Lesson Solved problems on surface area of cones

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Solved problems on surface area of cones


In this lesson you will find typical solved problems on surface area of cones.
The theoretical base for these problems is the lesson  Surface area of cones  under the topic  Area and surface area  of the section  Geometry  in this site.

Problem 1

Find the lateral surface area of a cone if the base radius of the cone is of  10 cm  and the height of the cone is of  5 cm.
Then find the total surface area of the cylinder.

Solution

First,  find the slant height of the cone.  It is

l = sqrt%28r%5E2+%2B+h%5E2%29 = sqrt%2810%5E2+%2B+5%5E2%29 = sqrt%28125%29 = 5sqrt%285%29.

The lateral surface area of the cone equals  pi  times the product of the base radius and the slant height

S%5Blateral%5D = pirl = 3.14159*10*5sqrt%285%29 = 351.24 cm%5E2 (approximately).

The area of the base is
S%5Bbase%5D = pir%5E2 = 3.14159*100 = 314.159 cm%5E2 (approximately).

So,  the total surface area of the cone is  S%5Blateral%5D + S%5Bbase%5D = 351.24 + 314.159 = 665.40 cm%5E2 (approximately).

Answer.  The lateral surface area of the cone is  351.24 cm%5E2 (approximately).
               The total surface area of the cone is  665.40 cm%5E2 (approximately).


Problem 2

Find the lateral surface area of a cone if the triangular axial section of the cone  (Figure 2)  has the area  a = 10 cm%5E2.

Solution

The lateral surface area of the cone equals pi times the product of the radius of                
the cone at the base and the height of the cone
S%5Blateral%5D = pirh.

The area of the triangle at the axial section is
S%5Bsection%5D = 1%2F2.2r.h = rh.

It implies that the lateral surface area of the cylinder is  pi  times the area
of the triangle at the axial section
S%5Blateral%5D = pi%2AS%5Bsection%5D.

Hence,  S%5Blateral%5D = pi%2Aa = 3.14159*10 = 31.4159 cm%5E2 (approximately).



Figure 2.  To the  Problem 2

Answer.  The lateral surface area of the given cone is  31.4159 cm%5E2 (approximately).


Problem 3

Find the surface area of a combined solid body which comprises of two identical cones joined base to base  (Figure 3),  if their common base radius is of  4 cm  and the height
of each cone is of  3 cm.

Solution

We are given a 3D body comprised of two identical cones whose bases are joined and                          
overposed each to the other  (Figure 3).

First,  find the slant height of the cone.  It is

l = sqrt%28r%5E2+%2B+h%5E2%29 = sqrt%284%5E2+%2B+3%5E2%29 = sqrt%2825%29 = 5.

The lateral surface area of the cone equals  pi  times the product of the base
radius and the slant height

S%5Blateral%5D = pirl = 3.14159*4*5 = 62.832 cm%5E2 (approximately).

The total surface area of the combined solid body is doubled the lateral area
of the single cone,  i.e.

S = 2*62.832 = 125.664 cm%5E2 (approximately).



Figure 3. To the Problem 3

Answer.  The surface area of the combined body is  125.664 cm%5E2 (approximately).


Problem 4

Find the lateral surface area of a body  (a truncated cone)  obtained from a cone with the base radius of  6 cm  and the height of  8 cm  after cutting off the part of the cone by the plane parallel to the base in a way that the cutting plane bisects the height of the original cone  (Figure 4).

Solution

The strategy solving this problem is to find first the lateral surface area of the                                
entire cone with the base radius of  6 cm  and the height of  8 cm  and then
to distract the lateral surface area of the cone with the base radius of  3 cm
and the height of  4 cm.

The lateral surface area of the original entire cone is   pi%2Ar%2Al,  where  r= 6 cm
is the cone base radius and  l  is the cone slant height.  The slant height is
l = sqrt%28r%5E2+%2B+h%5E2%29 = sqrt%286%5E2+%2B+8%5E2%29 = sqrt%28100%29 = 10,
therefore the lateral area of the original cone is  3.14159*6*10 = 188.495cm%5E2.

The small cone has the base radius of  3 cm  and the slant height of  1%2F210 = 5 cm.
Therefore,  the lateral surface area of the small cone is  3.14159*3*5 = 47.124cm%5E2.



    Figure 4. To the  Problem 4

Thus the lateral surface area of the truncated cone under consideration is  188.495 - 47.124 = 141.372 cm%5E2 (approximately).

Answer.  The lateral surface area of body under consideration is  141.372 cm%5E2 (approximately).


My lessons on surface area of cones and other 3D solid bodies in this site are

Lessons on surface area of prisms

Surface area of prisms
Solved problems on surface area of prisms
Overview of lessons on surface area of prisms                  

Lessons on surface area of pyramids

Surface area of pyramids
Solved problems on surface area of pyramids
Overview of lessons on surface area of pyramids

Lessons on surface area of cylinders

Surface area of cylinders
Solved problems on surface area of cylinders
Overview of lessons on surface area of cylinders              

Lessons on surface area of cones

Surface area of cones
Solved problems on surface area of cones
Overview of lessons on surface area of cones                

Lessons on surface area of spheres

Surface area of spheres
Solved problems on surface area of spheres
Overview of lessons on surface area of spheres


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