Solved Examples: Area of Parallelograms and Triangles (High School Level)

📚 Quick Study Guide

• 📐 Area of a Parallelogram: The area of a parallelogram is given by the formula: $A = b \times h$, where $b$ is the length of the base and $h$ is the perpendicular height.
• 📏 Area of a Triangle: The area of a triangle is given by the formula: $A = \frac{1}{2} \times b \times h$, where $b$ is the length of the base and $h$ is the perpendicular height.
• ✨ Area of a Triangle (Heron's Formula): If you know the lengths of all three sides ($a$, $b$, $c$), you can use Heron's formula: $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$ (the semi-perimeter).
• 🚧 Area of a Triangle (Two Sides and Included Angle): If you know two sides ($a$ and $b$) and the included angle ($C$), the area is: $A = \frac{1}{2}ab\sin(C)$.
• 🤝 Parallelograms and Triangles on the Same Base: If a parallelogram and a triangle are on the same base and between the same parallels, the area of the triangle is half the area of the parallelogram.
• 💡 Median of a Triangle: A median of a triangle divides it into two triangles of equal area.

Practice Quiz

1. A parallelogram has a base of 10 cm and a height of 5 cm. What is its area?
   1. A. 25 sq cm
   2. B. 50 sq cm
   3. C. 75 sq cm
   4. D. 100 sq cm
2. A triangle has a base of 8 cm and a height of 6 cm. What is its area?
   1. A. 14 sq cm
   2. B. 24 sq cm
   3. C. 48 sq cm
   4. D. 96 sq cm
3. A triangle has sides of length 5 cm, 6 cm, and 7 cm. What is its area (using Heron's formula)?
   1. A. 6$\sqrt{6}$ sq cm
   2. B. 12 sq cm
   3. C. 18 sq cm
   4. D. 36 sq cm
4. A triangle has sides of length 4 cm and 6 cm, with an included angle of 30 degrees. What is its area?
   1. A. 6 sq cm
   2. B. 12 sq cm
   3. C. 24 sq cm
   4. D. 48 sq cm
5. A parallelogram and a triangle are on the same base and between the same parallels. If the area of the parallelogram is 60 sq cm, what is the area of the triangle?
   1. A. 15 sq cm
   2. B. 30 sq cm
   3. C. 45 sq cm
   4. D. 120 sq cm
6. A triangle has an area of 40 sq cm. A median divides the triangle into two smaller triangles. What is the area of each smaller triangle?
   1. A. 10 sq cm
   2. B. 20 sq cm
   3. C. 30 sq cm
   4. D. 40 sq cm
7. What is the height of a triangle with area 72 sq cm and base 12 cm?
   1. A. 6 cm
   2. B. 12 cm
   3. C. 24 cm
   4. D. 36 cm