Mathematics
Grade 9
15 min
Perimeter: Set 1
Perimeter: Set 1
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1
Introduction & Learning Objectives
Learning Objectives
Calculate the perimeter of polygons with side lengths represented by numerical values, variables, and polynomial expressions.
Set up and solve algebraic equations to find unknown side lengths when the total perimeter is given.
Determine the perimeter of composite shapes by identifying and summing all exterior sides.
Calculate the perimeter of polygons on a coordinate plane by applying the distance formula.
Simplify expressions involving radicals to find the exact perimeter of a shape.
Translate real-world scenarios into perimeter problems and solve them.
Distinguish between the concepts of perimeter and area and apply the correct formula for each.
Ever wondered how much fencing you'd need for a uniquely shaped garden, or how to frame a custom piece...
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Key Concepts & Vocabulary
TermDefinitionExample
PerimeterThe total distance around the exterior of a two-dimensional closed figure. It is a one-dimensional measurement (e.g., cm, m, inches).A rectangle with sides of 5 cm and 3 cm has a perimeter of 5 + 3 + 5 + 3 = 16 cm.
PolygonA closed two-dimensional shape made up of straight line segments.Triangles, squares, pentagons, and octagons are all polygons.
Composite ShapeA figure made by combining two or more basic geometric shapes. Its perimeter is the sum of the lengths of its outer boundary only.An L-shaped figure can be seen as two rectangles joined together. To find its perimeter, you would add the lengths of the six outer sides.
Polynomial ExpressionAn algebraic expression consisting of variables, coefficients, and exponents, combined using addition, subtraction...
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Core Formulas
Perimeter of a General Polygon
P = s_1 + s_2 + s_3 + ... + s_n
The perimeter (P) of any polygon is the sum of the lengths of all its sides (s_1, s_2, etc.). Use this for any shape with straight sides, especially irregular ones.
Perimeter of a Rectangle
P = 2l + 2w
The perimeter (P) of a rectangle is twice the length (l) plus twice the width (w). This is a shortcut for adding all four sides.
Distance Formula
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
Used to find the distance (d) between two points (x_1, y_1) and (x_2, y_2) on a coordinate plane. This allows you to calculate the length of each side of a polygon before summing them for the perimeter.
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Challenging
Find the perimeter of a quadrilateral with vertices at A(-2, 1), B(3, 3), C(5, -2), and D(0, -4).
A.4√31
B.4√29
C.2√29 + 2√31
D.√29 + √34 + √29 + √41
Challenging
The perimeter of a pentagon is given by the polynomial 10x² + 5x - 7. Four of its sides are (2x²), (x² + 3x), (4x² - 1), and (x + 5). What is the expression for the fifth side?
A.3x² + 2x - 11
B.3x² + x + 1
C.7x² + 4x + 4
D.3x² + x - 11
Challenging
An L-shaped figure is formed from a larger rectangle by removing a smaller rectangle from one corner. The outer dimensions are 5x and 3x. The inner corner's dimensions are 2x and x, as measured from the opposite outer edges. If the total perimeter is 96 units, what is the value of x?
A.4
B.5
C.6
D.8
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