Mathematics
Grade 10
15 min
Area and perimeter: word problems
Area and perimeter: word problems
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1
Introduction & Learning Objectives
Learning Objectives
Translate a word problem describing a geometric shape into a coordinate plane representation.
Calculate the lengths of the sides of a polygon using the Distance Formula given the coordinates of its vertices.
Determine the perimeter of polygons in the coordinate plane by summing side lengths.
Calculate the area of triangles and quadrilaterals in the coordinate plane using appropriate formulas or methods.
Apply properties of parallel and perpendicular lines to identify types of polygons and simplify area calculations.
Solve multi-step word problems involving area and perimeter in real-world contexts represented on a coordinate plane.
Ever wondered how a city planner calculates the exact area of a new park using just a map? 🗺️ Let's explore how coordina...
2
Key Concepts & Vocabulary
TermDefinitionExample
Coordinate PlaneA two-dimensional plane formed by the intersection of a horizontal line called the x-axis and a vertical line called the y-axis.The point (3, -2) is located 3 units to the right of the origin and 2 units down.
VerticesThe corner points of a polygon. In the coordinate plane, each vertex is represented by an ordered pair (x, y).A triangle might have vertices at A(1, 5), B(4, 1), and C(-2, 1).
PerimeterThe total distance around the outside of a closed two-dimensional figure. It is found by adding the lengths of all its sides.For a triangle with side lengths 3, 4, and 5 units, the perimeter is 3 + 4 + 5 = 12 units.
AreaThe measure of the space enclosed within a two-dimensional figure, expressed in square units.A rectangle with a length of 5 units and a wi...
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Core Formulas
Distance Formula
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
Use this formula to find the length of a line segment between two points, (x₁, y₁) and (x₂, y₂). This is the primary tool for finding the side lengths of a polygon needed to calculate its perimeter.
Shoelace Formula (for Area of a Polygon)
Area = \frac{1}{2} |(x_1y_2 + x_2y_3 + ... + x_ny_1) - (y_1x_2 + y_2x_3 + ... + y_nx_1)|
Use this formula to find the area of any simple polygon when you know the coordinates of its vertices. List the vertices in counter-clockwise order, repeating the first vertex at the end of the list.
Slope Formula
m = \frac{y_2 - y_1}{x_2 - x_1}
Use this formula to determine the relationship between sides of a polygon. Parallel sides have equal slopes. Perpendicular sides have slopes tha...
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Challenging
A rectangular field is mapped on a coordinate grid where 1 unit represents 5 meters. The vertices are at A(0,0), B(20,0), C(20,10), and D(0,10). A landscaper charges $2 per square meter to plant grass. One student calculates the cost as 200 units² * 5 m/unit * $2/m² = $2000. What is their error?
A.They confused area with perimeter.
B.They used the linear scale factor for an area calculation.
C.They made a sign error in the distance formula.
D.They calculated the area of the field incorrectly in square units.
Challenging
A city park is a large rectangle with vertices at (0, 0), (10, 0), (10, 8), and (0, 8). Inside the park, a triangular pond has been built with vertices at (1, 1), (4, 1), and (1, 5). What is the area of the grass in the park, excluding the pond?
A.80 square units
B.74 square units
C.77 square units
D.6 square units
Challenging
A triangular plot of land has vertices at A(1, 5), B(6, 2), and C(k, 2). If the area of the plot is 12 square units, what is a possible value for k?
A.-2
B.0
C.4
D.8
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