In general, you can find the perimeter of a rectangle by adding its length and width, and then multiplying by 2:
perimeter = 2(length + width)
Perimeter
Addding up the sides
The perimeter of any figure is the distance around it. A football field is a rectangle that is 360 feet long and 160 feet wide. If you walk all the way around a football field, you walk a distance of 360 ft + 160 ft + 360 ft + 160 ft = 1040 ft.
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In general, you can find the perimeter of a rectangle by adding its length and width, and then multiplying by 2: perimeter = 2(length + width) |
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Sometimes you have to find the lengths of missing sides. When those sides are parallel to other sides, you can usually think of "shadowing" the missing sides on the other side.
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For example, in the figure on the left, you don't have to know the values of a, b, c, and d, since the heights b and d add up to 12 (the height on the left) and the lengths a and c add up to 30. So the perimeter is 12 + 30 + 12 + 30 = 84 units. |
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When a figure contains slanting lines, you may have to use the Pythagorean Theorem to find their lenghts. For example, to find the perimeter of the rhombus on the right, you would first have to find the length of the side marked x. Click on the image to see how this is done. The perimeter is 15 + 18 + 17 + 26 = 76 units.
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For example, to find the perimeter of the figure on the right, you have to add the perimeter of the semicircle on the right to the three straight sides. The semicircle's perimeter is half the circumference of a circle with diameter 8, so the perimeter of the figure is 14 + 8 + 14 + 4π = 36 + 4π = 48.57 units. Click on the image for details. |
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Using algebra
The following problems illustrate various roles of algebra in solving perimeter problems.
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The rectangle on the right consists of 5 squares of equal size. The area of the whole figure is 180 square centimeters. What is the perimeter of the rectangle? |
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If we let x represent the side of a square, then the area of the whole figure is 5x2. Since this is given as 180 cm2, 5x2 = 180 so x2 = 180 ÷ 5 = 36. Thus, x = 6 so the length of the rectangle is 5x6 = 30 cm and the width is 5 cm. Therefore the perimeter is 2(30 + 5) = 70 cm. |
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A thin wire 48 cm long is formed into a rectangle. If rectangle is twice as long as it is wide, then what is its area?
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If we let x be the width of the rectangle, then 2x is its length. Its perimeter is 2(x + 2x), and this is the length of the wire. So 6x = 48 cm and therefore x = 8 cm. The rectangle measures 8 cm by 16 cm, so its area is the product of these, or 128 cm2. |
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A side of one square is 5 m longer than twice a side of another square. The perimeter of the first square is 32 m greater than the perimeter of the second. What is area of the larger square?
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If we let x be the length of a side of the shorter square, then 2x + 5 is the length of a side of the larger square. The perimeters of these squares are 4(2x + 5) and 4x, so we can solve the following equation: 4(2x + 5) = 4x + 32 Distribute on the left: 8x + 20 = 4x + 32 Subtract 4x from both sides: 4x + 20 = 32 Subtract 20 from both sides: 4x = 12 Divide by 4: x = 3 A side of the larger square is 2(3) + 5 = 11 m, so its area is 121 m2 |
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The inside track of a circluar race track is exactly 1 mile long. The racetrack is 12 feet wide. How much longer is the outside track than the iside track?
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If the radius of the inside track is r then the radius of the outside track is r + 12. The circumference of the inside track is 5280 feet, so we know (since C = 2πr) 2πr = 5280. Thererore r = 5280/(2π) = 840.34 ft. So the radius of the outside track is 852.34 ft, and therefore its circumference is 2π(852.34)= 5355.40 ft. The outside track is therefore 5355.40 − 5280 = 75.40 feet longer than the inside track. A quicker approach is to observe that the difference of the circumferences is 2π(r + 12) − 2πr = 2πr + 24π − 2πr = 24π. The answer is independent of r! No matter what the circumference of the inside track, if the width to the outside track is 12 feet, then that track will be 75.4 feet longer than the inside track. |
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