Modeling with Heron's Formula: Gardens, Panels, Sails & More (HSG-MG.A.1)

1) A triangular flower bed has sides 6 ft, 8 ft, and 10 ft. How much soil is needed to fill it if 1 cubic foot of soil covers 1 ft² of area?

20 ft³

22 ft³

24 ft³

26 ft³

Right triangle ⇒ area = ½·6·8 = 24 ft² ⇒ 24 ft³ of soil.

Explanation

Right triangle ⇒ area = ½·6·8 = 24 ft² ⇒ 24 ft³ of soil.

2) A triangular roof support beam has sides 8 ft, 8 ft, and 10 ft. Find the area to the nearest tenth.

19.5 ft²

31.2 ft²

19.6 ft²

21.0 ft²

s = 13 ⇒ A = √(13·5·5·3) = √975 ≈ 31.2 ft²

Explanation

s = 13 ⇒ A = √(13·5·5·3) = √975 ≈ 31.2 ft²

3) A triangular garden has sides 9 m, 10 m, and 17 m. Is it possible to construct this garden?

Yes, and the area is 20 m²

Yes, and the area is 36 m²

Yes, and the area is 40 m²

No, the sides do not form a valid triangle

s = 18 ⇒ A = √(18·9·8·1) = √1296 = 36 m².

Explanation

s = 18 ⇒ A = √(18·9·8·1) = √1296 = 36 m².

4) A triangular billboard frame has sides 5 m, 6 m, and 7 m. What is its area to the nearest tenth?

13.5 m²

14.7 m²

16.0 m²

18.5 m²

s = 9 ⇒ A = √(9·4·3·2) = √216 ≈ 14.7 m².

Explanation

s = 9 ⇒ A = √(9·4·3·2) = √216 ≈ 14.7 m².

5) A triangular piece of land has sides 8 m, 15 m, and 17 m. A fence will be built around it costing $10 per meter. What will the fencing cost?

$200

$400

$300

$320

Perimeter = 8 + 15 + 17 = 40 ⇒ Cost = 40 × $10 = $400.

Explanation

Perimeter = 8 + 15 + 17 = 40 ⇒ Cost = 40 × $10 = $400.

6) A triangular steel plate has sides 5 cm, 5 cm, and 6 cm. Find its area to the nearest tenth.

11.5 cm²

12.0 cm²

12.8 cm²

13.0 cm²

s = 8 ⇒ A = √(8·3·3·2) = √144 = 12.0 cm².

Explanation

s = 8 ⇒ A = √(8·3·3·2) = √144 = 12.0 cm².

7) A triangular pond has sides 9 m, 10 m, and 11 m. What is the water surface area to the nearest tenth?

39.8 m²

42.4 m²

44.5 m²

48.0 m²

s = 15 ⇒ A = √(15·6·5·4) = √1800 ≈ 42.4 m².

Explanation

s = 15 ⇒ A = √(15·6·5·4) = √1800 ≈ 42.4 m².

8) A triangular glass panel has sides 6 ft, 10 ft, and 12 ft. Find its area.

24√11 ft²

8√4 ft²

6√35 ft²

8√20 ft²

s = 14 ⇒ A = √(14·8·4·2) = √896 = 8√14 ft²

Explanation

s = 14 ⇒ A = √(14·8·4·2) = √896 = 8√14 ft²

9) A triangular sail has sides 13 ft, 14 ft, and 15 ft. Each square foot of sailcloth costs $3. What is the cost?

$180

$252

$270

$300

Area = √(21·8·7·6) = 84 ft² ⇒ Cost = 84 × 3 = $252.

Explanation

Area = √(21·8·7·6) = 84 ft² ⇒ Cost = 84 × 3 = $252.

10) A triangular lot has sides 13 m, 14 m, and 15 m. Grass costs $2 per m². What is the total cost?

$120

$150

$168

$200

Area = 84 m² ⇒ Cost = 84 × 2 = $168.

Explanation

Area = 84 m² ⇒ Cost = 84 × 2 = $168.

11) A triangular roof tile has sides 7 cm, 8 cm, and 9 cm. What is its area?

25.0 cm²

26.8 cm²

28.0 cm²

29.5 cm²

s = 12 ⇒ A = √(12·5·4·3) = √720 ≈ 26.8 cm².

Explanation

s = 12 ⇒ A = √(12·5·4·3) = √720 ≈ 26.8 cm².

12) A triangular park has sides 3 m, 4 m, and 5 m. What is the park's area?

6 m²

8 m²

9 m²

10 m²

Right triangle ⇒ A = ½·3·4 = 6 m². Hence, 6 m².

Explanation

Right triangle ⇒ A = ½·3·4 = 6 m². Hence, 6 m².

13) A triangular flag with sides 7 in, 24 in, and 25 in will need printing on both sides. What total fabric area is required?

84 in²

120 in²

168 in²

200 in²

Right triangle ⇒ one side area = 84 in² ⇒ total = 168 in².

Explanation

Right triangle ⇒ one side area = 84 in² ⇒ total = 168 in².

14) A triangular section of a park has sides 10 m, 10 m, and 12 m. What is its area?

42 m²

44 m²

48 m²

54 m²

s = 16 ⇒ A = √(16·6·6·4) = √2304 = 48 m².

Explanation

s = 16 ⇒ A = √(16·6·6·4) = √2304 = 48 m².

15) A triangular courtyard has sides 25 m, 26 m, and 27 m. Approximate its area.

270 m²

290 m²

292 m²

300 m²

s = 39 ⇒ A = √(39·14·13·12) = √85176 ≈ 292 m².

Explanation

s = 39 ⇒ A = √(39·14·13·12) = √85176 ≈ 292 m².

16) A triangular deck has sides 9 ft, 10 ft, and 11 ft. A protective coating costs $5 per ft². Find the approximate total cost.

$200

$210

$212

$250

Area ≈ 42.4 ft² ⇒ Cost = 42.4 × 5 ≈ $212.

Explanation

Area ≈ 42.4 ft² ⇒ Cost = 42.4 × 5 ≈ $212.

17) A company wants to design a triangular logo with sides 9 cm, 12 cm, and 15 cm. What is the logo’s area?

54 cm²

56 cm²

60 cm²

64 cm²

Right triangle ⇒ A = ½·9·12 = 54 cm².

Explanation

Right triangle ⇒ A = ½·9·12 = 54 cm².

18) A triangular piece of fabric has sides 9 in, 9 in, and 9 in. What is the approximate area?

24√3 in²

21√3 in²

(81√3)/4 in²

30√3 in²

Equilateral ⇒ A = (√3/4)a² = (81√3)/4.

Explanation

Equilateral ⇒ A = (√3/4)a² = (81√3)/4.

19) A triangular garden has sides 5 m, 12 m, and 13 m. How much fertilizer covers the garden if 1 kg covers 10 m²?

0.4 kg

0.6 kg

3.0 kg

1.2 kg

Area = ½·5·12 = 30 m² ⇒ Fertilizer = 30/10 = 3.0 kg.

Explanation

Area = ½·5·12 = 30 m² ⇒ Fertilizer = 30/10 = 3.0 kg.

20) A builder designs a triangular patio with sides 10 m, 13 m, and 15 m. Which best describes the role of Heron's Formula?

It estimates the slope of the patio

It measures the perimeter

It checks if the shape is right-angled

It allows finding the area without knowing height

Heron’s formula allows area from side lengths only.

Explanation

Heron’s formula allows area from side lengths only.

Real-World Area Problems Using Heron’s Formula

1) A triangular flower bed has sides 6 ft, 8 ft, and 10 ft. How much soil is needed to fill it if 1 cubic foot of soil covers 1 ft² of area?

2)

What first name or nickname would you like us to use?

2) A triangular roof support beam has sides 8 ft, 8 ft, and 10 ft. Find the area to the nearest tenth.

3) A triangular garden has sides 9 m, 10 m, and 17 m. Is it possible to construct this garden?

4) A triangular billboard frame has sides 5 m, 6 m, and 7 m. What is its area to the nearest tenth?

5) A triangular piece of land has sides 8 m, 15 m, and 17 m. A fence will be built around it costing $10 per meter. What will the fencing cost?

6) A triangular steel plate has sides 5 cm, 5 cm, and 6 cm. Find its area to the nearest tenth.

7) A triangular pond has sides 9 m, 10 m, and 11 m. What is the water surface area to the nearest tenth?

8) A triangular glass panel has sides 6 ft, 10 ft, and 12 ft. Find its area.

9) A triangular sail has sides 13 ft, 14 ft, and 15 ft. Each square foot of sailcloth costs $3. What is the cost?

10) A triangular lot has sides 13 m, 14 m, and 15 m. Grass costs $2 per m². What is the total cost?

11) A triangular roof tile has sides 7 cm, 8 cm, and 9 cm. What is its area?

12) A triangular park has sides 3 m, 4 m, and 5 m. What is the park's area?

13) A triangular flag with sides 7 in, 24 in, and 25 in will need printing on both sides. What total fabric area is required?

14) A triangular section of a park has sides 10 m, 10 m, and 12 m. What is its area?

15) A triangular courtyard has sides 25 m, 26 m, and 27 m. Approximate its area.

16) A triangular deck has sides 9 ft, 10 ft, and 11 ft. A protective coating costs $5 per ft². Find the approximate total cost.

17) A company wants to design a triangular logo with sides 9 cm, 12 cm, and 15 cm. What is the logo’s area?

18) A triangular piece of fabric has sides 9 in, 9 in, and 9 in. What is the approximate area?

19) A triangular garden has sides 5 m, 12 m, and 13 m. How much fertilizer covers the garden if 1 kg covers 10 m²?

20) A builder designs a triangular patio with sides 10 m, 13 m, and 15 m. Which best describes the role of Heron's Formula?