Heron's Formula

This chapter teaches a powerful formula for finding the area of a triangle when the lengths of all three sides are known. The formula was given by Heron of Alexandria. If the sides of a triangle are a, b and c, the semi-perimeter is s = (a + b + c)/2, and the area is the square root of s(s - a)(s - b)(s - c). You will see that this formula gives the same result as the familiar formula, half of base times height, for right-angled triangles, and that it is especially useful for scalene triangles such as triangular plots, signboards and walls. You will apply Heron's formula to equilateral, isosceles and scalene triangles, to triangles whose sides are in a given ratio, and to practical problems such as finding the cost of painting a triangular wall or of laying grass on a triangular park. You will also find the area of a quadrilateral by dividing it into two triangles along a diagonal and applying the formula to each triangle.

Detailed Worksheet: Heron's Formula Section A - Definitions (10 marks) 1. What is the semi-perimeter of a triangle? Find the semi-perimeter of a triangle with sides 8 cm, 11 cm and 13 cm. (2 marks) 2. State Heron's formula for the area of a triangle. (2 marks) 3. Write the formula for the area of an equilateral triangle of side x. Find it for x = 4 cm in terms of the square root of 3. (2 marks) 4. When is Heron's formula more useful than the formula of half of base times height? (2 marks) 5. How can Heron's formula be used to find the area of a quadrilateral? (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the area of a triangle with sides 13 cm, 14 cm and 15 cm. (3 marks) 7. The sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 144 cm. Find its area. (3 marks) 8. Find the area of an isosceles triangle with equal sides 15 cm each and base 24 cm. Verify using base and height. (3 marks) 9. A triangular park has sides 26 m, 28 m and 30 m. Find the cost of laying grass on it at Rs 20 per square m. (3 marks) 10. The perimeter of a triangle is 42 cm and two of its sides are 13 cm and 14 cm. Find the third side and the area of the triangle. (3 marks) Section C - Diagrams (10 marks) 11. Draw a quadrilateral PQRS with PQ = 9 cm, QR = 40 cm, RS = 28 cm, SP = 15 cm and diagonal PR = 41 cm (a rough sketch is enough). Find its area using Heron's formula for both triangles. (4 marks) 12. Draw a rough figure of a triangular signboard with sides 7 cm, 24 cm and 25 cm. Find its area using Heron's formula and verify that it is right-angled. (3 marks) 13. Draw a rough figure of a rhombus with each side 10 cm and one diagonal 12 cm. Find its area by dividing it into two triangles. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. A triangular wall of a building has sides 122 m, 22 m and 120 m. Find its area. If it is painted at Rs 5 per square m, find the cost. Check whether the triangle is right-angled. (5 marks) 15. A field in the shape of a quadrilateral ABCD has AB = 13 m, BC = 14 m, CD = 18 m, DA = 15 m and diagonal AC = 15 m. Find the area of the field. (5 marks) 16. An umbrella is made by stitching 10 triangular pieces of cloth, each with sides 20 cm, 50 cm and 50 cm. Find the area of cloth needed, leaving the answer in terms of a square root. Then estimate it using the square root of 6 as about 2.45. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Write the formula, show the value of s and all steps in every question.