Surface Area of Geometric Solids

Surface Area of Geometric Solids

By Keith Webb

​Surface Area of Prisms and Cylinders

​is defined as the sum of the area measures of the faces of a geometric solid.

​General Formulas for Surface Area of a Prism or Cylinder

​SA = 2B + PH​ or SA = 2B + PL

B is the area of a Base of the prism ​

P is the Perimeter of the Base

H/L is the height/length of the prism​

Multiple Select

Which of the following could you use to calculate the surface area for prisms or cylinders. Select all that apply.

• SA = 2B + PB

• SA = SB + PH

• SA = 2B + LH

• SA = 2B + PL

• SA = 2B + BH

​Rectangular Prisms

​The figure to the left is called a NET. It helps us to visualize the six rectangular regions that make up the faces of a rectangular prism.

​The red segment is equal to the measure of the Perimeter of the Base and the purple segment is the height of the prism. Using the general form, we get

SA = 2B + Ph Perimeter--> P = L+W+L+W = 3 + 5 + 3 + 5 = 16

SA= 2(LW) + Ph = 2(3)( 5) + (16)(6) = 126 units²

​

​You can also find the surface area by finding the sum of the areas of the individual faces.

A rectangular prism has three pairs of congruent faces located on opposite sides of the prism​. Understanding this fact leads us to the formula below.

SA = 2LW + 2WH + 2LH

SA = 2(3)(6) + 2(6)(5) + 2(3)(5)

SA = 126 sq. units​

Multiple Choice

Find the surface area for the rectangular prism. $$Use\ \ SA\ =\ 2B\ +PH$$  

• 24

• 24 ft

• 24 ft²

• 24 ft³

Multiple Choice

Find the surface area of the rectangular prism. $$Use\ \ \ SA\ =\ 2B\ +PH$$  

• 93.6 ft²

• 284.96 ft²

• 378.56 ft²

• 486.72 ft²

​Triangular Prism

​A triangular prism has two triangular bases and three rectangular faces. The perimeter of the triangle is the width of the central rectangular region of the net.

SA = 2B + PL

SA = 2(b*h/2) + PL​

SA = 2(3*4/2) + (12)(3)

SA = 12 + 36

SA = 48 units²​

​

​Because the area of a triangle uses height, h, in its formula, we will use the general form SA = 2B + PL in place of SA = 2B + Ph to avoid confusion.

Multiple Choice

Find the surface area for this triangular prism. $$Use\ SA\ =\ 2B\ +\ PL$$  

• 48 ft²

• 54 ft²

• 60 ft²

• 66 ft²

Multiple Choice

Find the surface area of this triangular prism. $$Use\ \ \ SA\ =\ 2B\ +PL$$  

• 16.7 m

• 16.7 m²

• 57.4 m

• 57.4 m²

A ​Cube has six congruent square faces.

​

SA = 6a²​

​

where a represents the length of one side of the cube​.

​

​

Multiple Choice

Find the surface area for the cube using

Use SA = 6a² .

• 72 in²

• 432 in²

• 576 in²

• 3456 in²

Multiple Choice

A child's toy is cube shaped and covered in a plush material. The length of one side of the toy is 3 inches. What is the minimal amount of material required to cover the toy?

$$Use\ \ SA\ =\ 6s^2\ \ where\ s\ is\ the\ length\ of\ a\ side$$  .

• 54 in²

• 27 in²

• 18 in²

• 9 in²

​Circular Cylinders

​The surface area of a cylinder consists of two congruent circular bases with a radius, r, and a central rectangular region. The Perimeter of the Base, P, is the circumference of the circular base.

​SA = 2B + PH

SA= 2πr² + 2πrh

Multiple Choice

Calculate the surface area of the cylinder.

$$Let\ \pi\ =\ 3.14.\$$   $$SA\ =\ 2B\ PH\ =\ s\pi r^2\ +\ 2\pi rh$$  

• 250 cm²

• 628 cm²

• 1570 cm²

• 2198 cm²

Multiple Choice

What is the exact surface area of a cylinder with a radius of 5 inches and a height of 7 inches? $$SA\ =\ 2B\ +\ PH$$  

$$SA\ =\ 2\pi r^2\ +\ 2\pi rh$$  

• $$35\pi\ in^2$$  

• $$50\pi\ in^2$$  

• $$70\pi\ in^2$$  

• $$120\pi\ in^2$$