Surface Area of Cones Examples

Find the lateral and surface area of the cone.

Lateral and surface areas of cones? There's an app for that. And if there isn't an app, there's a formula.

L = πrl

Our radius is 4 cm and our slant height is 5 cm. Plug those bad boys in.

L = π × 4 cm × 5 cm
L = 62.8 cm²

That was easy enough. Surface area is just code for "add in the area of the circular base." For cones it is, anyway.

SA = L + πr²
SA = 62.8 cm² + π × (4 cm)²
SA = 113.1 cm²

We have a soft spot in our hearts when it comes to cones. Well, soft serve, anyway.

Find the surface area of the solid.

This shape is a little weird. Before we calculate anything, we should come up with an overall surface area formula. The total surface area is the lateral area of a cylinder, and the two lateral areas of the cones.

SA = L_Cylinder + 2L_Cone

The lateral area of a cylinder is the circumference of the circle times the length of the cylinder. Circumference is two times the radius times our favorite dessert.

L_Cylinder = 2πrh
L_Cylinder = 2π(0.5 ft)(4 ft)
L_Cylinder = 12.57 ft²

The lateral area of the cone has its own formula too.

L_Cone = πrl

We don't know the slant height yet, but we can figure it out. Pythagoras knows what's up.

a² + b² = c²
(0.5 ft)² + (2 ft)² = c²
c = l ≈ 2.06 ft

Now we're ready to calculate the lateral area of the cone.

L_Cone = πrl
L_Cone = π(0.5 ft)(2.06 ft)
L_Cone = 3.24 ft²

One final step remains.

SA = L_Cylinder + 2L_Cone
SA = (12.57 ft²) + 2(3.24 ft²)
SA ≈ 19 ft²

Hurrah. We've succeeded.