LengthLength and Perimeter

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Length

Length and Perimeter

LengthIn the beginning, to measure and record a length, we gauged and matched the measurement to a physical item such as a stick or a piece of rope (we all have dipped sticks in pools of water to measure their depths.)

Length and Perimeter

LengthIn the beginning, to measure and record a length, we gauged and matched the measurement to a physical item such as a stick or a piece of rope (we all have dipped sticks in pools of water to measure their depths.)

After units such as feet and meters were established, we were able to record lengths with numbers instead of physical objects.

Length and Perimeter

LengthIn the beginning, to measure and record a length, we gauged and matched the measurement to a physical item such as a stick or a piece of rope (we all have dipped sticks in pools of water to measure their depths.)

After units such as feet and meters were established, we were able to record lengths with numbers instead of physical objects. For example, to find the distance between two locations L and R with L at 4 and R at 16 as shown below,

L

42 160

R

Length and Perimeter

LengthIn the beginning, to measure and record a length, we gauged and matched the measurement to a physical item such as a stick or a piece of rope (we all have dipped sticks in pools of water to measure their depths.)

After units such as feet and meters were established, we were able to record lengths with numbers instead of physical objects.

L

42 160

R

we subtract: 16 – 4 = 12, so that they are 12 units apart.

Length and Perimeter

For example, to find the distance between two locations L and R with L at 4 and R at 16 as shown below,

LengthIn the beginning, to measure and record a length, we gauged and matched the measurement to a physical item such as a stick or a piece of rope (we all have dipped sticks in pools of water to measure their depths.)

After units such as feet and meters were established, we were able to record lengths with numbers instead of physical objects.

L

42 160

RIn general, to find the distance between two locations L and R on a ruler,

L R

we subtract: 16 – 4 = 12, so that they are 12 units apart.

we subtract: R – L (i.e. Right Point – Left Point).

Length and Perimeter

For example, to find the distance between two locations L and R with L at 4 and R at 16 as shown below,

LengthIn the beginning, to measure and record a length, we gauged and matched the measurement to a physical item such as a stick or a piece of rope (we all have dipped sticks in pools of water to measure their depths.)

After units such as feet and meters were established, we were able to record lengths with numbers instead of physical objects.

L

42 160

RIn general, to find the distance between two locations L and R on a ruler,

L R

The length between L and R is R – L

we subtract: 16 – 4 = 12, so that they are 12 units apart.

we subtract: R – L (i.e. Right Point – Left Point).

Length and Perimeter

For example, to find the distance between two locations L and R with L at 4 and R at 16 as shown below,

Example A. a. Find the mileage-markers for points A and B and the distance between A and B below.

A0 60 (miles) B

Length and Perimeter

Example A. a. Find the mileage-markers for points A and B and the distance between A and B below.

A

The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles.

0 60 (miles) B

Length and Perimeter

Example A. a. Find the mileage-markers for points A and B and the distance between A and B below.

A

The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles.

0

Hence, B is at 25 and A is at 45.

60 (miles) B

Length and Perimeter

Example A. a. Find the mileage-markers for points A and B and the distance between A and B below.

A

The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles.

0

Hence, B is at 25 and A is at 45.

60 (miles) B

25 45

Length and Perimeter

Example A. a. Find the mileage-markers for points A and B and the distance between A and B below.

A

The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles.

0

The distance between A and B is 45 – 25 = 20 miles. Hence, B is at 25 and A is at 45.

60 (miles) B

25 45

Length and Perimeter

Example A. a. Find the mileage-markers for points A and B and the distance between A and B below.

A

The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles.

0

The distance between A and B is 45 – 25 = 20 miles. Hence, B is at 25 and A is at 45.

60 (miles) B

b. The following is a straight road and the corresponding positions of towns A, B, and C, marked along the road.

A B C

We drove from A to B and found out they are 56 miles apart. How far is A to C? How many miles more are there to reach C?

25 45

Length and Perimeter

Example A. a. Find the mileage-markers for points A and B and the distance between A and B below.

A

The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles.

0

The distance between A and B is 45 – 25 = 20 miles. Hence, B is at 25 and A is at 45.

60 (miles) B

b. The following is a straight road and the corresponding positions of towns A, B, and C, marked along the road.

A B C

We drove from A to B and found out they are 56 miles apart. How far is A to C? How many miles more are there to reach C?There are 7 subdivisions from A to B which covers 56 miles, hence each sub-divider is 56/7 = 8 miles.

25 45

Length and Perimeter

Example A. a. Find the mileage-markers for points A and B and the distance between A and B below.

A

The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles.

0

The distance between A and B is 45 – 25 = 20 miles. Hence, B is at 25 and A is at 45.

60 (miles) B

b. The following is a straight road and the corresponding positions of towns A, B, and C, marked along the road.

A B C

We drove from A to B and found out they are 56 miles apart. How far is A to C? How many miles more are there to reach C?There are 7 subdivisions from A to B which covers 56 miles, hence each sub-divider is 56/7 = 8 miles. There are 12 subdivisions from A to C, i.e. so they are 8 x 12 = 96 miles.

25 45

Length and Perimeter

Example A. a. Find the mileage-markers for points A and B and the distance between A and B below.

A

The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles.

0

The distance between A and B is 45 – 25 = 20 miles. Hence, B is at 25 and A is at 45.

60 (miles) B

b. The following is a straight road and the corresponding positions of towns A, B, and C, marked along the road.

A B C

We drove from A to B and found out they are 56 miles apart. How far is A to C? How many miles more are there to reach C?There are 7 subdivisions from A to B which covers 56 miles, hence each sub-divider is 56/7 = 8 miles. There are 12 subdivisions from A to C, i.e. so they are 8 x 12 = 96 miles.There are 96 – 56 = 40 more miles to reach C.

25 45

Length and Perimeter

If we connect the two ends of a rope that’s resting flat in a plane, we obtain a loop.

Length and Perimeter

If we connect the two ends of a rope that’s resting flat in a plane, we obtain a loop.

Length and Perimeter

If we connect the two ends of a rope that’s resting flat in a plane, we obtain a loop.

Length and Perimeter

If we connect the two ends of a rope that’s resting flat in a plane, we obtain a loop.

Length and Perimeter

If we connect the two ends of a rope that’s resting flat in a plane, we obtain a loop.The loop forms a perimeter or border that encloses a flat area, or a plane-shape.

Length and Perimeter

If we connect the two ends of a rope that’s resting flat in a plane, we obtain a loop.The loop forms a perimeter or border that encloses a flat area, or a plane-shape.

The length of the border, i.e. the length of the rope, is also referred to as the perimeter of the area.

Length and Perimeter

If we connect the two ends of a rope that’s resting flat in a plane, we obtain a loop.The loop forms a perimeter or border that encloses a flat area, or a plane-shape.

The length of the border, i.e. the length of the rope, is also referred to as the perimeter of the area.All the areas above are enclosed by the same rope, so they have equal perimeters.

Length and Perimeter

If we connect the two ends of a rope that’s resting flat in a plane, we obtain a loop.The loop forms a perimeter or border that encloses a flat area, or a plane-shape.

The length of the border, i.e. the length of the rope, is also referred to as the perimeter of the area.All the areas above are enclosed by the same rope, so they have equal perimeters.

A plane-shape is a polygon if it is formed by straight lines.

Length and Perimeter

If we connect the two ends of a rope that’s resting flat in a plane, we obtain a loop.The loop forms a perimeter or border that encloses a flat area, or a plane-shape.

The length of the border, i.e. the length of the rope, is also referred to as the perimeter of the area.All the areas above are enclosed by the same rope, so they have equal perimeters.

Following shapes are polygons:A plane-shape is a polygon if it is formed by straight lines.

Length and Perimeter

If we connect the two ends of a rope that’s resting flat in a plane, we obtain a loop.The loop forms a perimeter or border that encloses a flat area, or a plane-shape.

The length of the border, i.e. the length of the rope, is also referred to as the perimeter of the area.All the areas above are enclosed by the same rope, so they have equal perimeters.

Following shapes are polygons: These are not polygons:A plane-shape is a polygon if it is formed by straight lines.

Length and Perimeter

Three sided polygons are triangles.

Length and Perimeter

Triangles with three equal sides are call equilateral triangles.

Three sided polygons are triangles.

Length and Perimeter

Triangles with three equal sides are call equilateral triangles. An equilateral triangle

ss

s

Three sided polygons are triangles.

Length and Perimeter

Triangles are different from other polygons because if all three sides of a triangle are known then the shape of the triangle is determined.

Triangles with three equal sides are call equilateral triangles. An equilateral triangle

ss

s

Three sided polygons are triangles.

Length and Perimeter

Triangles are different from other polygons because if all three sides of a triangle are known then the shape of the triangle is determined. This is not so if a polygon has four or more sides.

Triangles with three equal sides are call equilateral triangles. An equilateral triangle

ss

s

Three sided polygons are triangles.

Length and Perimeter

Triangles are different from other polygons because if all three sides of a triangle are known then the shape of the triangle is determined. For example, there is only one triangular shape with all three sides equal to 1.

1 1

1

This is not so if a polygon has four or more sides.

Triangles with three equal sides are call equilateral triangles. An equilateral triangle

ss

s

Three sided polygons are triangles.

Length and Perimeter

Triangles are different from other polygons because if all three sides of a triangle are known then the shape of the triangle is determined. For example, there is only one triangular shape with all three sides equal to 1.

1 1

1

This is not so if a polygon has four or more sides. Four-sided polygons with sides equal of 1 may be squashed into various shapes.

1

1 1

1

Triangles with three equal sides are call equilateral triangles. An equilateral triangle

ss

s

Three sided polygons are triangles.

Length and Perimeter

Triangles are different from other polygons because if all three sides of a triangle are known then the shape of the triangle is determined. For example, there is only one triangular shape with all three sides equal to 1.

1 1

1

This is not so if a polygon has four or more sides. Four-sided polygons with sides equal of 1 may be squashed into various shapes.

1

1 1

1

Triangles with three equal sides are call equilateral triangles. An equilateral triangle

ss

s

Three sided polygons are triangles.

Length and Perimeter

Triangles are different from other polygons because if all three sides of a triangle are known then the shape of the triangle is determined. For example, there is only one triangular shape with all three sides equal to 1.

1 1

1

This is not so if a polygon has four or more sides. Four-sided polygons with sides equal of 1 may be squashed into various shapes.

1

1 1

1

Because of this, we say that “triangles are rigid”,

Triangles with three equal sides are call equilateral triangles. An equilateral triangle

ss

s

Three sided polygons are triangles.

Length and Perimeter

Triangles are different from other polygons because if all three sides of a triangle are known then the shape of the triangle is determined. For example, there is only one triangular shape with all three sides equal to 1.

1 1

1

This is not so if a polygon has four or more sides. Four-sided polygons with sides equal of 1 may be squashed into various shapes.

1

1 1

1

Because of this, we say that “triangles are rigid”, and that in general “four or more sided polygons are not rigid”.

Triangles with three equal sides are call equilateral triangles. An equilateral triangle

ss

s

Three sided polygons are triangles.

Length and Perimeter

If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter.

Length and Perimeter

a

b

c

P = a + b + c

If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter.

Length and Perimeter

The perimeter of an equilateral triangle is P = s + s + s = 3s.

a

b

c

P = a + b + c

Rectangles are 4-sided polygons where the sides are joint at a right angle as shown.

If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter.

Length and Perimeter

a

b

c

P = a + b + c

The perimeter of an equilateral triangle is P = s + s + s = 3s.

Rectangles are 4-sided polygons where the sides are joint at a right angle as shown.

If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter.

Length and Perimeter

a

b

c

P = a + b + c

The perimeter of an equilateral triangle is P = s + s + s = 3s.

Length and Perimeter

s

s

ss

A square

Rectangles are 4-sided polygons where the sides are joint at a right angle as shown.

If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter.

a

b

c

P = a + b + c

The perimeter of an equilateral triangle is P = s + s + s = 3s.

A square is a rectangle with four equal sides.

Length and Perimeter

s

s

ss

The perimeter of a squares is P = s + s + s + s = 4s

A square

Rectangles are 4-sided polygons where the sides are joint at a right angle as shown.

If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter.

a

b

c

P = a + b + c

The perimeter of an equilateral triangle is P = s + s + s = 3s.

A square is a rectangle with four equal sides.

Length and Perimeter

s

s

ss

The perimeter of a squares is P = s + s + s + s = 4s

If we know two adjacent sides of a rectangle, we know the entire rectangle because the opposites sides are identical.

A square

Rectangles are 4-sided polygons where the sides are joint at a right angle as shown.

If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter.

a

b

c

P = a + b + c

The perimeter of an equilateral triangle is P = s + s + s = 3s.

A square is a rectangle with four equal sides.

Length and Perimeter

s

s

ss

The perimeter of a squares is P = s + s + s + s = 4s

If we know two adjacent sides of a rectangle, we know the entire rectangle because the opposites sides are identical.

A square

Rectangles are 4-sided polygons where the sides are joint at a right angle as shown.

If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter.

a

b

c

P = a + b + c

The perimeter of an equilateral triangle is P = s + s + s = 3s.

A square is a rectangle with four equal sides.

Length and Perimeter

s

s

ss

The perimeter of a squares is P = s + s + s + s = 4s

If we know two adjacent sides of a rectangle, we know the entire rectangle because the opposites sides are identical.

However the names of the two sides is a source of confusion.

A square

Rectangles are 4-sided polygons where the sides are joint at a right angle as shown.

If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter.

a

b

c

P = a + b + c

The perimeter of an equilateral triangle is P = s + s + s = 3s.

Length and PerimeterBy the dictionary, “length” is the longest side and “width” is the horizontal side.

Length and Perimeter

This causes conflicts in labeling rectangles whose width is the longer side. length? width?

By the dictionary, “length” is the longest side and “width” is the horizontal side.

Hence we will use the words “height” for the vertical side, and “width” for the horizontal side instead.

Length and Perimeter

This causes conflicts in labeling rectangles whose width is the longer side.

By the dictionary, “length” is the longest side and “width” is the horizontal side.

length? width?

width (w)

height (h) h

w

Hence we will use the words “height” for the vertical side, and “width” for the horizontal side instead.

Length and Perimeter

The perimeter of a rectangle is

This causes conflicts in labeling rectangles whose width is the longer side.

By the dictionary, “length” is the longest side and “width” is the horizontal side.

length? width?

width (w)

height (h)P = 2h + 2w (= h + h + w + w) h

w

Hence we will use the words “height” for the vertical side, and “width” for the horizontal side instead.

Length and Perimeter

The perimeter of a rectangle is

This causes conflicts in labeling rectangles whose width is the longer side.

By the dictionary, “length” is the longest side and “width” is the horizontal side.

Example B.

length? width?

width (w)

height (h)P = 2h + 2w (= h + h + w + w) h

w

5 mi

How many miles did we travel?Assume it’s an equilateral triangle on top.

a. We drove in a loop as shown.

3 mi

Hence we will use the words “height” for the vertical side, and “width” for the horizontal side instead.

Length and Perimeter

The perimeter of a rectangle is

This causes conflicts in labeling rectangles whose width is the longer side.

By the dictionary, “length” is the longest side and “width” is the horizontal side.

Example B.

length? width?

width (w)

height (h)P = 2h + 2w (= h + h + w + w) h

w

5 mi

How many miles did we travel?Assume it’s an equilateral triangle on top.

a. We drove in a loop as shown.

3 mi

There are three 3-mile sections and two 5-milesections.

Hence we will use the words “height” for the vertical side, and “width” for the horizontal side instead.

Length and Perimeter

The perimeter of a rectangle is

This causes conflicts in labeling rectangles whose width is the longer side.

By the dictionary, “length” is the longest side and “width” is the horizontal side.

Example B.

length? width?

width (w)

height (h)P = 2h + 2w (= h + h + w + w) h

w

5 mi

How many miles did we travel?Assume it’s an equilateral triangle on top.

a. We drove in a loop as shown.

3 mi

There are three 3-mile sections and two 5-milesections. Hence one round trip P is

= 3(3)+ 2(5)P = 3 + 3 + 3 + 5 + 5

Hence we will use the words “height” for the vertical side, and “width” for the horizontal side instead.

Length and Perimeter

The perimeter of a rectangle is

This causes conflicts in labeling rectangles whose width is the longer side.

By the dictionary, “length” is the longest side and “width” is the horizontal side.

Example B.

length? width?

width (w)

height (h)P = 2h + 2w (= h + h + w + w) h

w

5 mi

How many miles did we travel?Assume it’s an equilateral triangle on top.

a. We drove in a loop as shown.

3 mi

There are three 3-mile sections and two 5-milesections. Hence one round trip P is

= 3(3)+ 2(5) = 9 + 10 = 19 miles. P = 3 + 3 + 3 + 5 + 5

Length and Perimeterb. We want to rope off a 50-meter by 70-meter rectangular area and also rope off sections of area as shown. How many meters of rope do we need?

50 m

70 m

Length and Perimeter

50 m

70 m

b. We want to rope off a 50-meter by 70-meter rectangular area and also rope off sections of area as shown. How many meters of rope do we need?We have three heights where each requires 50 meters of rope,

Length and Perimeter

and three widths where each requires 70 meters of rope.

50 m

70 m

b. We want to rope off a 50-meter by 70-meter rectangular area and also rope off sections of area as shown. How many meters of rope do we need?We have three heights where each requires 50 meters of rope,

Length and Perimeter

and three widths where each requires 70 meters of rope.

Hence it requires3(50) + 3(70) = 150 + 210 = 360 meters of rope.

50 m

70 m

b. We want to rope off a 50-meter by 70-meter rectangular area and also rope off sections of area as shown. How many meters of rope do we need?We have three heights where each requires 50 meters of rope,

Length and Perimeter

and three widths where each requires 70 meters of rope.

Hence it requires3(50) + 3(70) = 150 + 210 = 360 meters of rope.

c. What is the perimeter of the following step-shape if all the short segments are 2 feet?

2 ft

50 m

70 m

b. We want to rope off a 50-meter by 70-meter rectangular area and also rope off sections of area as shown. How many meters of rope do we need?We have three heights where each requires 50 meters of rope,

Length and Perimeter

and three widths where each requires 70 meters of rope.

Hence it requires3(50) + 3(70) = 150 + 210 = 360 meters of rope.

c. What is the perimeter of the following step-shape if all the short segments are 2 feet?

2 ftThe perimeter of the step-shape is the same as the perimeter of the rectangle that boxes it in as shown.

50 m

70 m

b. We want to rope off a 50-meter by 70-meter rectangular area and also rope off sections of area as shown. How many meters of rope do we need?We have three heights where each requires 50 meters of rope,

Length and Perimeter

and three widths where each requires 70 meters of rope.

Hence it requires3(50) + 3(70) = 150 + 210 = 360 meters of rope.

c. What is the perimeter of the following step-shape if all the short segments are 2 feet?

2 ftThe perimeter of the step-shape is the same as the perimeter of the rectangle that boxes it in as shown. There are 3 steps going up, and 5 steps going across.

50 m

70 m

b. We want to rope off a 50-meter by 70-meter rectangular area and also rope off sections of area as shown. How many meters of rope do we need?We have three heights where each requires 50 meters of rope,

Length and Perimeter

and three widths where each requires 70 meters of rope.

Hence it requires3(50) + 3(70) = 150 + 210 = 360 meters of rope.

c. What is the perimeter of the following step-shape if all the short segments are 2 feet?

2 ftThe perimeter of the step-shape is the same as the perimeter of the rectangle that boxes it in as shown. There are 3 steps going up, and 5 steps going across.

50 m

70 m

b. We want to rope off a 50-meter by 70-meter rectangular area and also rope off sections of area as shown. How many meters of rope do we need?We have three heights where each requires 50 meters of rope,

So the height is 6 ft, the width is 10 ft, and the perimeter P = 2(6) +2(10) = 32 ft.