Coordinate Geometry Applications and Proofs
3. Area of Coordinate Figures
Learning outcomes
- I can calculate the area of polygons using coordinates.
- I can decompose complex figures into simpler shapes to find area.
- I can apply coordinate geometry techniques to area problems.
- I can compare areas of different coordinate figures.
- I can solve real-world problems involving area on the coordinate plane.
Area of Coordinate Figures
When a polygon is drawn on a coordinate plane, its coordinates can be used to calculate its area. Sometimes the dimensions can be read directly from the grid. In other situations, we need to use coordinate geometry to calculate lengths, heights, or missing dimensions.
Complex figures can also be divided, or decomposed, into simpler shapes whose areas are easier to calculate.
This makes coordinate geometry useful for solving area problems involving:
- rectangles
- triangles
- parallelograms
- trapezoids
- irregular polygons
- composite figures
- real-world maps and designs
What Area Measures
Area measures the amount of two-dimensional space enclosed by a figure.
Area is measured in square units.
Examples include:
- cm²
- m²
- km²
- square coordinate units
If a rectangle is 6 m long and 4 m wide:
A = 6 × 4
A = 24 m²
The squared unit is important because area measures a two-dimensional region.
Area vs Perimeter
Area and perimeter describe different properties.
Perimeter measures the distance around a figure.
Area measures the space inside a figure.
For a rectangle measuring 8 units by 5 units:
Perimeter:
P = 2(8) + 2(5)
P = 26 units
Area:
A = 8 × 5
A = 40 square units
Do not confuse the two measurements.
Reading Dimensions from Coordinates
Suppose a rectangle has vertices:
A(2, 3)
B(9, 3)
C(9, 8)
D(2, 8)
Because AB is horizontal:
length = 9 − 2
length = 7
Because BC is vertical:
width = 8 − 3
width = 5
Therefore:
A = 7 × 5
A = 35 square units
Horizontal and Vertical Distances
For points on the same horizontal line, subtract their x-coordinates.
For:
A(−3, 5)
B(7, 5)
AB = |7 − (−3)|
AB = 10
For points on the same vertical line, subtract their y-coordinates.
For:
C(4, −2)
D(4, 6)
CD = |6 − (−2)|
CD = 8
Absolute value ensures that the length is positive.
Area of a Rectangle
The area formula for a rectangle is:
A = lw
where:
- l = length
- w = width
Suppose a rectangle has vertices:
(−2, 1)
(6, 1)
(6, 5)
(−2, 5)
Length:
6 − (−2) = 8
Width:
5 − 1 = 4
Therefore:
A = 8 × 4
A = 32 square units
Rectangles Do Not Need to Start at the Origin
A common mistake is thinking the coordinates themselves are the dimensions.
Consider a rectangle extending from:
x = 7 to x = 15
and:
y = 4 to y = 10
Its dimensions are:
length = 15 − 7 = 8
width = 10 − 4 = 6
Therefore:
A = 8 × 6
A = 48 square units
The coordinates tell us where the rectangle is located. Their differences tell us its dimensions.
Area of a Square
The area of a square is:
A = s²
where s is the side length.
Suppose a square has vertices:
A(1, 2)
B(6, 2)
C(6, 7)
D(1, 7)
Side length:
6 − 1 = 5
Therefore:
A = 5²
A = 25 square units
Area of a Triangle
The area of a triangle is:
A = 1/2 bh
where:
- b = base
- h = perpendicular height
The height must be measured perpendicular to the base.
Triangle with a Horizontal Base
Consider:
A(1, 2)
B(9, 2)
C(5, 7)
The base AB is horizontal.
Base:
b = 9 − 1
b = 8
The perpendicular height is the vertical distance from C to the line y = 2:
h = 7 − 2
h = 5
Therefore:
A = 1/2(8)(5)
A = 20 square units
Triangle with a Vertical Base
Consider:
A(2, 1)
B(2, 9)
C(7, 5)
AB is vertical.
Base:
b = 9 − 1
b = 8
The perpendicular height is the horizontal distance from C to x = 2:
h = 7 − 2
h = 5
Therefore:
A = 1/2(8)(5)
A = 20 square units
The Height Is Not Necessarily a Side
Consider a triangle where the base is horizontal but the third vertex is not directly above either endpoint.
The vertical distance from the third vertex to the line containing the base is still the height.
The height is always:
perpendicular distance from a vertex to the opposite side or its extended line
It does not need to be one of the triangle's sides.
Area of a Right Triangle
For a right triangle, the two perpendicular sides can be used directly as the base and height.
Suppose:
A(1, 1)
B(7, 1)
C(1, 5)
AB = 6
AC = 4
Since AB ⟂ AC:
A = 1/2(6)(4)
A = 12 square units
Area of a Parallelogram
The area of a parallelogram is:
A = bh
where:
- b = base
- h = perpendicular height
The height is not generally the slanted side length.
Suppose a parallelogram has vertices:
A(1, 2)
B(7, 2)
C(9, 6)
D(3, 6)
Base AB:
b = 6
The vertical distance between:
y = 2
and:
y = 6
is:
h = 4
Therefore:
A = 6 × 4
A = 24 square units
Why the Slanted Side Is Not the Height
A parallelogram may have a slanted side longer than its perpendicular height.
Area depends on the perpendicular separation between the two parallel bases.
Therefore:
A = base × perpendicular height
not:
base × slanted side
unless that side happens to be perpendicular to the base.
Area of a Trapezoid
The area of a trapezoid is:
A = 1/2(b₁ + b₂)h
where:
- b₁ and b₂ are the parallel bases
- h is the perpendicular distance between them
Suppose a trapezoid has vertices:
A(1, 1)
B(9, 1)
C(7, 5)
D(3, 5)
The bases are:
AB = 8
CD = 4
Height:
h = 4
Therefore:
A = 1/2(8 + 4)(4)
A = 1/2(12)(4)
A = 24 square units
Decomposing Complex Figures
An irregular or complex polygon can often be divided into simpler shapes.
This is called decomposition.
Possible simpler shapes include:
- rectangles
- squares
- triangles
- parallelograms
- trapezoids
Find the area of each piece and then add the areas.
Example: Decomposing an L-Shaped Figure
Suppose an L-shaped figure can be divided into:
- Rectangle A: 8 × 3
- Rectangle B: 4 × 5
Rectangle A:
A₁ = 8 × 3
A₁ = 24
Rectangle B:
A₂ = 4 × 5
A₂ = 20
Total:
A = 24 + 20
A = 44 square units
Subtracting Missing Regions
Sometimes it is easier to imagine a complex figure as a large rectangle with a smaller piece removed.
Suppose an L-shaped figure fits inside a:
10 × 8
rectangle.
A missing rectangular section measures:
4 × 3
Outer rectangle:
A_outer = 10 × 8
A_outer = 80
Missing region:
A_missing = 4 × 3
A_missing = 12
Therefore:
A = 80 − 12
A = 68 square units
Addition or Subtraction?
There are often several correct ways to calculate the area of a composite figure.
You can:
decompose the figure and add the pieces
or:
enclose the figure in a larger shape and subtract unwanted regions
Choose whichever method produces the simplest calculations.
Bounding Rectangles
A bounding rectangle is a rectangle drawn around an irregular polygon.
The polygon's area can sometimes be found by:
- finding the area of the bounding rectangle
- finding the areas outside the polygon but inside the rectangle
- subtracting those unwanted areas
This is especially useful for irregular coordinate polygons.
Worked Bounding Rectangle Example
Suppose a triangle has vertices:
A(1, 1)
B(7, 1)
C(4, 5)
It fits inside a rectangle from:
x = 1 to x = 7
and:
y = 1 to y = 5
Rectangle dimensions:
6 × 4
Rectangle area:
24
The two regions outside the triangle are right triangles.
Each has:
base = 3
height = 4
Area of each:
A = 1/2(3)(4)
A = 6
Total outside area:
6 + 6 = 12
Therefore:
triangle area = 24 − 12
A = 12 square units
This agrees with the direct calculation:
1/2(6)(4) = 12.
Area of an Irregular Polygon
Suppose a polygon has vertices:
A(1, 1)
B(7, 1)
C(7, 4)
D(4, 7)
E(1, 4)
One strategy is to divide it into:
- a rectangle
- a triangle
The lower rectangle extends from:
x = 1 to x = 7
and:
y = 1 to y = 4
Rectangle dimensions:
6 × 3
Rectangle area:
18
The upper triangle has:
base = 6
height = 3
Triangle area:
1/2(6)(3)
= 9
Total:
18 + 9
A = 27 square units
Drawing Helpful Lines
When decomposing a coordinate figure, it is often useful to draw imaginary:
- horizontal lines
- vertical lines
- diagonals
These lines can create familiar shapes.
The added lines do not change the total area. They simply make the area easier to calculate.
Using Distance in Area Problems
Not every polygon has horizontal or vertical sides.
Suppose a square has adjacent vertices:
A(1, 1)
B(4, 5)
The side length is:
AB = √[(4 − 1)² + (5 − 1)²]
AB = √(3² + 4²)
AB = 5
Because the figure is known to be a square:
A = s²
A = 5²
A = 25 square units
Coordinate geometry provided the side length needed for the area calculation.
Using Slope Before Calculating Area
Sometimes we first need to verify that two segments are perpendicular.
Suppose:
A(0, 0)
B(4, 2)
C(2, 6)
Slope AB:
m_AB = 2/4
m_AB = 1/2
Slope BC:
m_BC = (6 − 2)/(2 − 4)
m_BC = 4/(−2)
m_BC = −2
Since:
(1/2)(−2) = −1
AB ⟂ BC.
Therefore, if these segments form adjacent sides of a rectangle, their lengths can be used as length and width.
This shows how slope and distance can work together in area problems.
Area of a Rotated Rectangle
Consider a rectangle with:
A(0, 0)
B(4, 2)
C(2, 6)
D(−2, 4)
Find AB:
AB = √(4² + 2²)
AB = √20
AB = 2√5
Find BC:
BC = √[(2 − 4)² + (6 − 2)²]
BC = √(4 + 16)
BC = 2√5
The adjacent sides are perpendicular because their slopes are:
1/2 and −2.
Therefore:
A = AB × BC
A = (2√5)(2√5)
A = 4(5)
A = 20 square units
A rotated figure can be analyzed just as precisely as an axis-aligned figure.
Area of a Rotated Parallelogram
For a rotated parallelogram, finding a perpendicular height may require more reasoning.
Possible strategies include:
- decomposing the figure
- enclosing it in a rectangle
- using coordinate relationships
- dividing it into triangles
For introductory coordinate problems, decomposition is often the simplest approach.
Dividing a Quadrilateral into Triangles
Any convex quadrilateral can be divided into two triangles by drawing a diagonal.
For quadrilateral ABCD, draw diagonal AC.
Then:
Area ABCD = Area ABC + Area ACD
This method is particularly useful when the two triangles have convenient bases and heights.
Dividing Larger Polygons into Triangles
A pentagon can often be divided into three triangles.
A hexagon can often be divided into four triangles.
In general, a convex polygon with n sides can be divided from one vertex into:
n − 2 triangles
For example:
Pentagon:
5 − 2 = 3 triangles
Hexagon:
6 − 2 = 4 triangles
This can provide a useful strategy for area calculations.
Comparing Areas
Coordinate geometry makes it possible to compare figures even when they have different shapes.
Suppose:
Rectangle A:
length = 8
width = 5
Area:
40 square units
Triangle B:
base = 10
height = 8
Area:
1/2(10)(8)
= 40 square units
The figures have different shapes but:
equal areas
Equal area does not mean that figures are congruent.
Same Perimeter, Different Area
Two figures can have the same perimeter but different areas.
Rectangle A:
6 × 4
Perimeter:
2(6) + 2(4) = 20
Area:
24
Rectangle B:
8 × 2
Perimeter:
2(8) + 2(2) = 20
Area:
16
Both have perimeter 20 units, but their areas are different.
Same Area, Different Perimeter
Two figures can also have equal areas but different perimeters.
Rectangle A:
8 × 3
Area:
24
Perimeter:
22
Rectangle B:
6 × 4
Area:
24
Perimeter:
20
Area and perimeter measure different geometric properties.
Comparing Areas Using Ratios
Suppose:
Figure A has area 30 m².
Figure B has area 45 m².
The ratio is:
45/30
= 1.5
Therefore:
Figure B has 1.5 times the area of Figure A.
We could also say:
Figure B is:
(45 − 30)/30 × 100%
= 50%
50% larger in area
Scale and Area
Scale has an especially important effect on area.
Suppose every length of a figure is doubled.
The area does not merely double.
For a square:
Original side = 3
Original area:
3² = 9
New side:
6
New area:
6² = 36
Therefore:
doubling every length multiplies area by 4
General Scale-Factor Rule
If every linear dimension is multiplied by a scale factor k:
area is multiplied by k²
Examples:
| Linear Scale Factor | Area Scale Factor |
|---|---|
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 1/2 | 1/4 |
This is important when interpreting maps, plans, diagrams, and models.
Map Example
Suppose a coordinate map uses:
1 coordinate unit = 20 m
A rectangular field measures:
6 coordinate units by 4 coordinate units.
Actual dimensions:
6 × 20 = 120 m
4 × 20 = 80 m
Actual area:
A = 120 × 80
A = 9600 m²
Notice that one square coordinate unit represents:
20 m × 20 m
= 400 m²
Why Area Scale Is Squared
A scale factor affects two dimensions.
If one coordinate unit represents 20 m, then one square coordinate unit represents:
20 m × 20 m
= 400 m²
Therefore, we must square the linear scale when converting area.
This is a common source of mistakes in real-world coordinate problems.
Floor Plans
Coordinate geometry can model rooms and buildings.
Suppose a room is represented by:
A(2, 1)
B(10, 1)
C(10, 7)
D(2, 7)
If one coordinate unit represents 1 m:
length = 8 m
width = 6 m
Area:
48 m²
This could be used to estimate:
- flooring
- carpet
- tiles
- paint coverage for floors
- construction materials
Land and Property Areas
Property boundaries can sometimes be represented by coordinate polygons.
For a simple rectangular lot with corners:
(10, 5)
(40, 5)
(40, 25)
(10, 25)
dimensions are:
30 m × 20 m
Therefore:
area = 600 m²
Irregular properties can be decomposed into simpler shapes.
Architecture and Construction
Area calculations are essential when estimating quantities such as:
- flooring
- roofing
- paving
- wall panels
- glass
- insulation
- landscaping materials
Coordinate models can represent complicated designs and allow their areas to be calculated systematically.
Example: Paved Courtyard
A courtyard is represented by a rectangle:
12 m × 9 m
A rectangular garden:
4 m × 3 m
is located inside it and will not be paved.
Total courtyard area:
12 × 9 = 108 m²
Garden area:
4 × 3 = 12 m²
Paved area:
108 − 12
= 96 m²
Material Cost
Suppose paving costs:
$35 per m²
For the 96 m² courtyard:
cost = 96 × 35
cost = $3360
Area calculations often become the first step in a larger real-world problem.
Technology and Computer Graphics
Coordinate areas are useful in:
- computer graphics
- digital mapping
- game design
- image analysis
- geographic information systems
- computer-aided design
Digital objects are often represented using vertices.
Software can then calculate the area enclosed by those coordinates.
Science Applications
Area calculations appear in science when studying:
- habitat regions
- cross-sectional areas
- surface measurements
- experimental plots
- maps
- ecological sampling areas
A coordinate model can simplify an irregular physical region into a polygon that can be measured mathematically.
Estimating Irregular Areas
Real-world boundaries are not always straight.
A lake, forest, or habitat may have a curved irregular boundary.
One approach is to approximate the boundary with a polygon.
More vertices usually allow the polygon to follow the real boundary more closely.
The polygon's area then provides an:
estimate of the actual area
The result should not be treated as perfectly exact.
Coordinate Area and Measurement Error
Real-world coordinate data may contain measurement uncertainty.
For example, GPS coordinates or surveying measurements may not be exact.
This means the calculated polygon area may also contain uncertainty.
Mathematical calculations can be exact for the coordinates given while the real-world model remains approximate.
A Complete Composite-Figure Example
Suppose an L-shaped floor plan has vertices:
A(0, 0)
B(10, 0)
C(10, 4)
D(6, 4)
E(6, 8)
F(0, 8)
Enclose the figure in a:
10 × 8 rectangle.
Outer area:
10 × 8 = 80
The missing upper-right rectangle has:
width = 10 − 6 = 4
height = 8 − 4 = 4
Missing area:
4 × 4 = 16
Therefore:
A = 80 − 16
A = 64 square units
Checking the Same Answer by Decomposition
The same L-shape can be divided into two rectangles.
Bottom rectangle:
10 × 4 = 40
Upper-left rectangle:
6 × 4 = 24
Total:
40 + 24
= 64 square units
Both methods produce the same result.
Using a second method can be an effective way to check an answer.
A Complete Coordinate Geometry Example
A square has vertices:
A(1, 2)
B(4, 6)
C(8, 3)
D(5, −1)
First find AB:
AB² = (4 − 1)² + (6 − 2)²
AB² = 9 + 16
AB² = 25
Therefore:
AB = 5
If the coordinate relationships confirm that ABCD is a square, its area is:
A = 5²
A = 25 square units
This illustrates how classification and area can work together.
A Complete Real-World Example
A park is represented on a coordinate map by:
A(0, 0)
B(12, 0)
C(12, 5)
D(8, 9)
E(0, 9)
One coordinate unit represents:
10 m
Decompose the park into:
- a lower rectangle
- an upper rectangle
- an upper-right triangle
Lower rectangle:
12 × 5 = 60 square coordinate units
Upper rectangle:
8 × 4 = 32 square coordinate units
Upper-right triangle:
base = 4
height = 4
Area:
1/2(4)(4) = 8 square coordinate units
Total coordinate area:
60 + 32 + 8
= 100 square coordinate units
Each square coordinate unit represents:
10² = 100 m²
Therefore:
actual area = 100 × 100
= 10,000 m²
Choosing an Efficient Area Strategy
Before calculating, examine the figure.
Ask:
- Is it a familiar shape?
- Are the sides horizontal or vertical?
- Is there an obvious base and height?
- Can it be divided into rectangles or triangles?
- Would a bounding rectangle make the problem easier?
- Do I need the distance formula?
- Do I need slope to verify perpendicularity?
- Is there a scale conversion?
- Is the answer exact or an estimate?
Choosing the right strategy often saves more time than performing calculations immediately.
A Reliable Strategy for Coordinate Area Problems
- Plot or visualize the vertices in order.
- Identify the type of figure.
- Look for horizontal and vertical distances.
- Decide whether a standard area formula can be used.
- If necessary, calculate lengths using the distance formula.
- Verify perpendicular relationships when needed.
- Decompose complex figures into simpler shapes.
- Calculate each area carefully.
- Add or subtract areas as appropriate.
- Apply any coordinate scale.
- Include square units.
- Check whether the answer is reasonable.
Checking Whether an Answer Is Reasonable
A calculated area should make geometric sense.
For example, if a polygon fits entirely inside a:
10 × 8 rectangle
then its area cannot be greater than:
80 square units
Similarly, area cannot be negative.
Bounding figures provide a useful way to estimate the expected size of an answer before completing detailed calculations.
Common Mistakes
Confusing area with perimeter
Area measures the region inside the polygon. Perimeter measures the distance around it.
Forgetting square units
Write:
24 m²
not 24 m.
Using coordinates directly as dimensions
Use differences between coordinates to find lengths.
Forgetting absolute value
A length cannot be negative.
Using a slanted side as the height
Height must be perpendicular to the selected base.
Forgetting the 1/2 in triangle area
Triangle area is:
A = 1/2 bh
Using only one base in the trapezoid formula
Use:
A = 1/2(b₁ + b₂)h
Double-counting decomposed regions
The simpler pieces should cover the original polygon exactly without overlapping.
Leaving gaps when decomposing
All parts of the original figure must be included.
Subtracting the wrong region
When using a bounding rectangle, subtract only regions outside the required polygon.
Taking unnecessary square roots
Squared distances can be useful when comparing lengths, although actual lengths may be needed for area.
Applying a linear scale directly to area
If lengths scale by k, areas scale by:
k²
Assuming an irregular real-world boundary is exact
Coordinate polygon models may only approximate the actual region.
Key Terms
Area — amount of two-dimensional space enclosed by a figure.
Square unit — unit used to measure area, such as cm² or m².
Perimeter — total distance around a polygon.
Base — selected side used when calculating the area of a triangle, parallelogram, or trapezoid.
Height — perpendicular distance from a base to the opposite vertex or parallel side.
Decomposition — dividing a complex figure into simpler shapes.
Composite figure — figure made from two or more simpler shapes.
Bounding rectangle — rectangle enclosing a figure and used to help calculate or estimate its area.
Coordinate polygon — polygon whose vertices are specified using coordinates.
Horizontal distance — absolute difference between x-coordinates when y-coordinates are equal.
Vertical distance — absolute difference between y-coordinates when x-coordinates are equal.
Scale factor — ratio relating dimensions in a model to actual dimensions.
Area scale factor — square of the corresponding linear scale factor.
Approximation — value close to the actual value rather than perfectly exact.
Irregular polygon — polygon that does not have all sides and angles equal.
Key Takeaways
- Coordinates can be used to calculate the areas of polygons.
- Horizontal lengths are found from differences in x-coordinates.
- Vertical lengths are found from differences in y-coordinates.
- Area is always expressed in square units.
- Rectangles use A = lw.
- Squares use A = s².
- Triangles use A = 1/2 bh.
- Parallelograms use A = bh.
- Trapezoids use A = 1/2(b₁ + b₂)h.
- Height must be perpendicular to the chosen base.
- Complex figures can be decomposed into simpler shapes.
- Another useful strategy is to subtract unwanted regions from a bounding rectangle.
- Distance can provide missing side lengths for rotated coordinate figures.
- Slope can verify whether sides are perpendicular before using them as length and width.
- Squared distances can simplify some coordinate calculations.
- Different shapes can have the same area.
- Figures with the same perimeter do not necessarily have the same area.
- If linear dimensions change by a factor of k, area changes by a factor of k².
- Coordinate scale must be squared when converting coordinate area to real area.
- Coordinate area methods are useful in architecture, surveying, mapping, science, engineering, and computer graphics.
- Real-world coordinate models may provide estimates rather than perfectly exact areas.
Check Your Understanding
- A rectangle has vertices (2, 3), (10, 3), (10, 8), and (2, 8). Calculate its area.
- A square has vertices (−3, 1), (3, 1), (3, 7), and (−3, 7). Calculate its area.
- A triangle has vertices A(1, 2), B(9, 2), and C(5, 8). Calculate its area.
- A triangle has vertices A(3, 1), B(3, 9), and C(10, 5). Calculate its area.
- A parallelogram has a horizontal base from (1, 2) to (8, 2) and its opposite side lies on y = 7. Calculate its area.
- A trapezoid has parallel horizontal sides of lengths 10 units and 6 units, separated by 5 units. Calculate its area.
- An L-shaped figure fits inside a 12 × 9 rectangle. A 5 × 4 rectangular section is removed. Calculate the remaining area.
- A square has adjacent vertices A(1, 1) and B(4, 5). Use the distance formula to determine its area.
- Explain why a slanted side of a parallelogram cannot automatically be used as its height.
- Rectangle A measures 8 × 6, while Rectangle B measures 12 × 4. Compare their areas and perimeters.
- A coordinate map uses 1 unit = 25 m. A rectangular field measures 8 coordinate units by 5 coordinate units. Calculate its actual area in square metres.
- A polygon has coordinate area 36 square units. The map scale is 1 coordinate unit = 10 km. Determine the polygon's actual area in square kilometres.
- A composite floor plan consists of a 10 m × 8 m rectangle and an attached 4 m × 3 m rectangle with no overlap. Calculate the total floor area.
- A student enlarges a coordinate polygon so that every length is three times as large. By what factor does its area increase? Explain.
- A park is modeled by the vertices (0, 0), (12, 0), (12, 6), (8, 10), and (0, 10). Decompose the figure into simpler shapes, calculate its area in square coordinate units, and describe a second method you could use to check your answer.