Question

How do you find the surface area of a composite figure?

Answer 1

A composite figure is a figure made up of simple shapes. To find the total surface area of a composite figure, you break up the composite figure into simple shapes whose surface areas you can find using existing formulas. Then you add all the areas of the simple shapes to find the total surface area of the composite figure.

The figure in this problem is a cylinder with a cone on top. The visible areas are: 1) The lower base of the cylinder; 2) The lateral area of the cylinder (the vertical wall all around the cylinder); 3) The lateral area of the cone.

The base of the cone and top base of the cylinder are not visible because they are interior to the composite figure. We only have the three surfaces described above to calculate and add together.

1) The lower base of the cylinder is a circle. We use the formula for the area of a circle using a radius of 4 cm.

$A_{circle} = \pi r^2 = \pi \times (4~cm)^2 = 16 \pi~cm^2$

2) The lateral area of the cylinder is the area of a rectangle whose length is the circumference of the base and whose width is the height of the cylinder.

$A_{lateral~rectanlge} = LW = 2 \pi r \times h = 2 \pi (4~cm)(8~cm) = 64\pi~cm^2$

3) The lateral area of the cone is given by the formula:

$A_{lateral~cone} = \pi r \sqrt{r^2 + h^2}$

$A_{lateral~cone} = \pi \times (4~cm)\sqrt{(4~cm)^2 + (5~cm)^2}$

$A_{lateral~cone} = 4 \pi \sqrt{41} ~cm^2$

To find the total surface area, we add the three surface areas we found above.

$A = 16 \pi~cm^2 + 64\pi~cm^2 + 4 \pi \sqrt{41} ~cm^2$

The exact area is:

$A = (80 + 4 \sqrt{41})\pi ~cm^2$

The approximate area in terms of pi is:

$A = 105.6 \pi~cm^2$