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SURFACE AREA and VOLUME

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SURFACE AREA and VOLUME

SOLID FIGURES

In a previous unit we discussed POLYGONS.

POLYGONS are 2-dimensional objects.

That means they have length and width, but no depth.

Like they are cut out of paper.

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SOLID FIGURES

In this unit, we will learn about SOLIDS.

A SOLID is a 3-dimensional (3D) object.

That means it has length, width AND depth.

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SOLID FIGURES

When the SOLID is made up of POLYGONS, it is called a POLYHEDRON

Each POLYHEDRON is made up of

Surfaces (called faces)

Segments (called edges)

Corners (called vertices)

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SOLID FIGURES

Before we can do anything with SOLIDS, we must understand the difference between these shapes

There are many types of solids,

we are going to study the 5 basic types:

PRISM

PYRAMID

CYLINDER

CONE

SPHERE

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SOLID FIGURES

To create a prism, draw any polygon you like.

PRISMS:

Next, draw another polygon, just like the first one

Connect the corresponding vertices

BASE

BASE

BASE

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SOLID FIGURES

PYRAMIDS:

Pyramids have one base

Instead of connecting to an identical base…

They connect to a point.

BASE

BASE

BASE

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SOLID FIGURES

When we name prisms and pyramids, we describe the base to tell what kind of prism or pyramid it is.

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SOLID FIGURES

CYLINDERS:

A cylinder is like a prism…

But the bases are circles

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SOLID FIGURES

SPHERES:

A sphere is a unique shape...

It has no base or sides or edges or vertices.

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SOLID FIGURES

Identify each of the following shapes as a

PRISM, PYRAMID, CYLINDER, CONE or SPHERE

#1

#2

#3

#4

SPHERE

CYLINDER

PRISM

#5

#6

PYRAMID

#7

CYLINDER

CONE

PYRAMID

#9

#10

#8

SPHERE

CONE & SPHERE

PRISM

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SOLID FIGURES

So what are we going to do with these shapes?

We are going to calculate

SURFACE AREA and VOLUME

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SOLID FIGURES

SUFARCE AREA:

Is literally the area of all the surfaces of a solid.

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SOLID FIGURES

SUFARCE AREA:

Is literally the area of all the surfaces of a solid.

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SURFACE AREA and VOLUME

SOLID FIGURES

SUFARCE AREA:

Is literally the area of all the surfaces of a solid.

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SOLID FIGURES

SUFARCE AREA:

Is literally the area of all the surfaces of a solid.

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SOLID FIGURES

VOLUME:

Is the measure of how much “space” an object occupies.

It is often thought of as how much water an object can hold.

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SOLID FIGURES

VOLUME:

Or how many cubic units it would take to fill an object

This is why volume is measured in CUBIC UNITS

Like:

Cubic feet:

ft3

Cubic centimeters cm3

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PRISMS

A prism is a solid with 2 congruent BASES connected by LATERAL FACES

Lateral

face

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BASE

SURFACE AREA and VOLUME

PRISMS

To find the surface area of a PRISM, you can do it 2 ways:

Method 1:

Find the area of each shape that makes up the

Solid, and add them together

The front

2 x 10

= 20

12

The back

2 x 10

= 20

The left

6 x 10

= 60

10

60

The right

6 x 10

= 60

20

The top

2 x 6

= 12

The bottom

2 x 6

= 12

6

= 184

2

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PRISMS

To find the surface area of a PRISM, you can do it 2 ways:

Method 2:

Use the formula for SURFACE AREA:

- Identify the BASEs

(in a rectangular prism any pair of opposite sides can be bases)

- Find the Area and Perimeter of the base

BASE

P; 16

A: 12

10

h: height of the prism

B: area of the base

10

6

P: perimeter of the base

2

12

16

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PRISMS

Find the surface area of this regular decagonal prism:

Find the perimeter of the base

6+6+6+6+6+6+6+6+6+6

=60

Area of the bases

Lateral Area

A: 270m2

P:60m

6m

4m

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PRISMS

Finding Volume is easier!

Find the VOLUME of this regular decagonal prism:

h: height of the prism

B: area of the base

A: 270m2

P:60m

6m

4m

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PRISMS

RIGHT PRISMS vs OBLIQUE PRISMS

A right prism is a prism where the lateral faces are all perpendicular to the bases.

An oblique prism is a prism where the lateral faces are NOT perpendicular to the bases.

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PRISMS

SUMMARY:

Things you need to know about a prism:

A prism has 2 bases

All the lateral faces are rectangles

Surface Area = 2B + ph

B is the area of the base

p is the perimeter of the base

h is the height of the prism

p x h gives the lateral area

Volume = B x h

B is the area of the base

h is the height of the prism

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PRISMS

Find the…

3cm

…Surface Area

4cm

2cm

…Lateral Area

First we have to find that missing side

3cm

4cm

…Volume

Use the Pythagorean theorem

5cm

A:

P:

6cm2

12cm

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PRISMS

Find the…

…Surface Area

3cm

4cm

5cm

2cm

…Lateral Area

…Volume

A:

P:

6cm2

12cm

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PRISMS

Find the surface area and volume.

10

8

112

24

14

6

Area:

Perimeter:

136ft2

52ft

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CYLINDERS

Since CYLINDERS are like prisms with circular bases,

We need to remember a few basic rules for circles.

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CYLINDERS

Find the Circumference and area of the circle:

6in

12in

The red segment is the DIAMETER

We need the RADIUS

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CYLINDERS

A CYLINDER is like a prism with circular bases

The surface area of a cylinder can be found with this formula

h: height of the cylinder

B: area of the base

Area of the bases

Lateral Area

C: circumference of the base

Most people find this formula easier:

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CYLINDERS

The VOLUME of a Cylinder:

The VOLUME of a cylinder can be found with this formula

h: height of the cylinder

B: area of the base

The expanded version of the formula:

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CYLINDERS

SUMMARY

What you need to know about a cylinder:

- A cylinder is a prism with circular bases
- The lateral area is a rectangle
- Surface Area:
- Volume:
- Lateral Area:

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CYLINDERS

Find the Lateral Area, Surface Area and Volume:

11mm

9mm

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CYLINDERS

Find the surface area and volume of this shape

31ft

8ft

16ft

31ft

If it was a complete cylinder:

What did we forget?

But we only want half of it

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CYLINDERS

Find the surface area and volume of this shape

31ft

8ft

If it was a complete cylinder:

But we only want half of it

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PYRAMIDS

A PYRAMID is a shape with a single base, where all the vertices of the base are connected to a single point (apex)

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PYRAMIDS

Just like with prisms, we will need to find:

- Base Perimeter
- Base Area
- Height of the pyramid

However, with pyramids, we will also need to find a different measurement…

height

Slant height

…SLANT HEIGHT

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PYRAMIDS

Slant height is the distance someone would travel walking up the OUTSIDE of a pyramid.

SLANT HEIGHT

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PYRAMIDS

Finding the SURFACE AREA of a PYRAMID

Rectangular Pyramid

L: Slant height

B: area of the base

P: perimeter of the base

25m

22m

22m

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PYRAMIDS

Finding the SURFACE AREA of a PYRAMID

Rectangular Pyramid

25m

Lateral Area

22m

22m

Base Area:

Base Perimeter:

484m2

88m

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PYRAMIDS

Finding the VOLUME of a PYRAMID

Rectangular Pyramid

h: height

B: area of the base

11ft

8ft

10ft

Base Area:

Base Perimeter:

80ft2

36ft

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PYRAMIDS

SUMMARY:

Things you need to know about a pyramid:

A pyramid has only 1 base

All the lateral faces are triangles

Surface Area =

B is the area of the base

p is the perimeter of the base

l is the slant height of the pyramid

½ x p x l is the lateral area

Volume =

B is the area of the base

h is the height of the prism

h

l

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PYRAMIDS

Hey, I need to know the height.

Hey, I need to know the slant height.

15

20

24

16

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PYRAMIDS

Find the Lateral Area, Surface area and Volume:

Lateral Area

B:

P:

400

80

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PYRAMIDS

Find the Lateral Area, Surface area and Volume:

B:

P:

400

80

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PYRAMIDS

Find the Lateral Area, Surface area and Volume:

h

10

B:

P:

400

80

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CONES

Find the surface area and volume for the cone shown:

Find the slant height with the Pythagorean Theorem

13

12in

5in

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SPHERES

A sphere is just a ball.

It has 1 important measurement, that is used to find all its other properties

R

The RADIUS R

Official definition of a SPHERE:

The collection of all points in space that are the same distance (radius) from a point (center).

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SPHERES

Find the volume and surface area of the hemisphere:

10in

Half the sphere

+

the circle

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SPHERES

Find the volume and surface area of the hemisphere:

10in

We only want half

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