Heron’s Formula Basics: Semiperimeter, Validity & Setup

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| Questions: 20 | Updated: Nov 10, 2025
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1) What is the formula for the semiperimeter of a triangle with sides a, b, and c?

Explanation

s = (a + b + c)/2. Hence, s = ½(a + b + c).

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About This Quiz
Heron Formula Quizzes & Trivia

Are you ready to get a triangle’s area using only its three side lengths? This quiz walks you through Heron’s idea: first check the triangle inequality, then take half the perimeter (the semiperimeter), and finally combine those pieces to compute area—no altitude or angles required. You’ll practice quick validity checks,... see moreclear setup, and clean arithmetic so your answers come out accurate and unit-correct every time. see less

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2) What must be true for any set of three lengths to form a triangle?

Explanation

Triangle inequality: a + b > c, a + c > b, b + c > a.

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3) The semiperimeter of a triangle is 12 cm. Its sides are 7 cm and 9 cm. What is the third side?

Explanation

s = 12 ⇒ perimeter = 24 ⇒ third side = 8 cm.

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4) Which triangle is not valid?

Explanation

7 + 2 = 9 which is not > 10, so invalid.

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5) Heron’s Formula for area is

Explanation

By definition, A = √[s(s − a)(s − b)(s − c)].

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6) Why is the semiperimeter used in Heron’s Formula?

Explanation

The area expression naturally uses s − a, s − b, s − c.

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7) Which triangle below will have the largest area using Heron’s Formula?

Explanation

Computed areas show 7,8,9 yields the largest ≈ 26.8.

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8) The term inside the square root in Heron’s Formula, s(s − a)(s − b)(s − c), represents:

Explanation

The product gives squared area before square root.

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9) What are the units of the result from Heron’s Formula?

Explanation

Area is measured in square units (cm²).

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10) What happens to area if one side of a triangle increases while others stay fixed?

Explanation

Area rises then collapses toward 0 as validity nears limit.

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11) In Heron’s Formula, if a = b = c, then area simplifies to:

Explanation

s = 3a/2 ⇒ A = √[s(s − a)³].

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12) Which set of sides yields the simplest integer area?

Explanation

3–4–5 right triangle ⇒ A = 6 (integer, simplest).

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13) If a triangle has sides 4, 9, 14, why can’t Heron’s Formula be applied?

Explanation

4 + 9 = 13 not > 14 ⇒ invalid triangle.

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14) Which of these gives the most efficient first step when using Heron’s Formula?

Explanation

Heron’s area needs s, so compute s first.

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15) A triangle’s sides are a = 10 cm, b = 10 cm, c = 12 cm. What is s?

Explanation

s = (10 + 10 + 12)/2 = 16 cm.

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16) Why is Heron’s Formula useful?

Explanation

It uses only side lengths a, b, c.

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17) If semiperimeter = 8 m and one side a = 7 m, what is (s − a)?

Explanation

s − a = 8 − 7 = 1 m.

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18) If s = 10 cm and a = 7 cm, what is (s − a)?

Explanation

s − a = 10 − 7 = 3 cm.

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19) If a triangle has sides 5, 12, 13, what special type is it?

Explanation

5² + 12² = 13² ⇒ right triangle.

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20) For a right triangle with legs a and b, Heron’s Formula should give the same result as:

Explanation

Area = ½·a·b for right triangle.

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What is the formula for the semiperimeter of a triangle with sides a,...
What must be true for any set of three lengths to form a triangle?
The semiperimeter of a triangle is 12 cm. Its sides are 7 cm and 9 cm....
Which triangle is not valid?
Heron’s Formula for area is
Why is the semiperimeter used in Heron’s Formula?
Which triangle below will have the largest area using Heron’s...
The term inside the square root in Heron’s Formula, s(s − a)(s −...
What are the units of the result from Heron’s Formula?
What happens to area if one side of a triangle increases while others...
In Heron’s Formula, if a = b = c, then area simplifies to:
Which set of sides yields the simplest integer area?
If a triangle has sides 4, 9, 14, why can’t Heron’s Formula be...
Which of these gives the most efficient first step when using...
A triangle’s sides are a = 10 cm, b = 10 cm, c = 12 cm. What is s?
Why is Heron’s Formula useful?
If semiperimeter = 8 m and one side a = 7 m, what is (s − a)?
If s = 10 cm and a = 7 cm, what is (s − a)?
If a triangle has sides 5, 12, 13, what special type is it?
For a right triangle with legs a and b, Heron’s Formula should give...
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