Herons Formula Quiz: Calculate Area with Herons Formula

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.

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.

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.

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.

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.

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.

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.

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.

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.

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)].

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.