Compute Triangle Areas with Heron's Formula

  • 10th Grade
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| Attempts: 17 | Questions: 20 | Updated: Dec 12, 2025
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1) For sides 5, 6, and 7, what is the semiperimeter s?

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.

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About This Quiz
Compute Triangle Areas With Herons Formula - Quiz

Build the foundation: verify side lengths form a triangle, compute the semiperimeter, and set up the product that leads to the area. You’ll decide when Heron’s Formula is the right tool, interpret units, and compare different side sets to predict which triangles enclose more area—before you even crunch the numbers.

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

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.

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

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.

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

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.

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

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.

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

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

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

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.

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

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.

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

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.

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

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.

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

Explanation

Use the semiperimeter formula:

s = (a + b + c) / 2

Substitute the values:

s = (13 + 14 + 15) / 2

= 42 / 2 = 21

Hence, s = 21.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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For sides 5, 6, and 7, what is the semiperimeter s?
A triangle has sides 7 m, 8 m, and 9 m. Approximate its area to one...
A triangle with sides 5 cm, 5 cm, and 6 cm has an area of...
Find the area of a triangle with sides 3, 4, and 5.
A triangle has sides 6 m, 8 m, and 10 m. What is its area?
Using Heron's Formula, find the area of a triangle with sides 5, 6,...
A triangle has sides 9 m, 10 m, and 17 m. Can it exist?
The semiperimeter of a triangle is 12 cm, and the sides are 5, 7, and...
Find the area of a triangle with sides 8, 15, and 17.
Find the area of a triangle with sides 9 ft, 12 ft, and 15 ft.
A triangle has sides 10 m, 13 m, and 15 m. Find its area.
For triangle sides 13 cm, 14 cm, and 15 cm, find the semiperimeter s.
Using Heron's Formula, find the area for sides 13, 14, and 15.
A triangle with sides 9 m, 10 m, and 11 m has an area of...
A triangular garden has sides 8 m, 8 m, and 10 m. What is the area?
A triangular flag has sides 10 ft, 10 ft, and 12 ft. Find its area.
A triangular park has sides 9 m, 9 m, and 9 m. What is the approximate...
A triangular ramp has sides 7 ft, 24 ft, and 25 ft. Find its area.
For triangle sides 6 cm, 8 cm, and 10 cm, which statement is true?
A triangle has sides 9 m, 10 m, and 15 m. Approximate its area.
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