Area of Polygons

14 Grade 10 Core Mathematics questions on Area of Polygons, with worked marking schemes. 14 questions are available on this topic in full.

1. Question 1

   1 mark

   A rhombus-shaped road sign has diagonals of 16 cm and 12 cm. What is its area?

   A. 96 cm²

   B. 192 cm²

   C. 48 cm²

   D. 28 cm²

   (1 mark)

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2. Question 2

   6 marks

   A plot ABCD is a quadrilateral in which AB = 40 m, BC = 30 m and angle ABC = 90°. The remaining sides are CD = 26 m and AD = 50 m. Outline how the plot may be split into triangles, and calculate the total area of the plot correct to one decimal place.

   (6 marks)

   Marking scheme

   Join AC to split the plot into triangle ABC and triangle ACD ✓$1AC=\sqrt{40^2+30^2}=\sqrt{2500}=50m$ ✓1 Area of ABC = $\frac{1}{2}$ × $40\times 30=600m^2$ ✓1 For triangle ACD: s = (50 + 26 + 50)/2 = 63 ✓1 Area of ACD = √[$63\times 13\times 37\times 13$] = $\sqrt{393}939=627.6m^2$ ✓1 (Accept: isosceles with height $\sqrt{50^2-13^2}=48.28$, area = $\frac{1}{2}$ × $26\times 48.28=627.6m^2$) Total area = 600 + 627.6 = 1 227.6 m² ✓1

3. Question 3

   1 mark

   A paving slab is cut in the shape of a regular octagon whose perimeter is 48 cm and whose apothem (the perpendicular distance from the centre to a side) is 7.2 cm. What is its area?

   A. 345.6 cm²

   B. 172.8 cm²

   C. 28.8 cm²

   D. 57.6 cm²

   (1 mark)

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4. Question 4

   2 marks

   A triangular plot has an area of 240 m². Two of its sides measure 30 m and 20 m, and the angle between these two sides is acute. Calculate the size of this angle, correct to one decimal place.

   (2 marks)

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5. Question 5

   1 mark

   A parallelogram-shaped banner has adjacent sides of 10 cm and 7 cm with an included angle of 60°. What is its area, correct to one decimal place?

   A. 70.0 cm²

   B. 60.6 cm²

   C. 35.0 cm²

   D. 30.3 cm²

   (1 mark)

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6. Question 6

   1 mark

   A five-sided plot is made up of a rectangle 20 m long and 12 m wide, with an isosceles triangle of perpendicular height 9 m standing on one of the 20 m sides. What is the area of the plot?

   A. 420 m²

   B. 240 m²

   C. 285 m²

   D. 330 m²

   (1 mark)

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7. Question 7

   1 mark

   A triangular piece of land has two sides measuring 9 m and 12 m, and the angle between these two sides is 50°. What is the area of the land, correct to one decimal place?

   A. 34.7 m²

   B. 64.4 m²

   C. 41.4 m²

   D. 82.8 m²

   (1 mark)

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8. Question 8

   4 marks

   A triangular shamba has sides of 35 m, 44 m and 53 m. Using Heron's formula, calculate its area correct to the nearest square metre.

   (4 marks)

   Marking scheme

   s = (35 + 44 + 53)/2 = 66 ✓1 Area = √[66(66−35)(66−44)(66−53)] = $\sqrt{66\times 31\times 22\times 13}$ ✓$1=\sqrt{585}156$ ✓$1=765m^2$ ✓1

9. Question 9

   1 mark

   A trapezium-shaped goat pen has parallel sides of 12 m and 8 m, with a perpendicular distance of 5 m between them. What is its area?

   A. 48 m²

   B. 100 m²

   C. 50 m²

   D. 25 m²

   (1 mark)

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10. Question 10

    1 mark

    A triangular tea nursery bed has sides of 13 m, 14 m and 15 m. Using Heron's formula, what is its area?

    A. 42 m²

    B. 21 m²

    C. 105 m²

    D. 84 m²

    (1 mark)

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11. Question 11

    5 marks

    A signboard is cut in the shape of a regular pentagon of side 12 cm. Calculate its area, correct to one decimal place.

    (5 marks)

    Marking scheme

    Divide into 5 congruent isosceles triangles with apex angle 360° ÷ 5 = 72° at the centre ✓1 Half the apex angle = 36°, and half the side = 6 cm, so apothem = $6÷\tan 36$° ✓1 Apothem = $6÷0.7265=8.258$ cm ✓1 Area of one triangle = $\frac{1}{2}$ × $12\times 8.258=49.55$ cm² ✓1 Total area = $5\times 49.55=247.7$ cm² ✓1

12. Question 12

    4 marks

    A cattle boma is built in the shape of a regular hexagon of side 9 m. Describe how the hexagon can be divided in order to find its area, and hence calculate the area correct to one decimal place.

    (4 marks)

    Marking scheme

    The hexagon is divided into 6 congruent equilateral triangles meeting at the centre, each of side 9 m ✓1 Area of one triangle = (√3/4) × 9² = 0.4330 × 81 ✓$1=35.07m^2$ ✓1 Total area = $6\times 35.07=210.4m^2$ ✓1

13. Question 13

    3 marks

    A triangular maize plot has two sides measuring 45 m and 60 m, with an angle of 68° between them. Calculate the area of the plot in hectares, correct to three decimal places. (1 hectare = 10 000 m²)

    (3 marks)

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14. Question 14

    3 marks

    A flower bed in the shape of a trapezium has parallel sides of 8.4 m and 5.6 m, with a perpendicular distance of 3.5 m between them. Calculate its area, and hence the cost of covering it with manure at KSh 60 per square metre.

    (3 marks)

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