28 Space and Shape

Introduction:

In this section you will learn about circumference, diameter, radius, area, surface area, and volume. You’ll be presented with concepts and examples, and given opportunities to practice throughout.

Chapter Contents:

• Topic 1: Perimeter and Circumference
  • 1.1 Polygons
  • 1.2 Perimeter
  • 1.3 Circumference/Diameter/Radius
  • 1.4 The Number π
  • 1.5 Circumference of a Circle
• Topic 2: Area
  • 2.1 Finding Areas of Some Common Geometric Figures
  • 2.2 Areas of Parallelograms and Triangles
• Topic 3: Surface Area
• Topic 4: Volume
• Topic 5: Pythagoras’ Theorem
• Test: Outcome 7

Shapes

Circle: A circle is a perfectly round shape with all points on its boundary equidistant from its center.
Square: A square is a four-sided shape where all sides are equal in length and all angles are right angles.
Triangle: A triangle is a shape with three sides and three angles.   A right triangle is a triangle with one angle measuring 90 degrees, called a right angle.
Rectangle: A rectangle is a four-sided shape with four right angles. Opposite sides of a rectangle are equal in length and parallel to each other.

Topic 1: Perimeter and Circumference

We can make use of conversion skills with denominate numbers to make measure­ments of geometric figures such as rectangles, triangles, and circles. To make these measurements we need to be familiar with several definitions.

1.1 Polygons

A polygon is a closed plane (flat) figure whose sides are line segments (portions of straight lines).

1.2 Perimeter

The perimeter of a polygon is the distance around the polygon. To find the perimeter of a polygon, we simply add up the lengths of all the sides.

Example 1

Example 2

The sides do not need to be going perpendicular or parallel to each other. In this irregular shape, to find the perimeter, you still need to add all the sides.

When adding measurements together, they must be in the same units. Notice some of the units are in mm, and some in cm. There are 10 mm in 1 cm.

Example 3

In some regular shapes you will need to figure out unknown measurements based on other given measurements. Find the sides labeled A, B and C.

If the measurements you need are across the shape, it is sometimes useful to visualize them all together like in this diagram:

Consider side A:

The unknown top side, A (blue), is 10 cm long.

Consider side B:

The unknown side B (yellow), is 8 cm.

Consider unknown side C:

The unknown side, C (blue), is 11 cm long.

1.3 Circumference/Diameter/Radius

Circumference

The circumference of a circle is the distance around the circle. In plain language, the circumference is the perimeter of a circle.

You could measure the circumference of a circle with a string, but if you wanted to calculate the circumference, you need to use a formula. The formula depends upon one of two measurements: the diameter, or the radius.

Diameter

A diameter of a circle is any line segment that passes through the center of the circle and has its endpoints on the circle.

Radius

A radius of a circle is any line segment having as its endpoints the center of the circle and a point on the circle. The radius is one half the diameter.

1.4 The Number π

The symbol π, read “pi,” represents the nonterminating, nonrepeating decimal number 3.14159 … . This number has been computed to millions of decimal places without the appearance of a repeating block of digits.

For computational purposes, π is often approximated as 3.14. We will write π≈3.14 to denote that π is approximately equal to 3.14. The symbol “≈” means “approximately equal to.”

When doing calculations on your calculator with a pi button you should use it, and round the answer after. Here is the pi button:

1.5 Circumference of a Circle

To find the circumference of a circle, we need only know its diameter or radius. We then use a formula for computing the circumference of the circle. Formulas are usually composed of letters that represent important, but possibly unknown, quantities.

C=2πr

OR

C=πD

Example 4

Find the circumference of the circle:

Example 5

Find the circumference of the circle:

1. Find the perimeter of the shapes below:

a)

b)

d)

  2. Extension Question: Find the perimeters of the following figures

a)

Questions for construction trades:

3. For the next questions, refer to this floor plan:

a) What is the perimeter of the living room in the plan shown? (Do not deduct for openings.)

b) What is the perimeter of the outside walls?

c) How many feet of baseboard are needed for the living room? (Deduct 3 feet for single door openings and 5 feet for double door openings.)

d) How many linear feet of shoe base are needed for the two bedrooms? (Deduct 3 feet for each door opening.)

e) How many linear feet of casing are needed for both sides of the double door between the dining room and the living room? (Allow 7 feet for each vertical piece and 5 feet 6 inches for each horizontal piece.

4. What is the perimeter of the semicircular form in the following figure?  (Express your answer to the nearest thousandth of a metre.)

5. Find the perimeter

Topic 2: Area

The area of a surface is the amount of square length units contained in the surface.

For example, 3 sq in. means that 3 squares, 1 inch on each side, can be placed precisely on some surface. (The squares may have to be cut and rearranged so they match the shape of the surface.)

2.1 Finding Areas of Some Common Geometric Figures 

We will examine the area of the following geometric figures:

We can determine the areas of these geometric figures using the following formulas.

Remember, when calculating “one half” or “over two” the way to type that in the calculator is using the divide button. For example, “one half the base times the width” in your calculator is base × width ÷ 2.

Example 1

Note that feet multiplied by feet give ft² pronounced as “feet squared.” (The unit is squared; this does not mean square the number.) Whenever a unit is squared, it is a unit of area.

Example 2

In this example, we need to convert 4 ft 2 in into units of inches. You cannot mix different units when finding area. There are 12 inches per foot.

Example 3

Example 4

In this case, call the bottom b1 and the top b2, but the labeling doesn’t matter in this case. The top could be called b1 and the bottom b2.

When using the calculator, “over 2” means divide by 2.

A note about order of operations: In this calculation, the 20.4 and 14.5 need to be added before the sum can be divided by 2. There are two ways to accomplish this:

• Use brackets on your calculator to calculate 20.4 + 14.5 before dividing by 2
• Use the equal sign after 20.4 + 14.5. Once the equal sign is pressed, the numbers are added, and you can divide the total by 2.

The sample calculation below uses brackets.

Example 5

In this example, the diameter is given. The area formula for a circle calls for radius. We must convert the diameter into a radius by dividing it in half before using it in the formula.

Example 6

In some questions, it is necessary to find a total area, and subtract an inner area. In this example, to find the shaded area, we can find the area of the larger rectangle and subtract the area of the smaller rectangle.

Example 6a for the construction trades

When figuring the surface are of the shape above, only with uniform width, it is a shortcut to consider the centre line length.  The centre line length is an imaginary line that travels through the centre of the shape.  This is useful for calculations involving footings and basement walls.

The short side centre line length is:

the outer dimension – half thickness – half thickness

or

the outer dimension – one thickness

(note the same can be achieved by taking the inner dimension and adding one thickness.)

Similarily, the long side centre line length is the outer dimension – one thickness.

Total centre line length is:

short side centre line length X 2 + long side centre line length X 2

For the above diagram, the total centre line length is:

To calculate the area, multiply the total centre line length by the thickness:

2.2 Areas of Parallelograms and Triangles

A parallelogram is a quadrilateral (it has four sides). The opposite sides of a parallelogram are parallel. It is also true that:

• the opposite sides of a parallelogram have equal length, and
• the opposite angles of a parallelogram have equal measure.

There are several strategies for finding the area of a parallelogram.

1. We can decompose and rearrange a parallelogram to form a rectangle.

2. We can enclose the parallelogram and then subtract the area of the two triangles in the corner.

Both ways will work for any parallelogram. However, for some parallelograms the process of decomposing and rearranging requires a lot more steps than if we use a formula. Using a formula is a good shortcut, but if you don’t remember the formula, you can always rely on your ability to manipulate the image and use simpler formulas that you remember.

A parallelogram can always be decomposed into two identical triangles by a segment that connects opposite vertices.

Going the other way around, two identical copies of a triangle can always be arranged to form a parallelogram, regardless of the type of triangle being used.

To produce a parallelogram, we can join a triangle and its copy along any of the three sides, so the same pair of triangles can make different parallelograms.

Here are examples of how two copies of both Triangle A and Triangle F can be composed into three different parallelograms.

This special relationship between triangles and parallelograms can help us reason about the area of any triangle. If the area of a parallelogram is base times height, then it makes sense that the area of a triangle (which is half a parallelogram) is one half of the base times height.

Practice Exercises: Area

1. Are these statements true or false?

– A parallelogram has six sides.

– Opposite sides of a parallelogram are parallel.

– A parallelogram can have one pair or two pairs of parallel sides.

– All sides of a parallelogram must have the same length.

– All angles of a parallelogram must have the same measure.

2. Five students labeled a base b and a corresponding height h for each of these parallelograms. Are all drawings correctly labeled? Explain how you know.

3. Find the area of the following shapes:

 a. b.

c.

d.e.

f.g.

h. (Assume the following shape is vertically symmetric about the center.

i.

Topic 3: Surface Area

Surface area is the combined area of all the sides of a three dimensional shape.

Polyhedra

A polyhedron is a three-dimensional figure composed of faces. Each face is a filled-in polygon and meets only one other face along a complete edge. The ends of the edges meet at points that are called vertices.

Polyhedra always encloses a three-dimensional region.

Here are pictures that represent polyhedra:

Here are pictures that do not represent polyhedra:

The surface area of a polyhedron is the sum of the areas of all the faces.

Because a net shows us all faces of a polyhedron at once, it can help us find the surface area. We can find the areas of all polygons in the net and add them.

A prism is a type of polyhedron with two identical faces that are parallel to each other and that are called bases. The bases are connected by a set of rectangles (or sometimes parallelograms).

A prism is named for the shape of its bases. For example, if the base is a pentagon, then it is called a “pentagonal prism.”

A pyramid is a type of polyhedron that has one special face called the base. All the other faces are triangles that all meet at a single vertex.

A pyramid is named for the shape of its base. For example, if the base is a pentagon, then it is called a “pentagonal pyramid.”

A net is a two-dimensional representation of a polyhedron. It is composed of polygons that form the faces of a polyhedron.

In a cube, every face shares its edges with 4 other squares. In a net of a cube, not all edges of the squares are joined with another edge. When the net is folded, however, each of these open edges will join another edge.

13. cube with 6 identical faces

14. The net of the cube. Each side is connected by one shared line.

A net can be cut out and folded to make a model of the polyhedron.

Example 1

Find the surface area of this square pyramid.

It is useful to help visualize all the sides flat. Then, the question just becomes finding the area of a bunch of basic shapes. Draw a net for the figure like so:

A square pyramid has a square and four triangles for its faces. Its surface area is the sum of the areas of the square base and the four triangular faces:

The surface area of this square pyramid is 96 square units.

A cylinder is a 3 dimensional shape that has a circular base.

As seen in the diagram, the sides of the cylinder can be drawn as two circles for the top and bottom, and a rectangular side. The surface area of a cylinder is total area of the two circles and rectangle.

Example 2

Find the surface area of the following cylinder

The cylinder can be broken down into the following circles (for top and bottom) with radius 5:

The area of one circle comes from the formula πr²

Area= πr²

Area= π×5²≈ 78.54

Finding the dimensions of the side involves a trick. The height is obviously 7, from the original diagram. The width is the circumference of the circle. (Remember, the formula for the circumference is

To calculate the width, us the formula for the circumference of the top or base: 2πr.

circumference=2πr

width=2×π×5≈31.42

The area of the rectangle is the length times the width:

7×31.42=219.94

The total area of the sides of the cylinder:

total surface area=area top+ area bottom +area side

total surface area=78.54+78.54+219.94

total surface area ≈ 377

Practice Exercises: Surface Area

1. Which of the following shapes represent a polyhedron?

2. Is this polyhedron a prism, a pyramid, or neither? Explain how you know.

3. Here are two polyhedra and their nets. Label all edges in the net with the correct lengths. Find the surface area of polyhedron A.

Challenge Question: Find the surface area of polyhedron B. (Hint: you will need to use Pythagoras’ Theorem to find the heights of the triangles.)

4. Calculate the surface area of the following cylinder

Topic 4: Volume

Volume Formulas

Example 1 

Example 2

Find the volume of this sphere:

Example 3

Find the volume of this cylinder:

Example 4

Find the volume of this cone:

Practice Exercises: Volume

1. Find the volume of the following shapes:

a.

b.

c.

d.

e.

f.

g.

h.

i.

Topic 5: Pythagoras’ Theorem

Introduction:

In this section you will learn about right-angle triangle properties, proportionality, and the Pythagorean Theorem. Throughout the section you will be presented with concepts and examples, as well as opportunities to practice.

5.1 Right Angle Triangle Properties

An angle is formed when two straight paths cross. Consider this story: you walk along a straight path, and so does your friend. At first you both are so far apart that you can’t see each other. But as you walk along you get closer and closer until you meet together. Your meeting place is called the vertex.

The interesting part about angles is at the beginning of the journey, you are far from your friend. Halfway through your journey, your friend is closer, and at the end of the journey there is no distance between you and your friend. Throughout the journey you were a changing distance from each other, but the angle of your paths did not change.