Heron’s Formula Basics: Semiperimeter, Validity & Setup

9th Grade

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| Attempts: 13 | Questions: 20 | Updated: Dec 11, 2025

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1) What is the formula for the semiperimeter of a triangle with sides a, b, and c?

S = a + b + c

S = ½(a + b + c)

S = (a + b + c)/3

S = (a × b × c)½

The semiperimeter is defined as half the perimeter:

s = (a + b + c) / 2

This can be written as:

s = ½(a + b + c)

Hence, s = ½(a + b + c).

Explanation

The semiperimeter is defined as half the perimeter:

s = (a + b + c) / 2

This can be written as:

s = ½(a + b + c)

Hence, s = ½(a + b + c).

Submit

About This Quiz

Herons Formula Basics: Semiperimeter, Validity & Setup - Quiz

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.
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2) Heron's Formula for area is

A = ab sin C

A = ½ bh

A = √[s(s − a)(s − b)(s − c)]

A = s²

Heron’s Formula for the area of a triangle with semiperimeter s is:

A = √[s(s − a)(s − b)(s − c)]

Hence, the correct formula is A = √[s(s − a)(s − b)(s − c)].

Explanation

Heron’s Formula for the area of a triangle with semiperimeter s is:

A = √[s(s − a)(s − b)(s − c)]

Hence, the correct formula is A = √[s(s − a)(s − b)(s − c)].

Submit

3) Why is the semiperimeter used in Heron's Formula?

It represents half the area

It simplifies writing (s − a)(s − b)(s − c)

It avoids decimals

It equals the radius

Heron’s Formula naturally uses the expressions (s − a), (s − b), and (s − c), so using s simplifies the expression inside the square root.

Hence, the semiperimeter is used because it simplifies (s − a)(s − b)(s − c).

Explanation

Submit

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

10 cm

12 cm

16 cm

18 cm

Compute the semiperimeter:

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

Hence, s = 16 cm.

Explanation

Compute the semiperimeter:

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

Hence, s = 16 cm.

Submit

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

½ab

2ab

(a + b)/2

For a right triangle with legs a and b, area = ½ab.

Heron’s Formula also gives the same result.

Hence, it matches ½ab.

Explanation

For a right triangle with legs a and b, area = ½ab.

Heron’s Formula also gives the same result.

Hence, it matches ½ab.

Submit

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

The volume

The product used to find squared area

The perimeter

The height

The expression s(s − a)(s − b)(s − c) is the product used before taking the square root. It gives the squared area of the triangle.

Hence, it represents the product used to find squared area.

Explanation

The expression s(s − a)(s − b)(s − c) is the product used before taking the square root. It gives the squared area of the triangle.

Hence, it represents the product used to find squared area.

Submit

7) What are the units of the result from Heron's Formula?

Linear (cm)

Cubic (cm³)

Square (cm²)

No units

Area is always measured in square units such as cm² or m². Hence, Heron’s Formula produces square units (cm²).

Explanation

Area is always measured in square units such as cm² or m². Hence, Heron’s Formula produces square units (cm²).

Submit

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

√[s(s − a)³]

√[3s]

√[a³]

½a²

For an equilateral triangle, a = b = c, and semiperimeter:

s = 3a/2

Substitute into Heron's Formula:

A = √[s(s − a)³]

Hence, area simplifies to √[s(s − a)³].

Explanation

For an equilateral triangle, a = b = c, and semiperimeter:

s = 3a/2

Substitute into Heron's Formula:

A = √[s(s − a)³]

Hence, area simplifies to √[s(s − a)³].

Submit

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

3, 4, 5

5, 5, 2008

7, 8, 2009

6, 8, 10

The triangle 3–4–5 is a right triangle, and its area is:

A = ½·3·4 = 6

This is a simple integer area.

Hence, the triangle 3, 4, 5 gives the simplest integer area.

Explanation

The triangle 3–4–5 is a right triangle, and its area is:

A = ½·3·4 = 6

This is a simple integer area.

Hence, the triangle 3, 4, 5 gives the simplest integer area.

Submit

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

Too small sides

Triangle inequality fails

Not right-angled

Only works for integer sides

Check the triangle inequality:

4 + 9 = 13, which is NOT greater than 14 → invalid triangle.

Hence, Heron’s Formula cannot be applied because the triangle is invalid.

Explanation

Check the triangle inequality:

4 + 9 = 13, which is NOT greater than 14 → invalid triangle.

Hence, Heron’s Formula cannot be applied because the triangle is invalid.

Submit

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

3 cm

17 cm

7 cm

10 cm

Compute (s − a):

s − a = 10 − 7 = 3

Hence, (s − a) = 3 cm.

Explanation

Compute (s − a):

s − a = 10 − 7 = 3

Hence, (s − a) = 3 cm.

Submit

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

All sides equal

Sum of any two sides > third side

Sum of any two sides < third side

Difference of any two sides > third side

For three lengths to form a triangle, they must satisfy the triangle inequality:

a + b > c

a + c > b

b + c > a

Hence, the sum of any two sides must be greater than the third side.

Explanation

For three lengths to form a triangle, they must satisfy the triangle inequality:

a + b > c

a + c > b

b + c > a

Hence, the sum of any two sides must be greater than the third side.

Submit

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

8 cm

10 cm

6 cm

9 cm

Given s = 12, the full perimeter is: perimeter = 2s = 24

Two sides are 7 and 9, so the third side is: third side = 24 − (7 + 9) = 24 − 16 = 8

Hence, the third side = 8 cm.

Explanation

Given s = 12, the full perimeter is: perimeter = 2s = 24

Two sides are 7 and 9, so the third side is: third side = 24 − (7 + 9) = 24 − 16 = 8

Hence, the third side = 8 cm.

Submit

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

Find s

Find height h

Find area with ½bh

Draw circle

Heron's Formula requires the semiperimeter s and the values (s − a), (s − b), (s − c).

The most efficient first step is to compute s.

Hence, the first step is to find s.

Explanation

Heron's Formula requires the semiperimeter s and the values (s − a), (s − b), (s − c).

The most efficient first step is to compute s.

Hence, the first step is to find s.

Submit

15) Why is Heron's Formula useful?

Finds area without altitude

Only works for right triangles

Replaces all formulas

Measures perimeter

Heron’s Formula is useful because it finds the area using only the side lengths, without needing the height.

Hence, it finds area without needing altitude.

Explanation

Heron’s Formula is useful because it finds the area using only the side lengths, without needing the height.

Hence, it finds area without needing altitude.

Submit

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

1 m

7 m

8 m

15 m

Compute (s − a):

s − a = 8 − 7 = 1

Hence, (s − a) = 1 m.

Explanation

Compute (s − a):

s − a = 8 − 7 = 1

Hence, (s − a) = 1 m.

Submit

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

Equilateral

Isosceles

Right

Obtuse

Check if the sides form a right triangle:

5² + 12² = 25 + 144 = 169 = 13²

So it satisfies the Pythagorean theorem.

Hence, the triangle is right-angled.

Explanation

Check if the sides form a right triangle:

5² + 12² = 25 + 144 = 169 = 13²

So it satisfies the Pythagorean theorem.

Hence, the triangle is right-angled.

Submit

18) Which triangle is not valid?

3, 4, 2005

7, 10, 2002

5, 12, 13

6, 8, 2009

Check triangle inequality.

For 7, 10, 2002:

7 + 10 = 17, which is NOT greater than 2002 → invalid.

Hence, 7, 10, 2002 is not a valid triangle.

Explanation

Check triangle inequality.

For 7, 10, 2002:

7 + 10 = 17, which is NOT greater than 2002 → invalid.

Hence, 7, 10, 2002 is not a valid triangle.

Submit

19) Which triangle below will have the largest area using Heron's Formula?

6, 6, 2006

10, 10, 2005

4, 5, 2009

7, 8, 9

Compute approximate areas:

Using Heron’s Formula, the triangle with sides 7, 8, 9 gives area ≈ 26.8, which is the largest among the choices.

Hence, the triangle 7, 8, 9 has the largest area.

Explanation

Compute approximate areas:

Using Heron’s Formula, the triangle with sides 7, 8, 9 gives area ≈ 26.8, which is the largest among the choices.

Hence, the triangle 7, 8, 9 has the largest area.

Submit

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

Area decreases

Area increases

Area constant

Increases then decreases

If one side increases while the others stay fixed, area increases up to a maximum, then decreases toward zero as the triangle becomes nearly flat. Hence, area increases then decreases.

Explanation

If one side increases while the others stay fixed, area increases up to a maximum, then decreases toward zero as the triangle becomes nearly flat. Hence, area increases then decreases.

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