Heron's Formula Practice: Exact & Approximate Triangle Areas (HSG-MG.A.1)

1) For sides 5, 6, and 7, what is the semiperimeter s?

8

9

10

11

Use the semiperimeter formula: s = (a + b + c) / 2

Substitute the side lengths: s = (5 + 6 + 7) / 2 = 18 / 2 = 9

Hence, s = 9.

Explanation

Use the semiperimeter formula: s = (a + b + c) / 2

Substitute the side lengths: s = (5 + 6 + 7) / 2 = 18 / 2 = 9

Hence, s = 9.

2) A triangle has sides 7 m, 8 m, and 9 m. Approximate its area to one decimal place.

24.0 m²

26.8 m²

28.0 m²

30.2 m²

Step 1: Find the semiperimeter s.

s = (7 + 8 + 9) / 2 = 24 / 2 = 12

Step 2: Use Heron’s Formula:

A = √[s(s − a)(s − b)(s − c)]

So, A = √[12 · (12 − 7) · (12 − 8) · (12 − 9)]

= √[12 · 5 · 4 · 3]

= √(720) ≈ 26.8

Hence, the area ≈ 26.8 m².

Explanation

Step 1: Find the semiperimeter s.

s = (7 + 8 + 9) / 2 = 24 / 2 = 12

Step 2: Use Heron’s Formula:

A = √[s(s − a)(s − b)(s − c)]

So, A = √[12 · (12 − 7) · (12 − 8) · (12 − 9)]

= √[12 · 5 · 4 · 3]

= √(720) ≈ 26.8

Hence, the area ≈ 26.8 m².

3) A triangle with sides 5 cm, 5 cm, and 6 cm has an area of approximately:

10.0 cm²

12.0 cm²

13.5 cm²

15.0 cm²

Step 1: Find the semiperimeter s.

s = (5 + 5 + 6) / 2 = 16 / 2 = 8

Step 2: Use Heron’s Formula:

A = √[s(s − a)(s − b)(s − c)]

So, A = √[8 · (8 − 5) · (8 − 5) · (8 − 6)]

= √[8 · 3 · 3 · 2]

= √(144) = 12

Hence, the area = 12 cm².

Explanation

Step 1: Find the semiperimeter s.

s = (5 + 5 + 6) / 2 = 16 / 2 = 8

Step 2: Use Heron’s Formula:

A = √[s(s − a)(s − b)(s − c)]

So, A = √[8 · (8 − 5) · (8 − 5) · (8 − 6)]

= √[8 · 3 · 3 · 2]

= √(144) = 12

Hence, the area = 12 cm².

4) Find the area of a triangle with sides 3, 4, and 5.

6

8

10

12

This is a 3–4–5 right triangle. Use the right-triangle area formula:

Area = ½ · base · height

Take base = 3 and height = 4:

Area = ½ · 3 · 4 = 6

Hence, the area = 6.

Explanation

This is a 3–4–5 right triangle. Use the right-triangle area formula:

Area = ½ · base · height

Take base = 3 and height = 4:

Area = ½ · 3 · 4 = 6

Hence, the area = 6.

5) A triangle has sides 6 m, 8 m, and 10 m. What is its area?

24 m²

36 m²

48 m²

The sides 6, 8, 10 form a 3–4–5 right triangle scaled by 2.

So we can use:

Area = ½ · base · height

Take base = 6 and height = 8:

Area = ½ · 6 · 8 = 24

Hence, the area = 24 m².

Explanation

The sides 6, 8, 10 form a 3–4–5 right triangle scaled by 2.

So we can use:

Area = ½ · base · height

Take base = 6 and height = 8:

Area = ½ · 6 · 8 = 24

Hence, the area = 24 m².

6) Using Heron's Formula, find the area of a triangle with sides 5, 6, and 7.

14.2

15.3

14.7

16

Step 1: Find the semiperimeter s.

s = (5 + 6 + 7) / 2 = 18 / 2 = 9

Step 2: Use Heron’s Formula:

A = √[s(s − a)(s − b)(s − c)]

So, A = √[9 · (9 − 5) · (9 − 6) · (9 − 7)]

= √[9 · 4 · 3 · 2]

= √(216) = 6√6 ≈ 14.7

Hence, the area ≈ 14.7.

Explanation

Step 1: Find the semiperimeter s.

s = (5 + 6 + 7) / 2 = 18 / 2 = 9

Step 2: Use Heron’s Formula:

A = √[s(s − a)(s − b)(s − c)]

So, A = √[9 · (9 − 5) · (9 − 6) · (9 − 7)]

= √[9 · 4 · 3 · 2]

= √(216) = 6√6 ≈ 14.7

Hence, the area ≈ 14.7.

7) A triangle has sides 9 m, 10 m, and 17 m. Can it exist?

Yes, because 9 + 10 = 19 > 17

No, because 9 + 10 = 18

Yes, because 9 + 10 = 20

No, because 9 + 10 = 19 < 17

Check the triangle inequality: the sum of any two sides must be greater than the third.

9 + 10 = 19, and we compare with 17:

19 > 17

So the inequality holds, and the triangle is valid.

Hence, the triangle can exist.

Explanation

Check the triangle inequality: the sum of any two sides must be greater than the third.

9 + 10 = 19, and we compare with 17:

19 > 17

So the inequality holds, and the triangle is valid.

Hence, the triangle can exist.

8) The semiperimeter of a triangle is 12 cm, and the sides are 5, 7, and 10. What is s − a?

7

6

5

2

We are given s = 12 and sides 5, 7, and 10.

Take a = 5. Then:

s − a = 12 − 5 = 7

Hence, s − a = 7.

Explanation

We are given s = 12 and sides 5, 7, and 10.

Take a = 5. Then:

s − a = 12 − 5 = 7

Hence, s − a = 7.

9) Find the area of a triangle with sides 8, 15, and 17.

56

60

64

72

The sides 8, 15, 17 form a right triangle (a 8–15–17 Pythagorean triple).

Use the area formula for a right triangle:

Area = ½ · base · height

Take base = 8 and height = 15:

Area = ½ · 8 · 15 = 60

Hence, the area = 60.

Explanation

The sides 8, 15, 17 form a right triangle (a 8–15–17 Pythagorean triple).

Use the area formula for a right triangle:

Area = ½ · base · height

Take base = 8 and height = 15:

Area = ½ · 8 · 15 = 60

Hence, the area = 60.

10) Find the area of a triangle with sides 9 ft, 12 ft, and 15 ft.

36 ft²

45 ft²

54 ft²

60 ft²

The sides 9, 12, 15 form a 3–4–5 triangle scaled by 3 (3·3, 3·4, 3·5). So this is a right triangle, and we use: Area = ½ · base · height,

Take base = 9 and height = 12: Area = ½· 9 · 12 = 54

Hence, the area = 54 ft².

Explanation

The sides 9, 12, 15 form a 3–4–5 triangle scaled by 3 (3·3, 3·4, 3·5). So this is a right triangle, and we use: Area = ½ · base · height,

Take base = 9 and height = 12: Area = ½· 9 · 12 = 54

Hence, the area = 54 ft².

11) A triangle has sides 10 m, 13 m, and 15 m. Find its area.

54 m²

64.1 m²

72 m²

90 m²

Step 1: Find the semiperimeter s.

s = (10 + 13 + 15) / 2 = 38 / 2 = 19

Step 2: Use Heron’s Formula:

A = √[s(s − a)(s − b)(s − c)]

So, A = √[19 · (19 − 10) · (19 − 13) · (19 − 15)]

= √[19 · 9 · 6 · 4]

= √(4104) ≈ 64.1

Hence, the area ≈ 64.1 m².

Explanation

Step 1: Find the semiperimeter s.

s = (10 + 13 + 15) / 2 = 38 / 2 = 19

Step 2: Use Heron’s Formula:

A = √[s(s − a)(s − b)(s − c)]

So, A = √[19 · (19 − 10) · (19 − 13) · (19 − 15)]

= √[19 · 9 · 6 · 4]

= √(4104) ≈ 64.1

Hence, the area ≈ 64.1 m².

12) For triangle sides 13 cm, 14 cm, and 15 cm, find the semiperimeter s.

19

20

21

22

Use the semiperimeter formula:

s = (a + b + c) / 2

Substitute the values:

s = (13 + 14 + 15) / 2

= 42 / 2 = 21

Hence, s = 21.

Explanation

Use the semiperimeter formula:

s = (a + b + c) / 2

Substitute the values:

s = (13 + 14 + 15) / 2

= 42 / 2 = 21

Hence, s = 21.

13) Using Heron's Formula, find the area for sides 13, 14, and 15.

78

84

90

96

Step 1: Find semiperimeter s.

s = (13 + 14 + 15) / 2 = 21

Step 2: Use Heron’s Formula:

A = √[s(s − a)(s − b)(s − c)]

So, A = √[21 · (21 − 13) · (21 − 14) · (21 − 15)]

= √[21 · 8 · 7 · 6]

= √(7056) = 84

Hence, the area = 84.

Explanation

Step 1: Find semiperimeter s.

s = (13 + 14 + 15) / 2 = 21

Step 2: Use Heron’s Formula:

A = √[s(s − a)(s − b)(s − c)]

So, A = √[21 · (21 − 13) · (21 − 14) · (21 − 15)]

= √[21 · 8 · 7 · 6]

= √(7056) = 84

Hence, the area = 84.

14) A triangle with sides 9 m, 10 m, and 11 m has an area of approximately:

40.0 m²

42.4 m²

45.0 m²

48.0 m²

Step 1: Find semiperimeter s.

s = (9 + 10 + 11) / 2 = 30 / 2 = 15

Step 2: Use Heron’s Formula:

A = √[15 · (15 − 9) · (15 − 10) · (15 − 11)]

= √[15 · 6 · 5 · 4]

= √(1800) ≈ 42.4

Hence, the area ≈ 42.4 m².

Explanation

Step 1: Find semiperimeter s.

s = (9 + 10 + 11) / 2 = 30 / 2 = 15

Step 2: Use Heron’s Formula:

A = √[15 · (15 − 9) · (15 − 10) · (15 − 11)]

= √[15 · 6 · 5 · 4]

= √(1800) ≈ 42.4

Hence, the area ≈ 42.4 m².

15) A triangular garden has sides 8 m, 8 m, and 10 m. What is the area?

5√39 m²

6√39 m²

12√5 m²

8√7 m²

Step 1: Find semiperimeter s.

s = (8 + 8 + 10) / 2 = 26 / 2 = 13

Step 2: Use Heron’s Formula: A = √[13 · (13 − 8) · (13 − 8) · (13 − 10)]

= √[13 · 5 · 5 · 3]

= √(975)

Factor 975:

975 = 25 · 39 = (5²) · 39

So: A = √(5² · 39) = 5√39

Hence, the area = 5√39 m².

Explanation

Step 1: Find semiperimeter s.

s = (8 + 8 + 10) / 2 = 26 / 2 = 13

Step 2: Use Heron’s Formula: A = √[13 · (13 − 8) · (13 − 8) · (13 − 10)]

= √[13 · 5 · 5 · 3]

= √(975)

Factor 975:

975 = 25 · 39 = (5²) · 39

So: A = √(5² · 39) = 5√39

Hence, the area = 5√39 m².

16) A triangular flag has sides 10 ft, 10 ft, and 12 ft. Find its area.

45 ft²

48 ft²

54 ft²

60 ft²

Step 1: Find semiperimeter s.

s = (10 + 10 + 12) / 2 = 32 / 2 = 16

Step 2: Use Heron’s Formula: A = √[16 · (16 − 10) · (16 − 10) · (16 − 12)]

= √[16 · 6 · 6 · 4]

= √(2304) = 48

Hence, the area = 48 ft².

Explanation

Step 1: Find semiperimeter s.

s = (10 + 10 + 12) / 2 = 32 / 2 = 16

Step 2: Use Heron’s Formula: A = √[16 · (16 − 10) · (16 − 10) · (16 − 12)]

= √[16 · 6 · 6 · 4]

= √(2304) = 48

Hence, the area = 48 ft².

17) A triangular park has sides 9 m, 9 m, and 9 m. What is the approximate area?

24.3 m²

35.1 m²

22.5 m²

27 m²

The triangle is equilateral with side a = 9.

Use the equilateral triangle area formula:

A = (√3 / 4) · a²

Substitute a = 9:

A = (√3 / 4) · 9²

= (√3 / 4) · 81

≈ 0.433 · 81 ≈ 35.1

Hence, the area ≈ 35.1 m².

Explanation

The triangle is equilateral with side a = 9.

Use the equilateral triangle area formula:

A = (√3 / 4) · a²

Substitute a = 9:

A = (√3 / 4) · 9²

= (√3 / 4) · 81

≈ 0.433 · 81 ≈ 35.1

Hence, the area ≈ 35.1 m².

18) A triangular ramp has sides 7 ft, 24 ft, and 25 ft. Find its area.

70 ft²

84 ft²

90 ft²

96 ft²

The sides 7, 24, 25 form a Pythagorean triple (7² + 24² = 25²), so it’s a right triangle.

Use the right triangle area formula:

Area = ½ · base · height

Take base = 7 and height = 24:

Area = ½ · 7 · 24 = 84

Hence, the area = 84 ft².

Explanation

The sides 7, 24, 25 form a Pythagorean triple (7² + 24² = 25²), so it’s a right triangle.

Use the right triangle area formula:

Area = ½ · base · height

Take base = 7 and height = 24:

Area = ½ · 7 · 24 = 84

Hence, the area = 84 ft².

19) For triangle sides 6 cm, 8 cm, and 10 cm, which statement is true?

It’s obtuse, area = 12 cm²

It’s invalid

It’s equilateral, area = 36 cm²

It’s a right triangle, so area = ½(6)(8) = 24 cm²

Check if the triangle is right:

Compute:

6² + 8² = 36 + 64 = 100

10² = 100

Since 6² + 8² = 10², it is a right triangle.

Area = ½ · 6 · 8 = 24 cm²

Hence, it’s a right triangle with area 24 cm².

Explanation

Check if the triangle is right:

Compute:

6² + 8² = 36 + 64 = 100

10² = 100

Since 6² + 8² = 10², it is a right triangle.

Area = ½ · 6 · 8 = 24 cm²

Hence, it’s a right triangle with area 24 cm².

20) A triangle has sides 9 m, 10 m, and 15 m. Approximate its area.

30.0 m²

34.0 m²

35.0 m²

43.6 m²

Step 1: Find semiperimeter s.

s = (9 + 10 + 15) / 2 = 34 / 2 = 17

Step 2: Use Heron’s Formula:

A = √[17 · (17 − 9) · (17 − 10) · (17 − 15)]

= √[17 · 8 · 7 · 2]

= √(1904) ≈ 43.6

Hence, the area ≈ 43.6 m².

Explanation

Step 1: Find semiperimeter s.

s = (9 + 10 + 15) / 2 = 34 / 2 = 17

Step 2: Use Heron’s Formula:

A = √[17 · (17 − 9) · (17 − 10) · (17 − 15)]

= √[17 · 8 · 7 · 2]

= √(1904) ≈ 43.6

Hence, the area ≈ 43.6 m².