Herons Formula Quiz: Calculate Area With Herons Formula

1) Find the area of a triangle with sides 9 cm, 10 cm, and 17 cm.

36 cm²

38.4 cm²

39 cm²

37 cm²

s = (9+10+17)/2 = 18. A = √(18×9×8×1) = √1296 = 36 cm². Since 36² = 1296, the result is exact. Triangle inequality check: 9+10 = 19 > 17 ✓. Options B, C, and D all require products larger than 1296 and do not match.

Explanation

s = (9+10+17)/2 = 18. A = √(18×9×8×1) = √1296 = 36 cm². Since 36² = 1296, the result is exact. Triangle inequality check: 9+10 = 19 > 17 ✓. Options B, C, and D all require products larger than 1296 and do not match.

2) Find the area of a triangle with sides 14 cm, 15 cm, and 13 cm.

84 cm²

85 cm²

86 cm²

87 cm²

s = (14+15+13)/2 = 21. A = √(21×7×6×8) = √7056 = 84 cm². Since 84² = 7056, the result is exact. This matches Q8 (sides 13, 14, 15) because the same three side lengths in any order yield the same semi-perimeter and area. Options B, C, and D require products larger than 7056 and do not match.

Explanation

s = (14+15+13)/2 = 21. A = √(21×7×6×8) = √7056 = 84 cm². Since 84² = 7056, the result is exact. This matches Q8 (sides 13, 14, 15) because the same three side lengths in any order yield the same semi-perimeter and area. Options B, C, and D require products larger than 7056 and do not match.

3) Find the area of a triangle with sides 9 m, 10 m, and 11 m.

42.4 m²

44 m²

45 m²

40 m²

s = (9+10+11)/2 = 15. A = √(15×6×5×4) = √1800 ≈ 42.43 m². Option B requires √1936 = 44. Option C requires √2025 = 45. Option D requires √1600 = 40. None of these match the correctly computed product of 1800.

Explanation

s = (9+10+11)/2 = 15. A = √(15×6×5×4) = √1800 ≈ 42.43 m². Option B requires √1936 = 44. Option C requires √2025 = 45. Option D requires √1600 = 40. None of these match the correctly computed product of 1800.

4) Select all statements that are true about Heron's formula.

It can find the area using only the three side lengths

It works for scalene triangles

It requires the height of the triangle

It does not require any angle measurements

Heron's formula A = √[s(s−a)(s−b)(s−c)] uses only side lengths a, b, c — no height or angles needed, confirming A and D. It applies to all triangle types including scalene, confirming B. Option C is false because Heron's formula specifically eliminates the need for height.

Explanation

Heron's formula A = √[s(s−a)(s−b)(s−c)] uses only side lengths a, b, c — no height or angles needed, confirming A and D. It applies to all triangle types including scalene, confirming B. Option C is false because Heron's formula specifically eliminates the need for height.

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5) Find the area of a triangle with sides 7 m, 24 m, and 25 m.

84 m²

83 m²

85 m²

80 m²

s = (7+24+25)/2 = 28. A = √(28×21×4×3) = √7056 = 84 m². Confirmed as a right triangle since 7²+24² = 49+576 = 625 = 25², giving (7×24)/2 = 84 m² as a second verification.

Explanation

s = (7+24+25)/2 = 28. A = √(28×21×4×3) = √7056 = 84 m². Confirmed as a right triangle since 7²+24² = 49+576 = 625 = 25², giving (7×24)/2 = 84 m² as a second verification.

6) Find the area of a triangle with sides 5 cm, 12 cm, and 13 cm.

31 cm²

30 cm²

32 cm²

33 cm²

s = (5+12+13)/2 = 15. A = √(15×10×3×2) = √900 = 30 cm². Confirmed as a right triangle since 5²+12² = 25+144 = 169 = 13², giving (5×12)/2 = 30 cm² as a second verification. Option A requires √961 = 31.

Explanation

s = (5+12+13)/2 = 15. A = √(15×10×3×2) = √900 = 30 cm². Confirmed as a right triangle since 5²+12² = 25+144 = 169 = 13², giving (5×12)/2 = 30 cm² as a second verification. Option A requires √961 = 31.

7) Find the area of a triangle with sides 10 m, 14 m, and 18 m.

69.6 m²

70 m²

71 m²

72 m²

s = (10+14+18)/2 = 21. A = √(21×11×7×3) = √4851 ≈ 69.65 m². Option B requires √4900 = 70. Option C requires √5041 = 71. Option D requires √5184 = 72. All three exceed the actual product of 4851.

Explanation

s = (10+14+18)/2 = 21. A = √(21×11×7×3) = √4851 ≈ 69.65 m². Option B requires √4900 = 70. Option C requires √5041 = 71. Option D requires √5184 = 72. All three exceed the actual product of 4851.

8) Which expression correctly states Heron's formula for the area of a triangle?

A = √[s(s−a)(s−b)(s−c)]

A = √[s(s−a)(s−b)]

A = s(s−a)(s−b)(s−c)

A = (s−a)(s−b)(s−c)

Heron's formula is A = √[s(s−a)(s−b)(s−c)]. Option B is missing the (s−c) factor. Option C removes the square root, producing units of length³ rather than area. Option D drops the s factor entirely, giving an incorrect result for every triangle except special cases.

Explanation

Heron's formula is A = √[s(s−a)(s−b)(s−c)]. Option B is missing the (s−c) factor. Option C removes the square root, producing units of length³ rather than area. Option D drops the s factor entirely, giving an incorrect result for every triangle except special cases.

9) The semi-perimeter is always equal to half the triangle's perimeter.

True

False

The answer is True. By definition, s = (a+b+c)/2, where a+b+c is the full perimeter. Dividing any perimeter by 2 always gives exactly half, regardless of triangle shape or size. This relationship is the foundation of Heron's formula since s must be calculated first before computing A = √[s(s−a)(s−b)(s−c)].

Explanation

The answer is True. By definition, s = (a+b+c)/2, where a+b+c is the full perimeter. Dividing any perimeter by 2 always gives exactly half, regardless of triangle shape or size. This relationship is the foundation of Heron's formula since s must be calculated first before computing A = √[s(s−a)(s−b)(s−c)].

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

6 cm²

7 cm²

8 cm²

10 cm²

s = (3+4+5)/2 = 6. A = √(6×3×2×1) = √36 = 6 cm². This is also a 3-4-5 right triangle, confirmed by (3×4)/2 = 6 cm². Options B, C, and D all exceed the correct value and do not satisfy either verification method.

Explanation

s = (3+4+5)/2 = 6. A = √(6×3×2×1) = √36 = 6 cm². This is also a 3-4-5 right triangle, confirmed by (3×4)/2 = 6 cm². Options B, C, and D all exceed the correct value and do not satisfy either verification method.

11) Select all correct forms of Heron's formula.

A = √[s(s−a)(s−b)(s−c)]

S = (a+b+c)/2

A = (a×b×c)/s

S = (a+b)/2

Option A is the correct area formula. Option B is the correct semi-perimeter definition. Option C, A = (a×b×c)/s, is not a valid area formula. Option D omits side c from the semi-perimeter.

Explanation

Option A is the correct area formula. Option B is the correct semi-perimeter definition. Option C, A = (a×b×c)/s, is not a valid area formula. Option D omits side c from the semi-perimeter.

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12) A triangle has sides 8 m, 15 m, and 17 m. Find its area.

60 m²

61 m²

62 m²

59 m²

s = (8+15+17)/2 = 20. A = √(20×12×5×3) = √3600 = 60 m². Confirmed as a right triangle since 8²+15² = 64+225 = 289 = 17², giving (8×15)/2 = 60 m² as a second verification.

Explanation

s = (8+15+17)/2 = 20. A = √(20×12×5×3) = √3600 = 60 m². Confirmed as a right triangle since 8²+15² = 64+225 = 289 = 17², giving (8×15)/2 = 60 m² as a second verification.

13) Find the area of a triangle with sides 13 cm, 14 cm, and 15 cm.

86 cm²

85 cm²

84 cm²

87 cm²

s = (13+14+15)/2 = 21. A = √(21×8×7×6) = √7056 = 84 cm². Since 84² = 7056, the result is exact. Option A requires √7396. Option B requires √7225. Option D requires √7569. None of these match the product 7056.

Explanation

s = (13+14+15)/2 = 21. A = √(21×8×7×6) = √7056 = 84 cm². Since 84² = 7056, the result is exact. Option A requires √7396. Option B requires √7225. Option D requires √7569. None of these match the product 7056.

14) Find the area of a triangle with sides 6 m, 8 m, and 10 m.

27 m²

25 m²

26 m²

24 m²

s = (6+8+10)/2 = 12. A = √(12×6×4×2) = √576 = 24 m². This is a 6-8-10 right triangle (scaled 3-4-5), confirmed by (6×8)/2 = 24 m². Options A, B, and C all overestimate the area.

Explanation

s = (6+8+10)/2 = 12. A = √(12×6×4×2) = √576 = 24 m². This is a 6-8-10 right triangle (scaled 3-4-5), confirmed by (6×8)/2 = 24 m². Options A, B, and C all overestimate the area.

15) Find the area of a triangle with sides 7 cm, 8 cm, and 9 cm.

26.83 cm²

28 cm²

29 cm²

27.5 cm²

s = (7+8+9)/2 = 12. A = √(12×5×4×3) = √720 = 26.83 cm². Option B requires √784 = 28. Option C requires √841 = 29. Option D requires √756.25 ≈ 27.5. None of these match the correctly computed product of 720.

Explanation

s = (7+8+9)/2 = 12. A = √(12×5×4×3) = √720 = 26.83 cm². Option B requires √784 = 28. Option C requires √841 = 29. Option D requires √756.25 ≈ 27.5. None of these match the correctly computed product of 720.

16) Which expression correctly gives the semi-perimeter s of a triangle with sides a, b, and c?

S = a+b+c

S = (a+b)/2

S = (a+b+c)/2

S = (a×b×c)/2

The semi-perimeter is defined as half the total perimeter. Since the perimeter = a+b+c, dividing by 2 gives s = (a+b+c)/2. Option A gives the full perimeter. Option B omits side c entirely. Option D multiplies the sides instead of adding them, which has no connection to the semi-perimeter definition.

Explanation

The semi-perimeter is defined as half the total perimeter. Since the perimeter = a+b+c, dividing by 2 gives s = (a+b+c)/2. Option A gives the full perimeter. Option B omits side c entirely. Option D multiplies the sides instead of adding them, which has no connection to the semi-perimeter definition.

17) Heron's formula can be used for any triangle when all three sides are known.

True

False

The answer is True. Heron's formula requires only the three side lengths to compute the area, with no need for angles or height. It works equally for scalene, isosceles, equilateral, and right triangles. The only requirement is that the sides satisfy the triangle inequality. As long as a valid triangle exists, Heron's formula will always produce the correct area.

Explanation

The answer is True. Heron's formula requires only the three side lengths to compute the area, with no need for angles or height. It works equally for scalene, isosceles, equilateral, and right triangles. The only requirement is that the sides satisfy the triangle inequality. As long as a valid triangle exists, Heron's formula will always produce the correct area.

18) A triangle has sides 5 m, 6 m, and 7 m. Find its area.

14.7 m²

16.2 m²

17 m²

14 m²

s = (5+6+7)/2 = 9. A = √(9×4×3×2) = √216 = 14.70 m². Option B gives 16.2, which would require the product under the root to equal ≈262.4. Options C and D are further from the correct calculation.

Explanation

s = (5+6+7)/2 = 9. A = √(9×4×3×2) = √216 = 14.70 m². Option B gives 16.2, which would require the product under the root to equal ≈262.4. Options C and D are further from the correct calculation.

19) When Heron's formula is applied to a right triangle, it gives the same result as ½ × base × height.

True

False

The answer is True. For any right triangle the two legs serve as base and height. Heron's formula simplifies to the same value as ½ × base × height for all right triangles. For example, the 3-4-5 triangle gives A = √36 = 6 using Heron's formula, and (3×4)/2 = 6 using the standard formula, confirming consistency.

Explanation

The answer is True. For any right triangle the two legs serve as base and height. Heron's formula simplifies to the same value as ½ × base × height for all right triangles. For example, the 3-4-5 triangle gives A = √36 = 6 using Heron's formula, and (3×4)/2 = 6 using the standard formula, confirming consistency.

20) All three side lengths must satisfy the triangle inequality for Heron's formula to produce a valid area.

True

False

The answer is True. If the triangle inequality is violated, the product s(s−a)(s−b)(s−c) becomes negative, making √[s(s−a)(s−b)(s−c)] undefined over the real numbers. Heron's formula only produces a meaningful positive area when the three sides form a valid triangle. Checking the triangle inequality is therefore a necessary first step before applying the formula.

Explanation

The answer is True. If the triangle inequality is violated, the product s(s−a)(s−b)(s−c) becomes negative, making √[s(s−a)(s−b)(s−c)] undefined over the real numbers. Heron's formula only produces a meaningful positive area when the three sides form a valid triangle. Checking the triangle inequality is therefore a necessary first step before applying the formula.