Surface Area

 

What is Surface Area?

Surface area is the sum of all the areas of the faces of a 3D shape.

It is a 2D concept applied to 3D objects. A face is any flat or curved surface of a shape.

 

Surface Area of Common 3D Shapes

Cube & Cuboid:

$$A_{\text{Total}} = 2(lw + lh +wh)$$

where:

• l = length

• w = width

• h = height

 

Prism:

$$A_{\text{Total}} = 2A + Pl$$

where:

• A = area of cross-section

• P = perimeter of cross-section

• l = length of the prism

 

Cylinder:

where:

• r = radius

• h = height

Cone:

where:

• r = radius

• l = slant height

 

Sphere:

where:

• r = radius

 

Hemisphere:

where:

• r = radius

 

Worked Example: Surface Area of a Composite Shape

Given: A cane is made up of a cylinder (radius 6 cm, height 15 cm) placed on top of a hemisphere (same radius).

 

 

Step 1: Find the Curved Surface Area of the Cylinder and the bottom area

$$A_{cylinder curved} = \pi d h= \pi (12)(15) = 180 \pi \\ A_{bottom} = \pi r^2 = \pi \times 6^2 = 36 \pi$$

 

Step 2: Find the Curved Surface Area of the Hemisphere

$$A_{hemisphere} = \frac{1}{2}(4\pi r^2) = \frac{1}{2}(4 \pi (6)^2) = 72\pi$$

 

Step 3: Find the Total Surface Area

$$A_{Total} = 36\pi + 72\pi = 108\pi$$

 

Step 4: Evaluate and Round to 3 Significant Figures

$$A_{Total} \approx 339.292 \Rightarrow 339 \text{cm}^2$$

Final Answer: $339 \text{cm}^2$ (3 s.f.)