What is the surface area of a cone that has a slant height of 18.5 inches and a radius of 11 inches?

Answer: The answer is C. 324.5

Step-by-step explanation:

correct on edge

Where,

r is the radius

h is the height

l is the slant height

The area of the curved (lateral) surface of a cone = πrl

Note:

A cone does not have uniform (or congruent) cross-sections. (more about conic section here)

Example 1: A cone has a radius of 3cm and height of 5cm, find total surface area of the cone.

Solution:

To begin with we need to find slant height of the cone, which is determined by using Pythagoras, since the cross section is a right triangle.

l2 = h2 + r2

l2 = 52 + 32

l2 = 25 + 9

l = √ (34)

l = 5.83 cm

And the total surface area of the cone is:

SA = πr2 + πrl

SA = π · r · (r + l)

SA = π · 3 · (3 + 5.83)

SA = 83.17 cm2

Therefore, the total surface area of the cone is 83.17cm2

Example 2: The total surface area of a cone is 375 square inches. If its slant height is four times the radius, then what is the base diameter of the cone? Use π = 3.

Solution:

The total surface area of a cone = πrl + πr2 = 375 inch2

Slant height: l = 4 * radius = 4r

Substitute l = 4r and π = 3

3 * r * 4 r + 3 * r2 = 375

12r2 + 3r2 = 375

15r2 = 375

r2 = 25

r = 25

r = 5

So the base radius of the cone is 5 inch.

And the base diameter of the cone = 2 * radius = 2 * 5 = 10 inch.

Example 3: What is the total surface area of a cone if its radius = 4cm and height = 3 cm.

Solution:

As mentioned earlier the formula for the surface area of a cone is given by:

SA = πr2 + πrl

SA = πr (r + l)

As in the previous example the slant can be determined using Pythagoras:

l2 = h2 + r2

l2 = 32 + 42

l2 = 9 + 16

l = 5

Insert l = 5 we will get:

SA = πr (r + l)

SA = 3.14 · 4 · (4+5)

SA = 113.04 cm2

Example 4: The slant height of a cone is 20cm. the diameter of the base is 15cm. Find the curved surface area of cone.

Solution:

Given that,

Slant height: l = 20cm

Diameter: d = 15cm

Radius: r = d/2 = 15/2 = 7.5cm

Curved surface area = πrl

CSA = πrl

CSA = π · 7.5 · 20

CSA = 471.24 cm2

Example 5: Height and radius of the cone is 5 yard and 7 yard. Find the lateral surface area of the given cone.

Solution:

Lateral surface area of the cone = πrl

Step 1:

Slant height of the cone:

l2 = h2 + r2

l2 = 72 + 52

l2 = 49 + 25

l = 8.6

Step 2: Lateral surface area:

LSA = πrl

LSA = 3.14 * 7 * 8.6

LSA = 189.03 yd2

So, the lateral surface area of the cone = 189.03 squared yard.

Example 6: A circular cone is 15 inches high and the radius of the base is 20 inches What is the lateral surface area of the cone?

Solution:

The lateral surface area of cone is given by:

LSA = π * r * l

LSA = 3.14 * 20 * 15

LSA = 942 inch2

Example 7: Find the total surface area of a cone, whose base radius is 3 cm and the perpendicular height is 4 cm.

Solution:

Given that:

r = 3 cm

h = 4 cm

To find the total surface area of the cone, we need slant height of the cone, instead the perpendicular height.

The slant height l can be found by using Pythagoras theorem.

l2 = h2 + r2

l2 = 32 + 42

l2 = 9 + 16

l = 5

The total surface area of the cone is therefore:

SA = πr (r + l)

SA = 3.14 · 3 · (3+5)

SA = 75.36 cm2