Surface Area

The Surface Area of a 3D shape is the sum of the areas of each face. Surface area is also measured in square units such as square centimeters ( $\text{cm}^2$ ), square inches ( $\text{in}^2$ ) etc.

Surface Area of Common Shapes

Let's look at how to calculate the surface area of the shapes below:

Cube

The surface area of a cube with side length $s$ is calculated as:

$SA = 6 s^2$

Cuboid

The surface area of a cuboid (rectangular prism) with length $l$, width $w$ and height $h$ is calculated as:

$SA = 2 (l w + l h + w h)$

Cone

The surface area of a cone with height $h$, slant $l$, and radius $r$ is calculated as:

$SA = \pi r (l + r)$

Cylinder

The surface area of a cylinder with radius $r$ and height $h$ is calculated as:

$SA = 2 \pi r (h + r)$

Triangular Prism

The surface area of a triangular prism with two equal side lengths $s$, base $b$, height $h$ and length $l$ is calculated as:

$SA = bh + 2ls + lb$

Sphere

The surface area of a sphere with radius $r$ is calculated as:

$SA = 4 \pi r^2 $

Surface Area Using Nets

The net of an 3D shape is formed when the shape is unfolded along its edges and its faces are laid out in a pattern in 2D. Nets are helpful to visualize the different faces of a 3D object to calculate the surface area.

Let's draw a net for the shape below.

Cuboid with a height of 6cm, length of 3cm, and width of 2cm.

We have to unfold the 3D object to get the net shape below:

Net shape of a cuboid with height of 6cm, length of 3cm, and width of 2cm.

You can see that the original 3D shape is made up of $6$ rectangles. We can calculate the area of each using $A = lw$. Lastly, we add them all up to get the surface area.

$SA = (2)(6) + (3)(6) + (2)(6) + (3)(6) + (3)(2) + (3)(2) $

$SA = 2 ( (2)(6) + (3)(6) + (3)(2) )$

$SA = 2 (12 + 18 + 6)$

$SA = 72 \; [\text{cm}^2]$

Therefore, we can determine that the surface area is $\boldsymbol{72 \; [\textbf{cm}^2]}$.

A sphere has a radius $6$ inches. What is its surface area?

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In order to determine the surface area of a sphere, we can plug its radius into its corresponding formula and solve:

$SA = 4 \pi r^2 $

$SA = 4 \pi (6)^2 $

$SA = 452.39 \; [\text{in}^2]$

Therefore, we can determine that the surface area of the sphere is $\boldsymbol{452.39 \; [\textbf{in}^2]}$.

The lateral area of a cone with radius $3\; [\text{cm}]$ is $72\; [\text{cm}^2]$. Determine the slant height of the cone, to the nearest centimetre.

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In order to determine the slant height of the cone, we can first substitute the appropriate values into its corresponding surface area formula:

$SA = \pi r (l + r)$

$72 = \pi \cdot 3 (l + 3)$

Next, we can expand the equation and rearrange it:

$72 = \pi (3l + 9)$

$72 = 9.425l + 28.275$

$9.425l = 72 - 28.275$

We can now simplify:

$9.425l = 43.725$

$\cfrac{\cancel{9.425}l}{\cancel{9.425}} = \cfrac{43.725}{9.425}$

$l \approx 5\; [\text{cm}]$

Therefore, we can determine that the slant height of the cone, to the nearest centimetre, is $\boldsymbol{5\; [\textbf{cm}]}$.

Juan is making a yard sign in the shape of a rectangular prism. The yard sign will be $60 \; [\text{cm}]$ long, $40 \; [{\text{cm}}]$ high and $5 \; [\text{cm}]$ thick. The cost of painting the yard sign is $$0.002$ per square $\text{cm}$. How much does Juan have to spend to paint the yard sign?

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First, we can determine the surface area of the sign. The surface area of a rectangular prism (which the sign represents) is calculated by adding the area of all $6$ sides of prism. We can plug in the appropriate values to solve:

$SA = 2 (l w + l h + w h)$

$SA = 2 ((60)(5) + (60)(40) + (5)(40)) $

$SA = 2 (300 + 2400 + 200)$

$SA = 2 (2900)$

$SA = 5800 \; [\text{cm}^2]$

Next, we can calculate the cost of painting by multiplying the surface area by the rate since Juan will need to paint all sides of the sign.

$C = SA \cdot r$

$C = 5800 \; \cancel{\text{cm}^2} \cdot \cfrac{$0.002}{\cancel{\text{cm}^2}}$

$C = $11.60$

Therefore, we can determine that it will cost $\boldsymbol{$11.60}$ to paint the sign.

Draw a net for the shape below and calculate the surface area.

Cylinder with a diameter of 4cm and a height of 10cm.

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First, we can draw the net:

A net of a cylinder with a diameter of 4cm and a height of 10cm.

The top and bottom of the cylinder are circles. The middle portion will unfold to a rectangle. The width of the rectanle is the circiumference of the circle! Recall the circiumference of a circle is $\pi d = 2 \pi r$.

The Surface Area of a cylinder consists of $2$ circles plus a rectangle:

$SA = 2 \pi r^2 + 2 \pi r h$

Next, we can determine the circle's radius, which is half its diameter:

$r = \cfrac{d}{2} = \cfrac{4}{2} = 2\; [\text{cm}]$

Then, we can use the formula below (which is a simplifed version), to determine the surface area of the cylinder:

$SA = 2 \pi r (h + r)$

$SA = 2 \pi 2(10 + 2)$

$SA = 4 \pi (12)$

$SA = 150.80 \; [\text{cm}^2]$

Therefore, we can determine that the surface area of the cylinder is $\boldsymbol{150.80 \; [\textbf{cm}^2]}$.

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Sufrace Area