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Introduction

This tutorial provides a straightforward guide on how to calculate the perimeter and area of an isosceles triangle. Understanding these calculations is essential for students and anyone working with geometric shapes in mathematics. This guide will break down the process into clear, actionable steps.

Step 1: Understanding the Properties of an Isosceles Triangle

An isosceles triangle has two equal sides and one base. Here’s what you need to know:

• Sides: Let the two equal sides be denoted as "a" and the base as "b."
• Height: The height (h) can be calculated using the Pythagorean theorem.

Step 2: Calculating the Perimeter

The perimeter (P) of an isosceles triangle is the sum of the lengths of all three sides. Use the formula:

• Perimeter Formula: [ P = 2a + b ]

Example Calculation

• If the sides are 5 cm each and the base is 6 cm: [ P = 2(5) + 6 = 10 + 6 = 16 \text{ cm} ]

Step 3: Calculating the Area

The area (A) of an isosceles triangle can be calculated using the following formula:

• Area Formula: [ A = \frac{1}{2} \times b \times h ]

Finding the Height

To find the height (h) when you know the lengths of the sides:

1. Use the Pythagorean theorem:

   • Divide the base (b) by 2 to find half of the base (b/2).
   • Create a right triangle with the height and half the base.

2. Apply the theorem: [ h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2} ]

Example Calculation

• Given sides of 5 cm and a base of 6 cm:
  1. Calculate half the base: [ \frac{b}{2} = \frac{6}{2} = 3 \text{ cm} ]
  2. Calculate the height: [ h = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \text{ cm} ]
  3. Calculate the area: [ A = \frac{1}{2} \times 6 \times 4 = 12 \text{ cm}^2 ]

Conclusion

In this tutorial, you learned how to calculate the perimeter and area of an isosceles triangle using clear formulas and examples. Remember to identify the lengths of the sides and the base before applying these formulas. Practice with various dimensions to strengthen your understanding and skills in geometry. For further learning, consider exploring other types of triangles, such as right-angled and scalene triangles.