area, imagine a line drawn from each corner to the centre, splitting it into 5 triangles. Work out the area of one of the triangles, and multiply by 5

Answer 2

The first two answerers are right about volume. And the only way I can answer the question about area is to assume you mean a regular pentagon, with all sides and angles equal.

If you join two of the (non-adjacent) vertices, you divide the pentagon into an isosceles triangle, with vertical angle 108 deg and base angels 36 deg, and an isoscles trapezium with angles 108 and 72.

The area of the triangle is 0.5*sin 72deg * a^2 (where a is the length of the side of the pentagon), and the area of the trapezium is (sin 72deg)*(1+cos 72deg)*(a^2). At least, I think that's right. You can check it by dropping perpendiculars between the ends of the parallel sides of the trapezium, dividing it into a rectangle with a rightangled triangle at each end, whose sides are a*sin72, a*cos72, and a.

You could invent some other pentagons which are not regular. e.g. if you start with a 3,4,5 rightangled triangle, and on the hypotenuse draw a rectangle with length 5 and width 3, you have formed a pentagon whose sides are 3,4,3,4,3 and whose angles are 90, 127, 90, 90, 143 deg, and you can work out the area by adding the area of the triangle and the rectangle.

Answer 3

Regular pentagon: Sides s

Draw circumscribing circle

Join the 5 vertices to the centre of the circle so that there are 5 isosceles triangles each with two sides equal to the radius of the circle and the third side (being the side of the pentagon equal to s)

So angle at vertex = 360°/5 = 72°

Area of triangle = ½absinC
= ½r²sin72°

Now draw an altitude for one of the triangles So

sin36° = s/2r

ie r = s/2sin36°

Therefore Area of triangle = ½(s/2sin36°)²sin72°
= ½ s²sin72°/(4sin²36°)

= ½ s²sin72°/(2(1 - cos72°)

= ¼ s²sin72°/(1 - cos72°)

So the area of the pentagon = 5 * area of triangle

= 5/4 s²sin72°/(1 - cos72°)

If you wish to generalise

Area of a regular n-gon = n/4 s²sin(360°/n)/(1 - cos(360°)/n)

Note: when n = 4 this reduces to
Area of a square = 4/4 s² sin(360°/4)/(1 - cos(360°)/4)
= 1 * s² * sin90°/( 1 - cos90°)

= s² * 1/( 1 - 0)

= s²

when n = 3 this reduces to

Area of equilateral triangle = 3/4 s²sin(360°/3)/(1 - cos(360°/3))

= 3/4 s² sin120°/ (1 - cos120°)

= 3/4 s² √3/2 /(1 - -½)

= 3/4 s² √3/2 *2/3

= √3/4 s²

Volumes of Platonic Solids with sides s

Tetrahedron V = (√2/12)s³
Cube: V = s³
Octahedron: V = (√2/3)s³
Dodecahedron: V = ¼(15 + 7√5)s³
Icosahedron: V = (1/12) (15 + 5√5)s³

AND Surface areas of Platonic solids:

Tetrahedron V = √3s²
Cube: V = s²
Octahedron: V = 2√3s²
Dodecahedron: V = 3√(25 + 10√5)s²
Icosahedron: V = 5√3s²

As you can see, these are pretty yukky formulae

Answer 4

A pentagon is not 3dimensional and has no volume.

To calculate the area of a regular pentagon(or any regular n-sided polygon), there are 3 different formulas you can use(depending on what you given)
If you know the length of one side(a)
OR
the distance from the middle of the the polygon to a corner(r)
OR
the distance from the middle of the the polygon to middle of an edge(R)

Area = n*a^2*/(4*tan(180/n) )
for a pentagon 5*a^2/(4*tan(180/5) = a^2 *1.25/tan(36) approx 1.720477401*a^2

OR
Area = n*r^2*tan(180/n)
for a pentagon 5*r^2*tan(180/5) = 5*r^2*tan(36) approx 3.63271264*r^2

OR
Area = n*R^2*sin(360/n)/2
for a pentagon 5*R^2*sin(360/5)/2 = 5*R^2*sin(72)/2 approx 2.377641291*R^2

NOTE-- If r =1 or R =1 as n gets larger the area will get closer to being pi

Answer 5

You need to pay particular attention to whether this is a regular pentagon (a pentagon with sides of all equal length and all equal angles) you're speaking of; otherwise, all interior triangles may have different areas.

The dodecahedron is a very interesting three-dimensional platonic solid with twelve pentagonal faces. The volume of this is 1/4 * (15 + 7sqrt(5)) * a^3 (where 'a' is the length of a given edge), though the derivation of which might be beyond the scope of your project.

Answer 6

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Answer 7

pentagon is a plane figure. it will not have any volume
for area calculate the area of 1 triangle and multiply by 5

Answer 8

would you like that in sq ft /sq yrds or sq in' and the salotion is ligth times width times height divided by 27 this is the sq yard formula because there is 27 sqft in a sq yard think of it as a rubics cube an d turn that in to sqft or down to sq in'using that magic number 27

Answer 9

area pentagon= you must break the pentagon to be triangle and trapezium. and then you calculate area triangle and trapezium . then you add both. you can get area of trapezium.......

volume of pentagon....= you multiply area pentagon with height...


i hope you can do it.......( ", )

Answer 10

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