Perimeters, areas and other measurements

• In many careers it is essential to have the ability to recognize 2-dimensional images as 3-dimensional objects. House builders use 2-dimensional plans to construct a 3-dimensional house.

Perimeters and Areas

How many measurements would it take to find the length of a wall in the

classroom? OneThis is one-dimensional measurement.

(linear)

How many measurements would it take to find the size (area) of the

floor? Two

This is two-dimensional measurement. (length and width)

Perimeter is the distance around a geometric figure. Perimeter is a linear measurement, in other words it is one-

dimension (length). To find the perimeter P of a polygon, you can add the

lengths of its sides.

Properties of a rectangle

Opposite sides of a rectangle are the same

length (congruent).The angles of a rectangle are all congruent (90°).

Remember that a 90° angle is called a “right angle.” So,

a rectangle has four right angles.

Opposite angles of a rectangle are congruent.

Opposite sides of a rectangle are parallel.

You can find the perimeter of a

rectangle by adding the lengths of its sides, Or, since

opposite sides of a rectangle are equal length, you can find

the perimeter by using the formula:

P = 2l + 2w

Area: the number of square units in a figure.

People talk about the area of all kinds of things. Area is always measured in some

size squares. They can be smaller or bigger, depending on the unit of

measurement. Here are two different types of measurement, kilometers and

inches.

Area is the number of square units needed to cover a figure.

Here we don’t have any particular unit for measuring the lengths of the sides of the

rectangle. The sides are 7 and 4. Can you figure out how many squares is the area?

Dimensions are measured in

squares so using the proper unit is

important.yd2 cm2 miles2

Area of a squareA = lw or A = S²

Area of a rectangleA = lw or A = bh

The area A of a rectangle is the product of its length (l) and its width (w)

A = lw or A = bh

Find the area of a rectangle:

Some problems may require you to find an additional piece of information before

finding the area.

This problem expects you to use the Pythagorean Theorem to find the base of

the rectangle before finding the area.

Area of a parallelogram

A = bh

The base of the parallelogram is the length of the rectangle. The height of the

parallelogram is the width of the rectangle.The area A of a parallelogram is the

product of its base b and its height h.

A = bhThe height of a

parallelogram is not the length of its slanted

side. The height of a figure is always

perpendicular to the base.

Find the area of a parallelogram: When working with

parallelogram problems, be sure the height you are using is in fact

perpendicular (makes a right angle) to the base (side) you are using. In this problem, 8 is the base and 9 is the height. The side

of 10 is not used in this area.

Area of a triangleA = ½bh

We have discussed how a square, rectangle or parallelogram can be divided by drawing a diagonal from one corner to the opposite

corner. This forms two congruent triangles.Finding the area of a triangle from here is fairly simple, just take ½ of the area of the square,

rectangle, or parallelogram, So,the area A of a triangle is half the product of its

base b and its height h

A = ½ bh

Find the area of the triangle:It may be necessary, when working with an

obtuse triangle, to look outside the triangle to find the height. Notice how the height is drawn to an extension of the base of the

triangle.

Remember, the height of a triangle or quadrilateral is a line perpendicular from the base to the opposite vertex

or side.

Some more triangles and their measures. Just a note to remember, the height of a figure is always perpendicular to the

base.

What do you have to know before you can find the area of the triangle to the

left?

Use the technique we just discussed and determine the area of the triangle

to the left.

The length of the height is 4 and the length of the base

is (AC) 8.

Now multiply(4 · 8) ⁄ 2 = 16

units2

Area of a trapezoidA = ½h(b¹ + b²)

A parallelogram can be divided into two congruent trapezoids. The area of each

trapezoid is one-half the area of the parallelogram.

The two parallel sides of a trapezoid are its bases. If we call the longer side b1 and the

shorter side b2, then the base of the parallelogram is (b1 + b2).

Find the area of the trapezoid:

When working with a trapezoid, the height may be measured anywhere between the two bases. Also, beware of “extra”

information. The 35 and 28 are not needed to compute this area.

Trapezoid formula:

Remember, perimeter is the distance around a figure and is measured in linear

units.Area is the space inside a figure and is

measured in square units.

Composite FiguresComposite FiguresComposite FiguresComposite Figures

Several shapes in oneSeveral shapes in one

Composite shapes offer a unique challenge. They can be several basic

shapes together that make up one larger shape.

Find the perimeter and area of this figure. Do you have enough information? Sometimes you need to use several different formulas to complete the

problem.

Find the perimeter and area of the figure.

Composite figures drawn on coordinate grids are easily label with dimensions.

Draw the figure and find the perimeter and area.

Find the perimeter and area of the

figure.

Find the perimeter and area of the

figure.

PythagoreanPythagoreanTheoremTheorem

PythagoreanPythagoreanTheoremTheorem

Using formulas to find Using formulas to find measurementsmeasurements

The Egyptians measured their fields with lengths of knotted rope. The size of the farmer’s field is used

to work out how big his yield would be, and how much tax he should pay. This knotted rope

indicates a triangle with sides of length 3, 4, and 5 units.

• Pythagoras’s theory is all about right-angled triangles. If squares are constructed on three sides of a right angled triangle, then these squares have a simple but very important connection.

Right angles and right-angled

triangles have been at the forefront of

building since ancient times and is

still used today.

Pythagorean Theorem -In any right triangle, the sum of the

squares of the lengths of the two legs is equal to the square of the length of the

hypotenuse.

Pythagorean Theorem

a² + b² = c²

What does that mean? It means that if I take leg a which is 3 units and square it (3²) and

add it to leg b which is 4 units and square it (4²), that together they will equal the square of the

hypotenuse c (5²)

a² + b² = c²3² + 4² = 5²9 + 16 = 25

25 = 25

First, let’s identify the parts of a right

triangle. The legs are the

two shorter sides of the triangle, and the hypotenuse is the longest side. The

hypotenuse is the side that is always opposite the right

angle.

Did you know?Sailboats have

triangular sails to capitalize on wind at

90° to the boat, thereby increasing its

maneuverability.

Find the length of the hypotenuse.

The points form a right triangle with

a = 8 and b = 6

a2 + b2 = c2

82 + 62 = c2

64 + 36 = c2

√100 = √c2

10 = c

(0, 9)

(6, 1)(0, 1)

Remember, the hypotenuse is the side opposite the right angle.

Find the length of the hypotenuse.

a2 + b2 = c2

82 + 152 = c2

64 + 225 = c2

289 = c2

√289 = √c2

17 ≈ c

Find the length of leg b.

a2 + b2 = c2

62 + b2 = 112

36 + b2 = 121 b2 = 121 – 36 b2 = 85 √b2 = √85 b ≈ 9.219544457…

Using Pythagorean Theorem to find area

a2 + b2 = c2

a2 + 82 = 122

a2 + 64 = 144a2 = 144 – 64

√a2 = √80A ≈ 8.94427191

A = ½bhA = ½(16)(8.9)A = 71.2 units2

Find the area and perimeter of the

rectangle.

a2 + b2 = c2

a2 + 122 = 132

a2 +144 = 169a2 = 169 – 144

a2 = 25√a2 = √25

a = 5A = bh

A = (12)(5)A = 60 units2

P = 2l + 2wP = 2(12) + 2(5)

P = 24 + 10P = 34 units

Use the Pythagorean Theorem to find the height of the triangle and

the distance across the pond. Could you use the height of the

triangle to find the area?

The Pythagorean Theorem can be a very useful tool to find

measurements.

Find the length between the two points, (1, 5), (3, 1).

Solving word problems

A jogger is taking his normal run for exercise.

He leaves home and jogs 8 miles north, then turns

and jogs 5 miles west. If he decides to jog straight

home, what is the shortest distance he must travel to

return to his original starting point.

Pythagorean Triples:The distinction between the Pythagorean Theorem and its converse are sometimes

over looked.The Pythagorean Theorem states that if a

triangle is a right triangle, then the lengths of the sides satisfy the equation a²+ b²=

c².

The converse says that if you have three numbers that satisfy the equation a²+ b² =

c², then those three numbers are side lengths of a right triangle.

The most common special sets of triples are below.

3, 4, 5a²+ b² = c²3² + 4² = 5²9 + 16 = 25

25 = 25

5, 12, 13a² + b² = c²

5² + 12² = 13²25 + 144 = 169

169 = 1698, 15, 17

a² + b² = c²8² + 15² = 17²

64 + 225 = 289289 = 289

Other triples do not work. Remember the three numbers must make the equation

a² + b² = c² true.

7, 8, 9a² + b² = c²7² + 8² = 9²

49 + 64 = 81113 = 81, no

12, 15, 20a² + b² = c²

12² + 15² = 20²144 + 225 = 400

369 = 400, no

Algebra in Algebra in MeasurementMeasurement

Algebra in Algebra in MeasurementMeasurement

Finding unknown measurementsFinding unknown measurements

Finding measurements using the coordinate graph make finding

the area much easier. It’s just a matter of

counting the number of units for each

length of the base and the height. Then all

you have to do is divide by 2.

Take another look at the previous question

and find the area.

Find the area of the triangle to the right.

What is your first step to determine the area.

There are 2 formulas you must know, what

are they?

Find the area of the rectangle below.What steps are needed to complete this

problem?

Remember me, the perimeter problem you needed to find the outside length. Well

now you can find the perimeter using the Pythagorean Theorem. Find the perimeter

and area of the figure below.

Loading this delivery truck is

very difficult when you must load

heavy containers, so the company

decided to build a ramp to make

loading easier. When the contractor started to build the ramp he needed to know the height of

the back of the ramp. What do you

needed to do to determine the

height of the ramp.