Grade 10 · Core Mathematics · Lesson 9 of 12

Determining Area of Irregular Polygons by Shape Subdivision

A Grade 10 Core Mathematics CBE lesson from the sub-strand Area of Polygons. In it, learners identify convenient geometric subdivisions for irregular composite polygons.

Strand Measurements and GeometrySub-strand Area of PolygonsLesson code 10CM-25-L940 minutes · 2 slides

KICD learning outcome (f)

Determine the area of irregular polygons in different situations.

By the end of this lesson, learners should be able to

Identify convenient geometric subdivisions for irregular composite polygons.
Calculate individual sub-region areas using trigonometric and geometric area formulas.
Determine the total area of irregular polygons in practical situations by summing component sub-region areas.

Key vocabulary

Irregular Polygon: A polygon with sides that are not all equal or angles that are not all congruent.; We subdivided the irregular polygon into a trapezium and a triangle to compute total area.
Decomposition: The mathematical process of breaking a complex geometric shape into simpler component sub-regions.; Decomposition allowed us to find the total floor area by adding component shape areas.
Perpendicular Height: The shortest distance measured at a right angle from a base to the opposite vertex.; We calculated the perpendicular height using the sine ratio to evaluate the trapezium area.

Lesson preview

Slide 1 · Subdividing Irregular Polygons into Geometrical ShapesOpening · 18 min

Figure 1: A pentagon is subdivided into a right-angled trapezium and a triangle by an internal dashed line.

To determine the area of irregular polygons in different situations, we subdivide the complex floor plan or boundary into convenient geometrical shapes. By drawing internal connecting lines between key vertices, we partition the irregular polygon into familiar figures like triangles, rectangles, or trapeziums. We then calculate the area of each component shape independently and sum the areas. The area of an irregular polygon is always equal to the sum of the areas of its subdivided component shapes. Use Figure 1 to see how to partition an irregular polygon into familiar figures.

An irregular polygon has non-equal side lengths or interior angles, requiring subdivision for area calculation.
Subdivide complex composite plans into convenient geometrical shapes such as rectangles, trapeziums, and triangles.
Calculate the area of each component sub-region using standard geometric or trigonometric area formulas.
The area of an irregular polygon equals the sum of the areas of its subdivided component shapes.

Worked example. Step 1: Calculate the area of right trapezium

A B C D

Area (A B C D) = \frac{1}{2} \times B C \times (A B + C D) = \frac{1}{2} \times 8 \times (10 + 16) = 4 \times 26 = 104 m^{2}

. Step 2: Calculate the area of triangle

A D E

using two sides and the included angle.

Area (A D E) = \frac{1}{2} \times A D \times A E \times \sin (∠ D A E) = \frac{1}{2} \times 10 \times 8 \times \sin (50^{\circ}) = 40 \times 0.7660444 = 30.6418 m^{2}

. Step 3: Sum the component areas to find the total area.

Total Area = 104 + 30.6418 = 134.6418 \approx 134.64 m^{2}

. Answer: $134.64 m^{2}$ .

Check question: A site layout $P Q R S X$ consists of rectangle $P Q R X$ and triangle $X R S$ , drawn outside the rectangle on side $X R$ . Rectangle $P Q R X$ has $P Q = 15 m$ and $Q R = 8 m$ . Triangle $X R S$ has base $X R = 15 m$ , side $X S = 10 m$ , and $∠ R X S = 40^{\circ}$ . What is the total area of $P Q R S X$ ?

$168.21 m^{2}$
$216.42 m^{2}$
$177.45 m^{2}$
$48.21 m^{2}$

Show answer

$168.21 m^{2}$ . Area of rectangle $P Q R X = 15 \times 8 = 120 m^{2}$ . Area of triangle $X R S = \frac{1}{2} \times 15 \times 10 \times \sin (40^{\circ}) = 75 \times 0.6427876 = 48.2091 m^{2}$ . Total area = $120 + 48.2091 = 168.2091 \approx 168.21 m^{2}$ .

Slide 2 · Computing Polygon Areas via Component SummationModel · 22 min

Figure 2: An irregular polygon partitioned into a trapezium and a triangle, with a dashed triangle showing perpendicular height.

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Both slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 11 lessons in this sub-strand

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Determining Area of Irregular Polygons by Shape Subdivision

By the end of this lesson, learners should be able to

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 12 lessons in Area of Polygons