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Published on February 6, 2014

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Length Length and Perimeter http://www.lahc.edu/math/frankma.htm

Length Length and Perimeter

Length and Perimeter Length In 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 Length In 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 Length In 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, 0 2 4 16 L R

Length and Perimeter Length In 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, we subtract: 16 – 4 = 12, so that they are 12 units apart. 0 2 4 16 L R

Length and Perimeter Length In 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, we subtract: 16 – 4 = 12, so that they are 12 units apart. 2 4 16 L 0 R In general, to find the distance between two locations L and R on a ruler, we subtract: R – L (i.e. Right Point – Left Point). L R

Length and Perimeter Length In 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, we subtract: 16 – 4 = 12, so that they are 12 units apart. 2 4 16 L 0 R In general, to find the distance between two locations L and R on a ruler, we subtract: R – L (i.e. Right Point – Left Point). L R The length between L and R is R – L

Length and Perimeter Example A. a. Find the mileage-markers for points A and B and the distance between A and B below. 0 B A 60 (miles)

Length and Perimeter Example A. a. Find the mileage-markers for points A and B and the distance between A and B below. The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles. 0 B A 60 (miles)

Length and Perimeter Example A. a. Find the mileage-markers for points A and B and the distance between A and B below. The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles. Hence, B is at 25 and A is at 45. 0 B A 60 (miles)

Length and Perimeter Example A. a. Find the mileage-markers for points A and B and the distance between A and B below. The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles. Hence, B is at 25 and A is at 45. 0 B A 25 45 60 (miles)

Length and Perimeter Example A. a. Find the mileage-markers for points A and B and the distance between A and B below. The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles. Hence, B is at 25 and A is at 45. The distance between A and B is 45 – 25 = 20 miles. 0 B A 25 45 60 (miles)

Length and Perimeter Example A. a. Find the mileage-markers for points A and B and the distance between A and B below. The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles. Hence, B is at 25 and A is at 45. The distance between A and B is 45 – 25 = 20 miles. B A 25 0 45 60 (miles) 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?

Length and Perimeter Example A. a. Find the mileage-markers for points A and B and the distance between A and B below. The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles. Hence, B is at 25 and A is at 45. The distance between A and B is 45 – 25 = 20 miles. B A 25 0 45 60 (miles) 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.

Length and Perimeter Example A. a. Find the mileage-markers for points A and B and the distance between A and B below. The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles. Hence, B is at 25 and A is at 45. The distance between A and B is 45 – 25 = 20 miles. B A 25 0 45 60 (miles) 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.

Length and Perimeter Example A. a. Find the mileage-markers for points A and B and the distance between A and B below. The 60-mile segment is divided into 12 pieces, so each subdivision is 60/12 = 5 miles. Hence, B is at 25 and A is at 45. The distance between A and B is 45 – 25 = 20 miles. B A 25 0 45 60 (miles) 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.

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

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. Following shapes are polygons: These are not polygons:

Length and Perimeter Three sided polygons are triangles.

Length and Perimeter Three sided polygons are triangles. Triangles with three equal sides are call equilateral triangles.

Length and Perimeter Three sided polygons are triangles. Triangles with three equal sides are call equilateral triangles. s s s An equilateral triangle

Length and Perimeter Three sided polygons are triangles. s s s Triangles with three equal sides An equilateral triangle are call equilateral triangles. Triangles are different from other polygons because if all three sides of a triangle are known then the shape of the triangle is determined.

Length and Perimeter Three sided polygons are triangles. s s s Triangles with three equal sides An equilateral triangle are call equilateral triangles. 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.

Length and Perimeter Three sided polygons are triangles. s s s Triangles with three equal sides An equilateral triangle are call equilateral triangles. 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. For example, there is only one triangular shape with all three sides equal to 1. 1 1 1

Length and Perimeter s Three sided polygons are triangles. s s Triangles with three equal sides An equilateral triangle are call equilateral triangles. 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. For example, there is only Four-sided polygons with sides one triangular shape with equal of 1 may be squashed into all three sides equal to 1. various shapes. 1 1 1 1 1 1 1

Length and Perimeter s Three sided polygons are triangles. s s Triangles with three equal sides An equilateral triangle are call equilateral triangles. 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. For example, there is only Four-sided polygons with sides one triangular shape with equal of 1 may be squashed into all three sides equal to 1. various shapes. 1 1 1 1 1 1 1

Length and Perimeter s Three sided polygons are triangles. s s Triangles with three equal sides An equilateral triangle are call equilateral triangles. 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. For example, there is only Four-sided polygons with sides one triangular shape with equal of 1 may be squashed into all three sides equal to 1. various shapes. 1 1 1 1 1 1 1 Because of this, we say that “triangles are rigid”,

Length and Perimeter s Three sided polygons are triangles. s s Triangles with three equal sides An equilateral triangle are call equilateral triangles. 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. For example, there is only Four-sided polygons with sides one triangular shape with equal of 1 may be squashed into all three sides equal to 1. various shapes. 1 1 1 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”.

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

Length and Perimeter If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter. The perimeter of an equilateral triangle is P = s + s + s = 3s. b c a P=a+b+c

Length and Perimeter If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter. The perimeter of an equilateral triangle is P = s + s + s = 3s. b Rectangles are 4-sided polygons where the sides are joint at a right angle c as shown. a P=a+b+c

Length and Perimeter If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter. The perimeter of an equilateral triangle is P = s + s + s = 3s. b Rectangles are 4-sided polygons where the sides are joint at a right angle c as shown. a P=a+b+c

Length and Perimeter If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter. The perimeter of an equilateral triangle is P = s + s + s = 3s. b Rectangles are 4-sided polygons where the sides are joint at a right angle c as shown. A square a s P=a+b+c s s s

Length and Perimeter If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter. The perimeter of an equilateral triangle is P = s + s + s = 3s. b Rectangles are 4-sided polygons where the sides are joint at a right angle c as shown. A square a s P=a+b+c s s s A square is a rectangle with four equal sides. The perimeter of a squares is P = s + s + s + s = 4s

Length and Perimeter If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter. The perimeter of an equilateral triangle is P = s + s + s = 3s. b Rectangles are 4-sided polygons where the sides are joint at a right angle c as shown. A square a s P=a+b+c s s s A square is a rectangle with four equal sides. 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.

Length and Perimeter If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter. The perimeter of an equilateral triangle is P = s + s + s = 3s. b Rectangles are 4-sided polygons where the sides are joint at a right angle c as shown. A square a s P=a+b+c s s s A square is a rectangle with four equal sides. 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.

Length and Perimeter If the sides of a triangle are labeled as a, b, and c, then a + b + c = P, the perimeter. The perimeter of an equilateral triangle is P = s + s + s = 3s. b Rectangles are 4-sided polygons where the sides are joint at a right angle c as shown. A square a s P=a+b+c s s s A square is a rectangle with four equal sides. 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.

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

Length and Perimeter By the dictionary, “length” is the longest side and “width” is the horizontal side. This causes conflicts in labeling rectangles whose width is the longer side. length? width?

Length and Perimeter By the dictionary, “length” is the longest side and “width” is the horizontal side. This causes conflicts in labeling rectangles length? width? whose width is the longer side. Hence we will use the words “height” for the vertical side, and “width” for the horizontal side instead. width (w) h height (h) w

Length and Perimeter By the dictionary, “length” is the longest side and “width” is the horizontal side. This causes conflicts in labeling rectangles length? width? whose width is the longer side. Hence we will use the words “height” for the vertical side, and “width” for the horizontal side instead. width (w) The perimeter of a rectangle is P = 2h + 2w (= h + h + w + w) h height (h) w

Length and Perimeter By the dictionary, “length” is the longest side and “width” is the horizontal side. This causes conflicts in labeling rectangles length? width? whose width is the longer side. Hence we will use the words “height” for the vertical side, and “width” for the horizontal side instead. width (w) The perimeter of a rectangle is P = 2h + 2w (= h + h + w + w) h height (h) Example B. a. We drove in a loop as shown. How many miles did we travel? w Assume it’s an equilateral triangle on top. 5 mi 3 mi

Length and Perimeter By the dictionary, “length” is the longest side and “width” is the horizontal side. This causes conflicts in labeling rectangles length? width? whose width is the longer side. Hence we will use the words “height” for the vertical side, and “width” for the horizontal side instead. width (w) The perimeter of a rectangle is P = 2h + 2w (= h + h + w + w) h height (h) Example B. a. We drove in a loop as shown. How many miles did we travel? w Assume it’s an equilateral triangle on top. There are three 3-mile sections and two 5-mile sections. 5 mi 3 mi

Length and Perimeter By the dictionary, “length” is the longest side and “width” is the horizontal side. This causes conflicts in labeling rectangles length? width? whose width is the longer side. Hence we will use the words “height” for the vertical side, and “width” for the horizontal side instead. width (w) The perimeter of a rectangle is P = 2h + 2w (= h + h + w + w) h height (h) Example B. a. We drove in a loop as shown. How many miles did we travel? w Assume it’s an equilateral triangle on top. 5 mi 3 mi There are three 3-mile sections and two 5-mile sections. Hence one round trip P is P=3+3+3+5+5 = 3(3)+ 2(5)

Length and Perimeter By the dictionary, “length” is the longest side and “width” is the horizontal side. This causes conflicts in labeling rectangles length? width? whose width is the longer side. Hence we will use the words “height” for the vertical side, and “width” for the horizontal side instead. width (w) The perimeter of a rectangle is P = 2h + 2w (= h + h + w + w) h height (h) Example B. a. We drove in a loop as shown. How many miles did we travel? w Assume it’s an equilateral triangle on top. 5 mi 3 mi There are three 3-mile sections and two 5-mile sections. Hence one round trip P is P=3+3+3+5+5 = 3(3)+ 2(5) = 9 + 10 = 19 miles.

Length and Perimeter 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? 70 m 50 m

Length and Perimeter 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? 70 m We have three heights where each requires 50 meters of rope, 50 m

Length and Perimeter 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? 70 m We have three heights where each requires 50 meters of rope, and three widths where each 50 m requires 70 meters of rope.

Length and Perimeter 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? 70 m We have three heights where each requires 50 meters of rope, and three widths where each 50 m requires 70 meters of rope. Hence it requires 3(50) + 3(70) = 150 + 210 = 360 meters of rope.

Length and Perimeter 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? 70 m We have three heights where each requires 50 meters of rope, and three widths where each 50 m requires 70 meters of rope. Hence it requires 3(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

Length and Perimeter 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? 70 m We have three heights where each requires 50 meters of rope, and three widths where each 50 m requires 70 meters of rope. Hence it requires 3(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 The perimeter of the step-shape is the same as the perimeter of the rectangle that boxes it in as shown.

Length and Perimeter 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? 70 m We have three heights where each requires 50 meters of rope, and three widths where each 50 m requires 70 meters of rope. Hence it requires 3(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 The 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.

Length and Perimeter 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? 70 m We have three heights where each requires 50 meters of rope, and three widths where each 50 m requires 70 meters of rope. Hence it requires 3(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 The 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. So the height is 6 ft, the width is 10 ft, and the perimeter P = 2(6) +2(10) = 32 ft.

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