Problem 2 Using Formulas to Find Surface Area of a Prism

What is the surface area of the prism at below?

• Step 1 Find the perimeter of a base.

  

• The perimeter of the base is the sum of the side lengths of the triangle. Since the base is a right triangle, the hypotenuse is 3 2 + 4 2 square root of 3 squared , plus , 4 squared end root cm, or 5 cm, by the Pythagorean Theorem. p = 3 + 4 + 5 = 12 p equals , 3 plus 4 plus 5 equals 12

• Step 2 Find the lateral area of the prism.

  L.A. = p h Use the formula for lateral area. = 12 · 6 Substitute  12  for  p  and  6  for  h . = 72 Simplify. table with 3 rows and 3 columns , row1 column 1 , cap l.cap a. , equals , column 2 p h , column 3 cap usetheformulaforlateralarea. , row2 column 1 , equals , column 2 12 middle dot 6 , column 3 cap substitute . 12 , for , p , and , 6 , for , h . , row3 column 1 , equals , column 2 72 , column 3 cap simplify. , end table

• Step 3 Find the area of a base.

  B = 1 2 b h Use the formula for the area of a triangle. = 1 2 ( 3 · 4 ) Substitute  3  for  b  and  4  for  h . = 6   table with 3 rows and 3 columns , row1 column 1 , b equals , column 2 1 half , b h , column 3 cap usetheformulafortheareaofatriangle. , row2 column 1 , equals , column 2 1 half . open 3 middle dot 4 , close , column 3 cap substitute . 3 , for , b , and , 4 , for , h . , row3 column 1 , equals , column 2 6 , column 3 , end table

• Think

  Which height do you need?

  For problems involving solids, make it a habit to note which height the formula requires. In Step 3, you need the height of the triangle, not the height of the prism.

• Step 4 Find the surface area of the prism.

  S.A. = L.A. + 2 B Use the formula for surface area. = 72 + 2 ( 6 ) Substitute  72  for L.A. and  6  for  B . = 84 Simplify. table with 3 rows and 3 columns , row1 column 1 , cap s.cap a. , equals , column 2 cap l.cap a. . plus 2 b , column 3 cap usetheformulaforsurfacearea. , row2 column 1 , equals , column 2 72 plus 2 , open 6 close , column 3 cap substitute . 72 . forcap l.cap a.and . 6 , for b . , row3 column 1 , equals , column 2 84 , column 3 cap simplify. , end table

The surface area of the prism is 84   cm 2 . 84 , cm squared , .

✓ Got It?

2. 1. What is the lateral area of the prism below?
   2. What is the area of a base in simplest radical form?

   3. What is the surface area of the prism rounded to a whole number?