Find the area of a triangle, given that its sides are 8 cm and 11 cm long, and its perimeter is 32 cm.
Answer: Third side = 32 − 8 − 11 = 13 cm; s = 16. Area = √(16 × 8 × 5 × 3) = √1920 = 8√30 ≈ 43.82 cm2.
Step-by-step solution
Given: Two sides 8 cm, 11 cm; Perimeter 32 cm
To find: Area of the triangle
To find: Area of the triangle
Idea: Find the third side from the perimeter, then use Heron’s formula, since no height is given.
- Third side = 32 − (8 + 11) = 13 cm. s = 32 ÷ 2 = 16 cm.1 mark
- s − a = 16 − 8 = 8, s − b = 16 − 11 = 5, s − c = 16 − 13 = 3.½ mark
- Area = √(16 × 8 × 5 × 3) = √1920 = √(64 × 30) = 8√30 ≈ 43.82 cm2.1½ marks
Area = 8√30 cm² ≈ 43.82 cm².
Check: 43.822 ≈ 1920 ✓. The answer is less than 12 × 8 × 11 = 44, the most two sides of 8 and 11 can enclose ✓.
Answer to write in the exam
Third side = 32 − 8 − 11 = 13 cm
s = 322 = 16 cm
Area = √(s(s − a)(s − b)(s − c)) = √(16 × 8 × 5 × 3)
= √1920 = 8√30
∴ Area ≈ 43.82 cm2
Common mistakes that cost marks
- Using s = 32 (the perimeter) instead of 16.
- Forgetting the third side and using only two sides.
- Simplifying √1920 wrongly; 1920 = 64 × 30.
How this can come in the exam
MCQ (1 mark)
Two sides of a triangle are 9 cm and 10 cm and its perimeter is 36 cm. Its area is
- 36 cm2
- 45 cm2
- 18√2 cm2
- 72 cm2
Show answer
(A) 36 cm2
Third side = 17; s = 18; √(18 × 9 × 8 × 1) = √1296 = 36 cm2.
Try one yourself
Two sides of a triangle are 13 cm and 14 cm and its perimeter is 42 cm. Find its area.
Show answer
Third side 15; s = 21; √(21 × 8 × 7 × 6) = √7056 = 84 cm2.
More questions like this
- The sides of a triangular plot are in the ratio 3: 5: 7; its perimeter is 300 m. Find its area.
- One diagonal of a rhombus is twice as long as the other diagonal. If the rhombus has area 128 cm2, find the length of the shorter diagonal.
- ABCD is a parallelogram. P and Q are any two points on side AB. What can you say about the ratio area (∆PCD): area (∆QCD)?
- O is any point on the diagonal PR of a parallelogram PQRS. Prove that the areas of triangles PSO and PQO are equal.
- If the mid-points of the sides of a 4-gon (also known as a quadrilateral, but we prefer to call it a ‘4-gon’) are joined in order, prove that the area of the parallelogram thus formed will be half of the area of the given 4-gon. (You may wonder whether the 4-gon thus formed is always a parallelogram, and if so, why? These questions will be tackled and answered later, when quadrilaterals are studied.)