Perimeter of the shape will be sum of measures of all sides around the figure.
Area of the shape can be figured out using the formula which related the given shape.
For example,
Area of rectangle Area of square Area of triangle Area of parallelogram Area of trapezoid |
Length x width side x side (1/2) x base x height base x height (1/2) x h (a + b) |
Problem 1 :
What is the distance around the rectangle if the length is
3x^{2} + 6x – 10
and the width is
3x + 5 ?
Solution :
The given shape is a rectangle, to find the distance around the shape rectangle is similar to find perimeter of the rectangle.
Length = 3x^{2} + 6x – 10
Width = 3x + 5
Perimeter of rectangle = 2(length + width)
= 2[(3x^{2} + 6x – 10) + (3x + 5)]
= 2[3x^{2} + 6x + 3x - 10 + 5]
= 2[3x^{2} + 9x - 5]
= 6x^{2} + 18x - 10
So, the length around the shape is 6x^{2} + 18x - 10.
Problem 2 :
If the perimeter of the pentagon below is 7x^{4} + 9x^{3} – 6x^{2} + 10, what is the length of the missing side?
Solution :
Number of sides of the shape pentagon = 5
Perimeter = 7x^{4} + 9x^{3} – 6x^{2} + 10
Adding the side lengths,
(3x^{4} - 4) + (5x^{3} - 2x^{2}) + (4x^{3} + 6x) + (4x^{4} - 6x^{2 }- 3x) + unknown side = 7x^{4} + 9x^{3} – 6x^{2} + 10
(3x^{4}+ 4x^{4}+ 5x^{3} + 4x^{3}- 2x^{2} - 6x^{2} + 6x - 3x - 4) + unknown side = 7x^{4} + 9x^{3} – 6x^{2} + 10
(7x^{4}+ 9x^{3} - 8x^{2} + 3x - 4) + unknown side = 7x^{4} + 9x^{3} – 6x^{2} + 10
unknown side
= 7x^{4} + 9x^{3} – 6x^{2} + 10 - (7x^{4}+ 9x^{3} - 8x^{2} + 3x - 4)
= 7x^{4} - 7x^{4}+ 9x^{3 }- 9x^{3}– 6x^{2} - 8x^{2} + 3x + 10 - 4
= -14x^{2} + 3x + 6
Problem 3 :
If the perimeter of the square below is 12x^{5} – 8x^{2} + 20x – 4, what is the length of one side?
Solution :
Perimeter of square = 12x^{5} – 8x^{2} + 20x – 4
Number of equal sides in the shape square = 4
Side length of square = Perimeter of square / 4
= (12x^{5} – 8x^{2} + 20x – 4) / 4
Factoring 4 from the numerator, we get
= 4(3x^{5} – 2x^{2} + 5x – 1) / 4
= 3x^{5} – 2x^{2} + 5x – 1
Problem 4 :
Ana knows that the perimeter of her backyard is (6x^{2} + 14x) feet. If the length of her backyard is (2x^{2} + 3x – 7) feet, what is the width of her backyard?
Solution :
Let w be the width of backyard. But it is not clear that what shape it is exactly.
Perimeter = 6x^{2} + 14x
Length = 2x^{2} + 3x – 7
2[(2x^{2} + 3x – 7) + w] = 6x^{2} + 14x
Dividing by 2 on both sides
[(2x^{2} + 3x – 7) + w] = (6x^{2} + 14x)/2
[(2x^{2} + 3x – 7) + w] = 3x^{2} + 7x
w = (3x^{2} + 7x) - (2x^{2} + 3x – 7)
w = 3x^{2} + 7x - 2x^{2} - 3x + 7
By combining like terms , we get
w = 3x^{2 }- 2x^{2} + 7x - 3x + 7
w = x^{2}+ 4x + 7
So, the required width of the rectangle is x^{2}+ 4x + 7.
Problem 5 :
The area of the square below is represented by the expression 4x^{2} + 4x + 1. The area of the rectangle is represented by the expression x^{2} – 5x + 6. Using the diagram below, find the area of the shaded region.
Solution :
Area of the square = 4x^{2} + 4x + 1
Area of the rectangle = x^{2} – 5x + 6
Area of unshaded region
= Area of the square - area of rectangle
= (4x^{2} + 4x + 1) - (x^{2} – 5x + 6)
= 4x^{2} - x^{2}+ 4x + 5x + 1 - 6
= 3x^{2} + 9x - 5
Problem 6 :
A rectangular piece of wood has an area of
5x^{4} + 3x^{2} – 6x + 8
If two identical circles are cut out of the wood and the area of EACH circle is
x^{2} – 2
find the area of the remaining piece of wood. (Hint: Use the picture below.)
Solution :
Area of rectangular piece of wood = 5x^{4} + 3x^{2} – 6x + 8
Area of one circle = x^{2} – 2
Area of the remaining piece of wood
= Area of rectangle - 2(area of circle)
= 5x^{4} + 3x^{2} – 6x + 8 - 2(x^{2} – 2)
= 5x^{4} + 3x^{2} – 6x + 8 - 2x^{2} + 4
= 5x^{4} + 3x^{2} - 2x^{2}– 6x + 8 + 4
= 5x^{4} + x^{2} – 6x + 12
So, area of the shaded part is 5x^{4} + x^{2} – 6x + 12.
Problem 7 :
A circular plot of land has an area of
7x^{5} – x^{3} + 4x^{2} + 9
If the walkway around this piece of land has an area of
x^{4} – 4x^{3} + 2x
what is the area of the land and walkway combined?
Solution :
Area of circular land = 7x^{5} – x^{3} + 4x^{2} + 9
Area of walkway = x^{4} – 4x^{3} + 2x
Combined area = (7x^{5} – x^{3} + 4x^{2} + 9) +(x^{4} – 4x^{3} + 2x)
= 7x^{5} + x^{4}– x^{3} – 4x^{3 }+ 4x^{2 }+ 2x + 9
= 7x^{5} + x^{4}– 5x^{3} + 4x^{2 }+ 2x + 9
Problem 8 :
The width of Adrian’s bedroom is (x – 5) feet. He knows that the length is four times the width. Find the perimeter of Adrian’s bedroom.
Solution :
Width of the bedroom = x - 5
Length = 4(width)
= 4(x - 5)
Perimeter of the bedroom = 2(length + width)
= 4(x - 5) + (x - 5)
= 4x - 20 + x - 5
= (5x - 25) feet
So, the required perimeter is (5x - 25) feet.
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