Area And Perimeter Of Triangles Worksheet

Area and Perimeter of Triangles Worksheet: A Comprehensive Guide

This comprehensive guide delves into the world of triangles, specifically focusing on calculating their area and perimeter. We'll explore the different formulas, provide practical examples, and offer a wealth of exercises to solidify your understanding. This resource serves as a complete worksheet solution, ideal for students, educators, and anyone looking to master these fundamental geometric concepts.

Understanding Triangles: A Quick Recap

Before diving into the calculations, let's refresh our understanding of triangles. A triangle is a polygon with three sides and three angles. The sum of the interior angles of any triangle always equals 180 degrees. Triangles are classified based on their sides and angles:

• Equilateral Triangle: All three sides are equal in length, and all three angles are equal (60 degrees each).
• Isosceles Triangle: Two sides are equal in length, and the angles opposite those sides are also equal.
• Scalene Triangle: All three sides are of different lengths, and all three angles are different.
• Right-Angled Triangle: One angle is a right angle (90 degrees).
• Acute Triangle: All three angles are less than 90 degrees.
• Obtuse Triangle: One angle is greater than 90 degrees.

Calculating the Perimeter of a Triangle

The perimeter of any polygon is simply the total length of its sides. For a triangle, this means adding the lengths of all three sides.

Formula:

Perimeter (P) = side a + side b + side c

Where 'a', 'b', and 'c' represent the lengths of the three sides of the triangle.

Example:

Consider a triangle with sides measuring 5 cm, 7 cm, and 9 cm.

Perimeter = 5 cm + 7 cm + 9 cm = 21 cm

Calculating the Area of a Triangle

Calculating the area of a triangle depends on the information available. We'll explore the two most common methods:

1. Using Base and Height

This is the most widely used method, especially for right-angled triangles.

Formula:

Area (A) = (1/2) * base * height

Where 'base' is the length of one side of the triangle, and 'height' is the perpendicular distance from the base to the opposite vertex (the highest point).

Example:

Consider a triangle with a base of 10 cm and a height of 6 cm.

Area = (1/2) * 10 cm * 6 cm = 30 cm²

2. Using Heron's Formula

Heron's formula is particularly useful when you know the lengths of all three sides but not the height.

Formula:

Area (A) = √[s(s-a)(s-b)(s-c)]

Where:

• 'a', 'b', and 'c' are the lengths of the three sides.
• 's' is the semi-perimeter, calculated as s = (a + b + c) / 2

Example:

Consider a triangle with sides of 5 cm, 6 cm, and 7 cm.

1. Calculate the semi-perimeter: s = (5 + 6 + 7) / 2 = 9 cm
2. Apply Heron's formula: A = √[9(9-5)(9-6)(9-7)] = √[9 * 4 * 3 * 2] = √216 ≈ 14.7 cm²

Worksheet Exercises: Area and Perimeter of Triangles

Now let's put our knowledge into practice with a series of exercises. Solve the following problems, showing your work for each.

Section 1: Perimeter Calculations

1. Find the perimeter of a triangle with sides measuring 8 cm, 12 cm, and 15 cm.
2. An equilateral triangle has a side length of 10 inches. What is its perimeter?
3. A triangle has sides of length 4.5 m, 6.2 m, and 7.8 m. Calculate its perimeter.
4. An isosceles triangle has two equal sides of 11 cm each. The third side is 8 cm. What is the perimeter?
5. If the perimeter of a triangle is 36 cm and two of its sides measure 10 cm and 12 cm, what is the length of the third side?

Section 2: Area Calculations (Using Base and Height)

1. Calculate the area of a triangle with a base of 14 cm and a height of 8 cm.
2. A right-angled triangle has legs (sides forming the right angle) of 5 cm and 12 cm. Find its area.
3. A triangle has a base of 20 mm and a height of 15 mm. What is its area?
4. The area of a triangle is 48 m² and its base is 12 m. What is its height?
5. A triangular garden has a base of 18 feet and a height of 10 feet. What is its area?

Section 3: Area Calculations (Using Heron's Formula)

1. Find the area of a triangle with sides of length 6 cm, 8 cm, and 10 cm.
2. A triangle has sides measuring 7 cm, 9 cm, and 11 cm. Calculate its area using Heron's formula.
3. Calculate the area of a triangle with sides of length 5 m, 12 m, and 13 m.
4. A triangle has sides of length 15 inches, 20 inches, and 25 inches. Find its area.
5. If a triangle has sides of 4 cm, 13 cm, and 15 cm, what is its area?

Section 4: Mixed Problems

1. A triangular plot of land has sides of 20 meters, 30 meters, and 40 meters. Find its perimeter and area.
2. A right-angled triangle has a hypotenuse (the side opposite the right angle) of 13 cm and one leg of 5 cm. Find the perimeter and area. (Hint: Use the Pythagorean theorem to find the length of the other leg).
3. An isosceles triangle has a base of 16 cm and equal sides of 17 cm each. Calculate its perimeter and area.
4. The perimeter of an equilateral triangle is 45 cm. What is its area?
5. A triangle has an area of 72 square units and a base of 18 units. Find its height and then calculate its perimeter if the other two sides are 13 units and 17 units long.

Answer Key (For Self-Checking)

Remember: These answers are approximate. Rounding differences may occur depending on the method used. Always show your workings to receive full credit. This answer key is provided for self-checking only; understanding the process is far more important than simply getting the correct answer.

(The answer key would be added here, providing the solutions to each problem in the worksheet.)

Advanced Concepts and Further Exploration

This worksheet provides a strong foundation in calculating the area and perimeter of triangles. For more advanced studies, consider exploring:

• Trigonometric functions: Using sine, cosine, and tangent to calculate areas and side lengths of triangles.
• Vectors: Applying vector mathematics to solve problems involving triangles.
• Coordinate geometry: Finding the area and perimeter of triangles using coordinates of their vertices.
• Three-dimensional geometry: Extending the concepts to pyramids and other three-dimensional shapes with triangular faces.

By mastering the basics covered in this worksheet and then venturing into these more advanced topics, you'll build a comprehensive understanding of geometry and its applications in various fields. Remember that consistent practice and problem-solving are key to success.