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Heron’s Formula

Heron’s Formula. The Law of Cosines can be used to derive a formula for the area of a triangle based on its side lengths. This formula is called Heron ’ s Formula. Example 3: Landscaping Application .

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Heron’s Formula

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  1. Heron’s Formula

  2. The Law of Cosines can be used to derive a formula for the area of a triangle based on its side lengths. This formula is called Heron’s Formula.

  3. Example 3: Landscaping Application A garden has a triangular flower bed with sides measuring 2 yd, 6 yd, and 7 yd. What is the area of the flower bed to the nearest tenth of a square yard? Step 1 Find the value of s. Use the formula for half of the perimeter. Substitute 2 for a, 6 for b, and 7 for c.

  4. A = A = Example 3 Continued Step 2 Find the area of the triangle. Heron’s formula Substitute 7.5 for s. A = 5.6 Use a calculator to simplify. The area of the flower bed is 5.6 yd2.

  5. m C ≈ 112.0° Solve for m c. Find the area of the triangle by using the formula area = ab sin c. area Example 3 Continued Check Find the measure of the largest angle, C. c2 = a2+ b2– 2ab cos C Law of Cosines 72 = 22+ 62– 2(2)(6) cos C Substitute. Solve for cos C. cos C ≈ –0.375 

  6. Check It Out! Example 3 The surface of a hotel swimming pool is shaped like a triangle with sides measuring 50 m, 28 m, and 30 m. What is the area of the pool’s surface to the nearest square meter? Step 1 Find the value of s. Use the formula for half of the perimeter. Substitute 50 for a, 28 for b, and 30 for c.

  7. A = Check It Out! Example 3 Continued Step 2 Find the area of the triangle. Heron’s formula Substitute 54 for s. Use a calculator to simplify. A = 367 m2 The area of the flower bed is 367 m2.

  8. m A ≈ 119.0° Solve for m A. Find the area of the triangle by using the formula area = ab sin A. Check It Out! Example 3 Continued Check Find the measure of the largest angle, A. 502 = c2+ b2– 2cb cos A Law of Cosines 502 = 302+ 282– 2(30)(28) cos A Substitute. Solve for cos A. cos A ≈ –0.4857 

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