PRISMS: SURFACE AREA AND VOLUME |

**3-Dimensional Solids**

** Three Dimensional Figures are called solids.
**We

We use

**There are two kinds** of 3-dimensional **solids**:

those **based on polygons**, and those **based on circles**.

The first category includes **prisms** of all shapes and pyramids.

The **circular solids** include **cylinders**, **cones**, and **spheres**.

This lesson covers surface area and volume of the polygonal solids -- prisms and pyramids.

**Prisms or Polygonal Solids**

**Prisms or Polygonal solids**** **are composed of** ****polygon**-shaped ** faces**,

A** prism **has

and pairs of

so

A Prism is

**Area and Volume Formulas**

Let's describe these solids and investigate the formulas for their surface area and volume.

**1 Rectangular Prism:** is a pile of cubes or rectangles. In common language, it's a box.

It has 6 square or rectangular faces:

** Surface Area** (usually called Area) is measured in square units. It tells us how much paper or material we'd need to cover the surface of the solid.

So, **The Surface Area** = the **sum** of the **areas** of **all the faces**.

And since there are 3 **pairs** of **congruent faces**, we take 2 times the sum of the areas of 3 faces.

If as shown, the dimensions are *l, w, *and *h* for length, width and height,

the total **Surface area** = 2( *lw* + * lh* + *wh* )

** Volume** (also called Capacity) measured in cubic units tells us how much material a shape can hold, or how much material it contains.

**Volume** = Area of the base multiplied by the height so

**Volume **= *l × w × **h*

**2 Triangular Prism**: is a pile of triangles if we stand it on end. This shape is what most people think of when they hear the word prism. It has 3 square or rectangular faces called "

and 2 trangular bases.

**3 Pyramid:** the number of sides depends on the shape of the base. It could be a square or a triangle or some other polygon. The sides or faces meet at a point called the vertex or apex.

**** Warning:** there are 2 different heights to consider in a pyramid: the __v____ertical height__ -- from vertex to the center point of the base, and the

**Surface Area** = sum of the areas of the base and sides or faces.

**square pyramid**: area of the square base plus the areas of the 4 triangular sides.

**triangular pyramid**: areas of the 4 triangles -- the base and 3 sides.

**Volume** = (Area of base × height).

**Note:**

for straight sided solids (prisms, cylinders): **Volume** = (area of base) × height

for pointy slanted sided solids (pyramids, cones): **Volume** = (area of base × height).

Now get a pencil, an eraser and a note book, copy the questions,

do the practice exercise(s), then check your work with the solutions.

If you get stuck, review the examples in the lesson, then try again.

.

**Practice Exercise**

Remember to include the proper units in your answers.

**Reminder:** the area of a triangle is ½ base × height.

1) For each figure shown:

a) name the figure .......... b) find the total surface area .......... c) find the volume.

2)

.

**Solutions**

1) a) rectangular prism .......... cube (square prism) .......... triangular prism.

b) **rectangular prism**: Area = 2(4 × 9 + 4 × 6 + 6 × 9) = **228 cm²**

**cube** (square prism): Area = 6(16²) = **1536 mm²**

**triangular prism**: Area = 2(3 × 5) + 3(11 × 5.8) = **221.4 cm²**.

c) **rectangular prism**: Volume = 4 × 6 × 9 = **324 cm**³

**cube** (square prism): Volume = 16³ = **4096 mm³**

**triangular prism**: Volume = 3 × 5 × 11 = **165 cm³**.

2)

a) each side is 12 × 5 = 60 m² less the 6 m² for the window.

Total dark green paint area = 2(60 – 6) = **108 m²**.

b) The windows are each 6 m² and each skylight is 2.3 m²

total area of windows and skylights is 2(6) + 3(2.3) = 18.9 m².

c) the back of the barn is a rectangle (7 × 5) and a triangle (*b* = 7, *h* = 4)

Area of the back of the barn is 35 + 14 = 49 m².

The front of the barn is 12 m² less because of the door, so

**total area** of light green paint to cover the front and back = 49 + 49 – 12 = **86 m²**.

d) the volume of the rectangular prism is 7 × 12 × 5 = 220 m³

the volume of the triangular prism is ½ (7 × 4 × 12) = 168 m³

**total volume** of the inner barn space is 220 + 168 = **388 m³**

( *Plane Geometry MathRoom Index* )

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