An isosceles triangle has perimeter 40 cm; the equal sides are 15 cm each. Find the area of the triangle.
Answer: Base = 40 − 30 = 10 cm; height = √(152 − 52) = √200 = 10√2 cm. Area = 12 × 10 × 10√2 = 50√2 ≈ 70.71 cm2.
Step-by-step solution
Given: Perimeter 40 cm; Equal sides 15 cm
To find: Area
To find: Area
Idea: Find the base from the perimeter. The height from the apex meets the base at its midpoint, so use the Baudhāyana–Pythagoras theorem (or Heron’s formula).
- Base = 40 − 15 − 15 = 10 cm.½ mark
- The height bisects the base: h = √(152 − 52) = √(225 − 25) = √200 = 10√2 cm.1 mark
- Area = 12 × 10 × 10√2 = 50√2 ≈ 70.71 cm2.1 mark
- Check by Heron: s = 20; √(20 × 5 × 5 × 10) = √5000 = 50√2 ✓.½ mark
Area = 50√2 cm² ≈ 70.71 cm².
Answer to write in the exam
Base = 40 − 2 × 15 = 10 cm
h = √(152 − 52) = √200 = 10√2 cm
Area = 12 × 10 × 10√2 = 50√2
∴ Area ≈ 70.71 cm2
Common mistakes that cost marks
- Using the whole base 10 instead of half of it (5) in the Pythagoras step.
- Using the slant side 15 cm as the height (giving 75 cm2).
How this can come in the exam
MCQ (1 mark)
An isosceles triangle has perimeter 32 cm and equal sides of 10 cm. Its area is
- 48 cm2
- 60 cm2
- 80 cm2
- 40 cm2
Show answer
(A) 48 cm2
Base 12, height √(100 − 36) = 8, area 12 × 12 × 8 = 48 cm2.
Try one yourself
An isosceles triangle has perimeter 50 cm and equal sides of 13 cm each. Find its area.
Show answer
Base 24, height √(169 − 144) = 5, area 60 cm2.
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