Grade 10 · Core Mathematics · Lesson 11 of 12

Exploring Area of Polygons in Real-Life Situations

A Grade 10 Core Mathematics CBE lesson from the sub-strand Area of Polygons. In it, learners identify triangular and quadrilateral shapes in immediate everyday environments.

Strand Measurements and GeometrySub-strand Area of PolygonsLesson code 10CM-25-L1140 minutes · 2 slides

KICD learning outcome (g)

Explore the area of polygons as used in real-life situations.

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

Identify triangular and quadrilateral shapes in immediate everyday environments.
Apply the trigonometric area formula ½ab sin C and Heron's formula to model triangular land plots and ad boards.
Calculate areas of real-life parallelograms and kites using side lengths, included angles, and diagonal dimensions.

Key vocabulary

Included angle: The interior angle located directly between two given meeting sides of a polygon.; The included angle of 60° lay between the 8 m and 11 m sides.
Semi-perimeter: Half the sum of all the side lengths surrounding a closed geometric shape.; The semi-perimeter of the 13 m, 13 m, and 10 m triangle was 18 m.
Diagonal: A straight line segment connecting two non-adjacent vertices of a geometric polygon.; Half the product of the diagonals seven and four gave the kite area of 14 m².
Parallelogram: A quadrilateral with two pairs of parallel opposite sides of equal length.; The arboretum formed a parallelogram with sides 14 m and 9 m.

Lesson preview

Slide 1 · Real-Life Applications: Triangular Objects and StructuresModel · 18 min

Figure 1: A triangular advertisement board is shown as an isosceles triangle, with a base of 10 metres and a perpendicular height of 12 metres.

Everyday objects and physical structures such as advertisement boards, triangular garden flower beds, and school field sections can be modelled using geometric polygons. By measuring available dimensions like side lengths, perpendicular heights, or interior angles, we can select the appropriate formula—such as $A r e a = \frac{1}{2} b h$ , $A r e a = \frac{1}{2} a b \sin C$ , or Heron's formula—to accurately evaluate total surface area and solve practical financial problems like painting or fencing costs. Use Figure 1 to see how to identify the base and perpendicular height of an advertisement board.

Real-life triangular structures like ad boards and flower beds can be measured using appropriate area formulas.
When two side lengths and an included angle are known, apply the formula Area = ½ ab sin C.
For triangles with three known side lengths, calculate semi-perimeter s and apply Heron's formula for area.
Empirical measurements using perpendicular height yield matching area values to trigonometric and algebraic methods.

Worked example. Step 1: Find the length of the base. Base = Perimeter − 2(equal side) = 36 − 2(13) = 10 m. Step 2: Calculate semi-perimeter s = 36 / 2 = 18 m. Step 3: Apply Heron's formula: Area =

\sqrt{18 (18 - 13) (18 - 13) (18 - 10)} = \sqrt{18 \times 5 \times 5 \times 8} = \sqrt{3600} = 60 m^{2}

. Step 4: Calculate total painting cost: Cost = 60 m² × Ksh 250/m² = Ksh 15,000. Answer: The area of the advertisement board is 60 m² and the total cost to paint it is Ksh 15,000..

Check question: A triangular flower bed in a school compound has side lengths of 8 m and 11 m with an included angle of 60°. Calculate its area to 2 decimal places.

22.00 m²
38.11 m²
44.00 m²
76.21 m²

Show answer

38.11 m². To find the area of the triangular flower bed when two side lengths and the included angle are given, use the formula $A r e a = \frac{1}{2} a b \sin C$ . Substituting the given values: $A r e a = \frac{1}{2} \times 8 \times 11 \times \sin (60^{\circ})$ $A r e a = 44 \times 0.866025...$ $A r e a \approx 38.11 m^{2}$ .

Slide 2 · Real-Life Applications: Quadrilateral Gardens and PlotsPractice · 22 min

Figure 2: Geometric models of a kite-shaped kitchen garden and a parallelogram arboretum are shown with their diagonal lengths, side lengths, and included angles labelled.

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Teacher notes, SNE adaptations and assessment checks
Homework and the other 11 lessons in this sub-strand

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Exploring Area of Polygons in Real-Life Situations

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