Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Find the area of a triangle with sides 13 cm, 14 cm, and 15 cm. Area = cm2

Correct! Heron's formula for scalene triangles.

(2) Find the area of a rhombus with sides 10 cm and one diagonal 16 cm. Area = cm2

Perfect! Used Pythagorean theorem to find missing diagonal.

(3) Find the area of a trapezium with parallel sides 15 cm and 25 cm and height 12 cm. Area = cm2

Excellent! Trapezium formula application.

(4) Find the area of a square whose diagonal is 142 cm. Area = cm2

Great! Or use Area = 12 × d2 = 12 × 1422 = 196 cm2.

(5) Find the area of a parallelogram with base 20 cm and height 15 cm. Area = cm2

Perfect! Straightforward parallelogram calculation.

Short Answer Questions (2 Marks Each)

Answer each question with clear calculations

(1) The sides of a triangle are 7 cm, 24 cm, and 25 cm. Find its area. Area = cm2

Perfect! Pythagorean triplet: 7-24-25.

(2) A rhombus has diagonals 18 cm and 24 cm. Find its area. Area = cm2

Excellent! Direct diagonal formula application.

(3) Find the area of a trapezium with parallel sides 12 cm and 20 cm and height 10 cm. Area = cm2

Great work!

(4) A rectangle has perimeter 60 cm and length 18 cm. Find its area. Area = cm2

(5) The sides of a parallelogram are 15 cm and 10 cm, with height 8 cm corresponding to the longer side. Find its area. Area = cm2

Perfect! Used the height corresponding to the longer side.

Long Answer Questions (4 Marks Each)

(1) A triangular park has sides 50 m, 120 m, and 130 m. Find its area. Area = m2

This is a scaled version of the 5-12-13 Pythagorean triplet (×10).

(2) A rhombus has sides 17 cm and one diagonal 30 cm. Find its area. Area = cm2

(3) A trapezium has parallel sides 18 cm and 30 cm and non-parallel sides 12 cm and 14 cm. Find its area. Area = cm2

(4) A rectangle has length 25 cm and breadth 15 cm. A square of side 10 cm is cut from one corner. Find the remaining area. Remaining area = cm2

(5) The sides of a triangle are 21 cm, 28 cm, and 35 cm. Find its area using Heron's formula. Area = cm2

This is a scaled 3-4-5 triangle (×7), so area = 12 × 21 × 28 = 294 cm2

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) The area of a triangle with sides 5 cm, 12 cm, and 13 cm is:

(a) 30 cm2 (b) 32 cm2 (c) 35 cm2 (d) 36 cm2

Correct! It's a 5-12-13 right triangle, so area = 12 × 5 × 12 = 30 cm2.

(2) The area of a rhombus with diagonals 16 cm and 12 cm is:

(a) 96 cm2 (b) 92 cm2 (c) 100 cm2 (d) 98 cm2

Correct! Area = 12 × 16 × 12 = 96 cm2.

(3) The area of a trapezium with parallel sides 10 cm and 14 cm and height 6 cm is:

(a) 72 cm2 (b) 70 cm2 (c) 68 cm2 (d) 74 cm2

Correct! Area = 12 × (10 + 14) × 6 = 72 cm2.

(4) The area of a square with diagonal 10 cm is:

(a) 50 cm2 (b) 48 cm2 (c) 52 cm2 (d) 54 cm2

Correct! Area = 12 × d2 = 12 × 102 = 50 cm2.

(5) A parallelogram has base 18 cm and height 12 cm. Its area is:

(a) 216 cm2 (b) 210 cm2 (c) 220 cm2 (d) 218 cm2

Correct! Area = 18 × 12 = 216 cm2.

(6) The area of a triangle with sides 8 cm, 15 cm, 17 cm is:

(a) 60 cm2 (b) 58 cm2 (c) 62 cm2 (d) 64 cm2

Correct! It's an 8-15-17 right triangle, so area = 12 × 8 × 15 = 60 cm2.

(7) A rectangle has perimeter 40 cm and length 12 cm. Its area is:

(a) 96 cm2 (b) 92 cm2 (c) 100 cm2 (d) 98 cm2

Correct! Breadth = (40÷2) - 12 = 8 cm. Area = 12 × 8 = 96 cm2.

(8) The area of a trapezium with parallel sides 16 cm and 24 cm, height 10 cm is:

(a) 200 cm2 (b) 190 cm2 (c) 210 cm2 (d) 205 cm2

Correct! Area = 12 × (16 + 24) × 10 = 200 cm2.

(9) A rhombus has diagonals 20 cm and 15 cm. Its area is:

(a) 150 cm2 (b) 155 cm2 (c) 145 cm2 (d) 160 cm2

Correct! Area = 12 × 20 × 15 = 150 cm2.

(10) A triangle has sides 9 cm, 12 cm, and 15 cm. Its area is:

(a) 54 cm2 (b) 50 cm2 (c) 52 cm2 (d) 56 cm2

Correct! It's a 9-12-15 right triangle (3-4-5 ×3), so area = 12 × 9 × 12 = 54 cm2.

Advanced Area Challenge

Determine whether these statements are True or False:

Advanced Area Quiz

🎉 Congratulations! What You've Mastered:

You have successfully completed the "Advanced Area and Perimeter" hard worksheet and learned:

(1) Heron's Formula Mastery: Calculating areas of scalene triangles using semi-perimeter and all three sides

(2) Pythagorean Triplet Recognition: Identifying and utilizing right triangles like 5-12-13, 8-15-17, and their multiples

(3) Advanced Rhombus Calculations: Finding missing diagonals using Pythagorean theorem and side lengths

(4) Diagonal-Based Area Formulas: Converting between diagonal and side measurements for squares and rhombi

(5) Complex Trapezium Problems: Solving trapeziums with all four sides given using advanced geometric methods

(6) Composite Area Calculations: Finding remaining areas after removing sections from larger shapes

(7) Scaled Triangle Recognition: Identifying relationships like 13-14-15 and 21-28-35 triangles

(8) Coordinate Geometry Applications: Using analytical methods for complex quadrilateral area calculations

(9) Multi-step Problem Solving: Combining multiple geometric concepts in single complex problems

(10) Perimeter-Area Relationships: Converting perimeter information to find missing dimensions and areas

(11) Advanced Formula Applications: Using Area = ½d² for squares and ½d₁d₂ for rhombi from diagonal measurements

(12) Geometric Verification: Checking solutions using multiple methods and verifying Pythagorean relationships

(13) Real-world Problem Modeling: Applying area calculations to practical scenarios like parks and construction

(14) Mathematical Reasoning: Understanding why formulas work and when to apply different methods

(15) Precision in Complex Calculations: Managing multi-step calculations involving square roots and advanced operations

Outstanding work! You now have mastery over advanced area and perimeter calculations with sophisticated problem-solving skills!