Question

if a pyramid has a rectangular base having side 32 cm and 20 cm .. the height of the pyramid us 12 cm... find the slant height and curved surface area​

Answer 1

Step-by-step explanation:

since the pyramid is having Rectangular base therefore

Since slant height acts as hypotenuse

Using Pythagoras theorem

(Slant Height )^2 = (Height)^2 + (Width/2)^2

Slant Height= (12*12 + 10*10)^1/2

= (144+100)^1/2

= square root(244)cm

=15.62 cm

Total Surface area is the area of the rectangular base (length* width) + areas of each of the four triangular faces.

Curved Surface Area= Area of all four triangles

Area of Front and Back Triangles

The area of a triangle is found through the formula A=1/2("base")("height")

Here, the base is length. To find the height of the triangle, we must find the slant height on that side of the triangle.

The slant height can be found through solving for the hypotenuse of a right triangle on the interior of the pyramid.

The two bases of the triangle will be the height of the pyramid, #h#, and one half the width, #w/2#. Through the Pythagorean theorem, we can see that the slant height

=sqrt(h^2+(w/2)^2)

= sqrt(12*12+ 10*10)= sqrt(244)

This is the height of the triangular face. Thus, the area of front triangle is

1/2lsqrt(h^2+(w/2)^2).

Since the back triangle is congruent to the front, their combined area is twice the previous expression, or

= lsqrt(h^2+(w/2)^2)= 32*sqrt(244)

Area of the Side Triangles

The side triangles' area can be found in a way very similar to that of the front and back triangles, except for that their slant height is = sqrt(h^2+(l/2)^2)= sqrt(12*12+ 16*16)

= sqrt(144+256)= sqrt(400)= 20

Thus, the area of one of the triangles is

= 1/2 * Width*sqrt(h^2+(l/2)^2)

and both the triangles combined is

= wsqrt(h^2+(l/2)^2)= 20*20= 400cm^2

Curved Surface area=

400+ 32*sqrt(244)= 400+ 500=

900 cm^2