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Chapter 10 Geometry 10-2 Finding Perimiters and Circumferences The perimeter of a closed figure is the distance around its outside borders. - To find the perimeters of triangles, squares, rectangles, parallelograms, and trapezoids, simply add up the lengths of the sides.
The distance around a circle is not called perimiter. Instead, it is called the circumference. - You need to use a special formula, C = 2pr, to find the circumference of a circle. This is described later in this lesson.
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| The Perimeter of Triangles The perimieter of a triangle is equal to the sum of the lengths of its three sides. Equation  | The equation for the perimeter of any triangle is: P = a + b + c Where: - a, b, and c are the lengths of the three sides
- P is the perimeter
| | Example | Problem Find the perimeter of a triangle that has sides equal to 3 inches, 3 inches , and 4 inches. | 
| Solution The perimeter of any triangle is equal to the sum of the lengths of the sides. For this particular triangle: - Perimeter = 3 inches + 3 inches + 4 inches
- Perimeter = 10 inches
| Example The sides of a certain triangle measure 24, 14, and 20 units. What is the perimeter of this triangle? | Problem | | A triangle has sides equal to 24, 14, and 20 units in lengths. Find the perimeter. | Cite the appropriate equation | P = a + b + c | | Assign the given values | P = 24 + 14 + 20 | | Solve the equation | P = 58 | | | Solution | | The perimeter of this triangle is 58 units. | Exercises Click the ? symbol to see the correct answer. | 1. | The sides of a certain triangle measure 2 inches, 5 inches, and 3 inches. What is the perimeter of this triangle ? | | 2. | What is the perimeter of a trangle that has sides of 2.2 cm, 3 cm, and 4.5 cm ? | | 3. | Each side of an equilateral triangle measures 2¼ inches. What is the perimeter ? | | 4. | Two of the sides of an isosceles triangle are found to be 10 ft long. If the third side is 5 feet long, what is the perimeter of the triangle ? | | 5. | Side a of a certain triangle is 10 inches long. One of the other sides is twice as long as side a, and the third is half the length of side a. Whatis the perimeter of this triangle ? | | Occasionally you will have a situation where you already know the perimeter of a triangle, but not the length of one of the sides. How can you determine the length of that side? The general equation for the perimeter of a triangle is: P = a + b + c If you already know the perimeter (P) and the lengths of sides a and b, then you can solve the general equation for side c: c = P – (a + b) Example The perimeter of a certain triangle is 400 ft. One of the sides is 100 ft long and another is 25 feet long. What is the length of the third side? - c = P – (a + b)
- c = 400 – (100 + 25)
- c = 400 – 125
- c = 275 ft
| Thinking Mathematically  A triangle may have three sides of equal length. This is called an equilateral triangle. If each side is s units long, we can substitute s for each of the sides in the general equation for the perimeter of a triangle:
- P = a + b + c
- P = s + s + s
- P = 3s
 A triangle that has two equal sides is called an isosceles triangle. If we assign s to the two equal sides and w to the third side, we can transform the general equation for the perimeter of a triagle into one that applies only to isosceles triangles:
- P = a + b + c
- P = s + s + w
- P = 2s + w
| | Exercises Click the ? symbol to see the correct answer. | 1. | Two sides of a triangle measure 3 inches and 5 inches. If the perimeter is 10 inches, what is the length of the third side ? | | 2. | A closed plane figure with three sides has a known perimeter of 8 feet. If the lengths of two of the sides add up to 6 inches, what is the length of the third side ? | | 3. | What is the distance around a triangular space that measures 500 m, 180 m, 220 m ? | | 4. | The perimeter of an equilateral triangle is know to be 12 cm. What is the length of each side ? | | 5. | Two of the sides of an isosceles triangle are 4 mm long. If the perimeter is 15 mm, what is the length of the third side ? | The Perimeter of a Square All four sides of a square have the same length. You can find the perimeter of any plane figure by simply adding up the lengths of the sides: P = s + s + s + s. But since the four sides of a square have the same length, it is even simpler to find the perimter by multiplying the length of a side by 4: P = 4s. Equation 
| The equation for the perimeter of any square is: P = 4s Where: - s is the length of the sides
- P is the perimeter
| | Example A certain square measures 6 inches on each side. What is the perimeter of this square? | Problem | | Find the perimeter of a square when the sides measure 6 inches. | Cite the appropriate equation | P = 4s | | Assign the given values | P = 4 · 6 | | Solve the equation | P = 24 | | | Solution | | P = 24 inches | Exercises Click the ? symbol to see the correct answer. | 1. | Each side of a certain square if 4 inches long. What is the perimeter of this square ? | | 2. | What is the perimeter of a square that measured 10mm on each side ? | | 3. | A certain city block forms a perfect square. If you find you need 560 steps to walk along one side of this city square, how many steps do you need to walk all the way around the block ? | | 4. | How much wire do you need if you want to bend it into a square that measures 8 inches on each side ? | | 5. | What is the perimeter of a square that measures 4.7 mm per side ? | | Suppose you already know the perimeter of a square, but not the lengths of the sides. Is it possible to determine the length of each side of a square when you only know the perimeter? The general equation for the perimeter of a square is: P = 4s Solving this equation for s: - s = P/4
- or
- s = 0.25P
Example A certain square has a perimeter of 180 inches. What is the length of each side? - s = 0.25P
- s = 0.25 · 180
- s = 45
So each side of this particular square is 45 inches long | Thinking Mathematically Where does the equation P = 4s come from? - The perimeter of any closed four-sided figure is given by
P = a + b + c + d,
where a, b, c, and d are the lengths of the four sides. - The four sides of a square have the same length. Call this s.
- Substituting s for each of the terms in Step 1:
P = s + s + s + s - Gathering like terms:
P = 4s | | Exercises Click the ? symbol to see the correct answer. | 1. | = ? | | 2. | = ? | | 3. | = ? | | 4. | = ? | | 5. | = ? | The Perimeter of Rectangles Equation 
| The equation for the perimeter of any rectangle is: P = 2(l + w) Where: - l is the length of one pair of parallel sides
- w is the length of the second pair of parallel sides
- P is the perimeter
| | Example For a certain rectangle, one set of parallel sides is 8 cm long and the second set is 12 cm long. What is the perimeter of this rectangle? | Problem | | Find the perimeter of a rectangle when the pairs of parallel sides measure 8 cm and 12 cm. | Cite the appropriate equation | P = 2(l + w) | | Assign the given values | P = 2( 8 + 12 ) | | Solve the equation and simplify | P = 2( 20 )
P = 40 | | | Solution | | P = 40 cm | Exercises Click the ? symbol to see the correct answer. | 1. | Calculate the perimeter of a square that measures 2 inches on each side. ? | | 2. | Calculate the perimiter of a square that measures 10 feet to a side. ? | | 3. | For a certain square, s = 7. Calculate P. ? | | 4. | If one side of a square is 12½ feet long, determine the perimeter. ? | | 5. | One side of a square is 2.2 mm long. Calculate the perimeter. ? | | If you happen to know the perimeter of a rectangle and the length of just one side, you can determine the lengths of all four sides. How can this be? The general equation for a rectangle is: P = 2(a + b) If you know P and a, you can solve this equation for side b. Expanding the general equation: P = 2a + 2b Soving for b and simplifying: - P - 2a = 2b
- b = (P – 2a) / 2
- b = P/2 – 2a/2
- b = P/2 – a
| Thinking Mathematically Where does the equation P = 2(l + w) come from? - The perimeter of any closed four-sided figure is given by
P = a + b + c + d,
where a, b, c, and d are the lengths of the four sides. - If two sides are equal to l and two sides are equal to w, the equation becomes:
P = l + l + w + w - Gathering like terms and simplifying:
P = 2l + 2w
P = 2(l + w) | | Exercises Click the ? symbol to see the correct answer. | 1. | = ? | | 2. | = ? | | 3. | = ? | | 4. | = ? | | 5. | = ? | The Perimeter of Parallelograms Equation 
| The equation for the perimeter of any parallelogram is: P = 2(a + b) Where: - a is the length of one pair of parallel sides
- b is the length of the second pair of parallel sides
- P is the perimeter
| | Example The parallel sides of a certain parallelogram measure 16 ft and 24 ft. What is the perimeter of this particular parallelogram? | Problem | | Find the perimeter of a parallelogram when the pairs of parallel sides measure 16 ft and 24 ft. | Cite the appropriate equation | P = 2(a + b) | | Assign the given values | P = 2( 16 + 24) | | Solve the equation and simplify | P = 2( 40 )
P = 80 | | | Solution | | P = 80 ft | Exercises Click the ? symbol to see the correct answer. | 1. = ? | 2. = ? | 3. = ? | 4. = ? | 5. = ? | | 6. = ? | 7. = ? | 8. = ? | 9. = ? | 10. = ? | The Perimeter of Trapezoids Equation 
| The equation for the perimeter of any trapezoid is: P = a + b + c + d Where: - a, b, c, d are the lengths of the four sides
- P is the perimeter
| | Example The sides of a trapezoid measure 4 in, 6 in, 8 in, and 2 in. What is the perimeter of this trapezoid? | Problem | | Find the perimeter of a trapezoid when the sides measure 4 in, 6 in, 8 in, and 2 in | Cite the appropriate equation | P = a + b + c + d | | Assign the given values | P = 4 + 6 + 8 + 2 | | Solve the equation and simplify | P = 20 | | | Solution | | P = 20 in | Exercises Click the ? symbol to see the correct answer. | 1. = ? | 2. = ? | 3. = ? | 4. = ? | 5. = ? | | 6. = ? | 7. = ? | 8. = ? | 9. = ? | 10. = ? | The Circumference of Circles Equation 
| The equation for the circumference of any circle is: C = 2pr or C = pd Where: - p = approximately 3.14 or 22/7
- r is the radius of the circle
- d is the diameter of the circle
- C is the circumferance
| | Example What is the circumference of a circle that has a radius of 15 cm? | Problem | | What is the circumference of a circle that has a radius of 15 cm? | Cite the appropriate equation | C = 2pr | | Assign the given values | C = 2 · 3.14 · 15 | | Solve the equation and simplify | C = 94.2 | | | Solution | | C = 94.2 cm | Example What is the circumference of a circle that has a diameter of 100 ft? | Problem | | What is the circumference of a circle that has a diameter of 100 ft? | Cite the appropriate equation | C = pd | | Assign the given values | C = 3.14 · 100 | | Solve the equation and simplify | C = 314 | | | Solution | | C = 314 ft | Exercises Click the ? symbol to see the correct answer. | 1. = ? | 2. = ? | 3. = ? | 4. = ? | 5. = ? | | 6. = ? | 7. = ? | 8. = ? | 9. = ? | 10. = ? |
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