1. Heron's formula for the area of a triangle with side lengths a, b, and c is:
A. √[s(s-a)(s-b)(s-c)]
B. s(s-a)(s-b)(s-c)
C. ½ × base × height
D. √[(s-a)(s-b)(s-c)]
Show Answer
Answer: A. √[s(s-a)(s-b)(s-c)]
Explanation: Heron's formula states that Area = √[s(s - a)(s - b)(s - c)], where s is the semi-perimeter.
2. In Heron's formula, 's' represents the:
A. Perimeter
B. Semi-perimeter
C. Square of perimeter
D. Side length
Show Answer
Answer: B. Semi-perimeter
Explanation: 's' stands for semi-perimeter, which is equal to (a + b + c) / 2.
3. If the sides of a triangle are 3 cm, 4 cm, and 5 cm, its semi-perimeter (s) is:
A. 12 cm
B. 6 cm
C. 8 cm
D. 10 cm
Show Answer
Answer: B. 6 cm
Explanation: s = (3 + 4 + 5) / 2 = 12 / 2 = 6 cm.
4. The area of a triangle with sides 3 cm, 4 cm, and 5 cm is:
A. 6 cm²
B. 12 cm²
C. 15 cm²
D. 20 cm²
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Answer: A. 6 cm²
Explanation: s = 6; Area = √[6 × (6-3) × (6-4) × (6-5)] = √[6 × 3 × 2 × 1] = √36 = 6 cm².
5. The area of an equilateral triangle with side 'a' is given by:
A. (√3 / 2) a²
B. (√3 / 4) a²
C. (√3 / 4) a
D. (1 / 2) a²
Show Answer
Answer: B. (√3 / 4) a²
Explanation: Using Heron's formula or standard trigonometry, the area of an equilateral triangle of side 'a' is (√3 / 4)a².
6. What is the perimeter of an equilateral triangle of side 10 cm?
A. 20 cm
B. 30 cm
C. 15 cm
D. 40 cm
Show Answer
Answer: B. 30 cm
Explanation: Perimeter = 3 × side = 3 × 10 = 30 cm.
7. The area of an equilateral triangle with side 8 cm is:
A. 16√3 cm²
B. 32√3 cm²
C. 64√3 cm²
D. 8√3 cm²
Show Answer
Answer: A. 16√3 cm²
Explanation: Area = (√3 / 4) × 8² = (√3 / 4) × 64 = 16√3 cm².
8. An isosceles triangle has equal sides of 5 cm each and a base of 6 cm. Its area is:
A. 12 cm²
B. 24 cm²
C. 18 cm²
D. 15 cm²
Show Answer
Answer: A. 12 cm²
Explanation: s = (5 + 5 + 6) / 2 = 8 cm. Area = √[8 × (8-5) × (8-5) × (8-6)] = √[8 × 3 × 3 × 2] = √144 = 12 cm².
9. If the perimeter of a triangle is 60 cm, then its semi-perimeter is:
A. 120 cm
B. 30 cm
C. 20 cm
D. 15 cm
Show Answer
Answer: B. 30 cm
Explanation: Semi-perimeter s = Perimeter / 2 = 60 / 2 = 30 cm.
10. The perimeter of a triangular field is 450 m and its sides are in the ratio 13 : 12 : 5. Its smallest side is:
A. 75 m
B. 150 m
C. 180 m
D. 90 m
Show Answer
Answer: A. 75 m
Explanation: 13x + 12x + 5x = 450 ⇒ 30x = 450 ⇒ x = 15. Smallest side = 5 × 15 = 75 m.
11. The area of a triangle having base 12 cm and height 5 cm is:
A. 60 cm²
B. 30 cm²
C. 15 cm²
D. 45 cm²
Show Answer
Answer: B. 30 cm²
Explanation: Area = ½ × base × height = ½ × 12 × 5 = 30 cm².
12. If each side of an equilateral triangle is doubled, its area becomes how many times?
A. 2 times
B. 3 times
C. 4 times
D. 8 times
Show Answer
Answer: C. 4 times
Explanation: Area is proportional to (side)². If side becomes 2a, Area becomes (2)² = 4 times.
13. The area of a right-angled triangle with base 8 cm and hypotenuse 10 cm is:
A. 24 cm²
B. 40 cm²
C. 48 cm²
D. 80 cm²
Show Answer
Answer: A. 24 cm²
Explanation: Height = √(10² - 8²) = √(100 - 64) = √36 = 6 cm. Area = ½ × 8 × 6 = 24 cm².
14. If the perimeter of an equilateral triangle is 180 cm, its area is:
A. 900√3 cm²
B. 1800√3 cm²
C. 3600√3 cm²
D. 450√3 cm²
Show Answer
Answer: A. 900√3 cm²
Explanation: Side = 180 / 3 = 60 cm. Area = (√3 / 4) × 60² = (√3 / 4) × 3600 = 900√3 cm².
15. Heron was a famous mathematician born in:
A. India
B. Alexandria (Egypt)
C. Greece
D. Rome
Show Answer
Answer: B. Alexandria (Egypt)
Explanation: Heron was born in Alexandria in Egypt.
16. The sides of a triangle are in the ratio 3 : 5 : 7 and its perimeter is 300 m. Its area is:
A. 1500√3 m²
B. 3000 m²
C. 750√3 m²
D. 1200√3 m²
Show Answer
Answer: A. 1500√3 m²
Explanation: 3x+5x+7x = 300 ⇒ 15x = 300 ⇒ x = 20. Sides are 60m, 100m, 140m. s = 150m. Area = √[150 × 90 × 50 × 10] = 1500√3 m².
17. An isosceles right triangle has area 200 cm². The length of its hypotenuse is:
A. 20 cm
B. 20√2 cm
C. 40 cm
D. 10√2 cm
Show Answer
Answer: B. 20√2 cm
Explanation: Area = ½ × x² = 200 ⇒ x² = 400 ⇒ x = 20. Hypotenuse = √(20² + 20²) = √800 = 20√2 cm.
18. The height corresponding to the longest side of a triangle with sides 13 cm, 14 cm, and 15 cm is:
A. 11.2 cm
B. 12 cm
C. 10 cm
D. 8.4 cm
Show Answer
Answer: A. 11.2 cm
Explanation: s = 21. Area = √[21 × 8 × 7 × 6] = 84 cm². Longest side = 15. Area = ½ × 15 × h = 84 ⇒ h = 168 / 15 = 11.2 cm.
19. If each side of a triangle is halved, its area becomes:
A. One-half
B. One-fourth
C. One-third
D. One-eighth
Show Answer
Answer: B. One-fourth
Explanation: Scaling sides by 1/2 changes the area by (1/2)² = 1/4 of the original area.
20. In an isosceles triangle, the length of two equal sides is 13 cm each and perimeter is 36 cm. The area is:
A. 60 cm²
B. 30 cm²
C. 120 cm²
D. 90 cm²
Show Answer
Answer: A. 60 cm²
Explanation: Third side = 36 - 26 = 10 cm. s = 18. Area = √[18 × 5 × 5 × 8] = √3600 = 60 cm².
21. The side lengths of a rhombus are 10 cm and one diagonal is 12 cm. The area of the rhombus is:
A. 96 cm²
B. 48 cm²
C. 120 cm²
D. 192 cm²
Show Answer
Answer: A. 96 cm²
Explanation: Diagonal bisects it into two triangles with sides 10, 10, 12. Area of 1 triangle = 48 cm². Total area = 2 × 48 = 96 cm².
22. The sides of a triangle are 11 cm, 12 cm, and 13 cm. Its semi-perimeter is:
A. 36 cm
B. 18 cm
C. 24 cm
D. 12 cm
Show Answer
Answer: B. 18 cm
Explanation: s = (11 + 12 + 13) / 2 = 36 / 2 = 18 cm.
23. If the area of an equilateral triangle is 16√3 cm², its perimeter is:
A. 24 cm
B. 12 cm
C. 48 cm
D. 36 cm
Show Answer
Answer: A. 24 cm
Explanation: (√3 / 4) a² = 16√3 ⇒ a² = 64 ⇒ a = 8 cm. Perimeter = 3 × 8 = 24 cm.
24. The area of a triangle with sides 8 cm, 11 cm, and perimeter 32 cm is:
A. 8√30 cm²
B. 16√30 cm²
C. 4√30 cm²
D. 12√30 cm²
Show Answer
Answer: A. 8√30 cm²
Explanation: Third side = 32 - 19 = 13 cm. s = 16. Area = √[16 × 8 × 5 × 3] = √1920 = 8√30 cm².
25. Unit of area in SI system is:
Show Answer
Answer: B. m²
Explanation: The SI unit of area is square meter (m²).
26. If a triangle has sides a, b, c and (s - a) = 5 cm, (s - b) = 10 cm, (s - c) = 1 cm, then semi-perimeter s is:
A. 16 cm
B. 32 cm
C. 20 cm
D. 18 cm
Show Answer
Answer: A. 16 cm
Explanation: Adding (s-a)+(s-b)+(s-c) = 3s - (a+b+c) = 3s - 2s = s. Thus s = 5 + 10 + 1 = 16 cm.
27. Using data from Q26, the area of the triangle is:
A. 20√2 cm²
B. 20√10 cm²
C. 40 cm²
D. 10√5 cm²
Show Answer
Answer: B. 20√10 cm²
Explanation: Area = √[16 × 5 × 10 × 1] = √800 = 20√2 cm² (Wait, √800 = √(400×2) = 20√2 cm²).
28. The altitude of an equilateral triangle of side 12 cm is:
A. 6√3 cm
B. 12√3 cm
C. 3√3 cm
D. 6 cm
Show Answer
Answer: A. 6√3 cm
Explanation: Altitude of equilateral triangle = (√3 / 2) × side = (√3 / 2) × 12 = 6√3 cm.
29. Can Heron's formula be used to find the area of a quadrilateral?
A. Yes, by dividing it into two triangles
B. No, never
C. Only if it is a square
D. Only if it is a circle
Show Answer
Answer: A. Yes, by dividing it into two triangles
Explanation: Any quadrilateral can be divided into two triangles by drawing a diagonal, and Heron's formula can be applied to both.
30. If the area of a right-angled isosceles triangle is 32 cm², the length of its hypotenuse is:
A. 8√2 cm
B. 8 cm
C. 16 cm
D. 4√2 cm
Show Answer
Answer: A. 8√2 cm
Explanation: ½ × x² = 32 ⇒ x² = 64 ⇒ x = 8. Hypotenuse = √(8² + 8²) = √128 = 8√2 cm.
