Calculate area of triangle and rectangle
Scenario
Aapke paas ek drawing sheet hai aur aapko pata karna hai ki kitna color lagega usko paint karne mein. Yeh directly sheet ke area par depend karta hai.
Basic shapes jaise rectangle aur triangle ka area fast kaise nikalte hain?
Mental model
Rectangle ka area dono sides ka product hota hai (`l * b`). Triangle rectangle ka aadha hota hai, isliye uska area `1/2 * base * height` hota hai.
Area ka matlab hai ki koi shape surface par kitni jagah cover kar rahi hai. Rectangle mein base aur height seedhe multiply hote hain. Kyunki ek diagonal line rectangle ko do equal triangles mein divide karti hai, isliye triangle ka area usi rectangle ke area ka aadha (`1/2`) ho jaata hai.
Explanation
Physics mein, especially kinematics wale chapter mein, graphs ke neeche ka area nikalna ek bohot important tool hota hai. Velocity-time graph ke under ka area aapko displacement (distance) deta hai. Lekin us area ko nikalne ke liye aapko basic geometry aani chahiye.
Sabse common shape jo graphs mein banti hai, woh ek rectangle ya square hoti hai (jab object ki velocity constant ho). Rectangle ka area calculate karna sabse simple hai: aap bas uski length ko uski width se multiply kar dete ho. `Area = length × width`. Graph mein iska matlab hai x-axis ki value aur y-axis ki value ko aapas mein multiply karna.
Doosri common shape triangle hoti hai (jab object ki velocity uniform rate se increase ya decrease ho rahi ho). Ek right-angled triangle ka area nikalne ke liye formula hota hai: `Area = 1/2 × base × height`. Agar aap dhyan se dekhein, toh ek triangle rectangle ko beech se cut karke banta hai. Isliye iska formula bhi rectangle ke formula ka exact half hota hai.
Kabhi kabhi graph ke neeche ki shape trapezium (ek combination of rectangle and triangle) jaisi lagti hai. Aise complex shapes ko nikalne ke liye, aap unhe ek triangle aur ek rectangle mein tod (split) sakte hain, dono ka area alag nikal kar unhe aapas mein jod (add) sakte hain.
Worked example
Problem: Calculate the area of a rectangle with length `8 m` and width `4 m`.
- Rectangle ke area ka formula use karte hain.
- Values put karke multiply karte hain.
Answer: `32 m^2`
Problem: Find the area of a right-angled triangle whose base is `10 cm` and height is `6 cm`.
- Triangle ke area ka formula apply karte hain.
- Base (`10`) aur height (`6`) daalte hain.
- Multiply and simplify karein.
Answer: `30 cm^2`
Key points
- Area of Rectangle: `Area = length × width`. (In graphs, this is usually `base × height`).
- Area of Triangle: `Area = 1/2 × base × height`.
Common mistakes
- Forgetting the 1/2 in the triangle formula: Bache triangle ka area nikalte waqt bhi jaldi mein `base × height` kar dete hain aur half karna bhool jaate hain. Hamesha dhyan rakho triangle ek rectangle ka aadha hissa hai.
- Wrong base or height: Height hamesha perpendicular distance (straight `90-degree` wali line) hoti hai, slanting (tirchi) line nahi.
Recall questions
- What is the formula for the area of a rectangle?
- What is the formula for the area of a triangle?
- If a rectangle has base 5 and height 10, what is its area?
Questions & answers
A right triangle has a base of 4 m and a perpendicular height of 3 m. What is its area?
6 m^2
Approach: Area = 1/2 * base * height = 1/2 * 4 * 3 = 2 * 3 = 6.
Calculate the area of a rectangular field with dimensions 20 m by 15 m.
300 m^2
Approach: Area = length * width = 20 * 15 = 300.