How to calculate area if you know the perimeter? Full analysis of geometric calculation formulas
In mathematics and practical applications, perimeter and area are two fundamental properties of geometric figures. Many people will encounter this problem during the learning process: How to calculate the area of a figure when its perimeter is known? This article will focus on this topic, combined with the hot topics on the Internet in the past 10 days, systematically sort out the relationship between the perimeter and area of common graphics, and provide structured data tables for easy reference.
1. Background of hot topics
Recently, the calculation of geometric figures has become very popular in the fields of education and popular science, especially the practical technique of "finding the area of a given perimeter". The following is the statistics of related hot topics in the past 10 days:
| hot topics | focus of discussion | heat index |
|---|---|---|
| Mathematics Education Innovation | How to derive area from perimeter | 85% |
| Practical Mathematics for Life | Garden fence and land area calculation | 78% |
| High-frequency test points | Conversion of perimeter and area of circle and square | 92% |
2. The relationship between the perimeter and area of common shapes
Different shapes have different calculation formulas for perimeter and area. The following is a detailed comparison of 5 common shapes:
| graphics | Perimeter formula | area formula | Steps to find area if perimeter is known |
|---|---|---|---|
| square | P = 4a (a is the side length) | S = a² | 1. Find the side length a = P/4 through P 2. Substitute the area formula S = (P/4)² |
| round | P = 2πr (r is the radius) | S = πr² | 1. Find the radius r = P/(2π) through P 2. Substitute the area formula S = π(P/2π)² |
| Equilateral triangle | P = 3a (a is the side length) | S = (√3/4)a² | 1. Find the side length a = P/3 through P 2. Substitute the area formula S = (√3/4)(P/3)² |
| Rectangle | P = 2(a+b) (a and b are length and width) | S = a×b | Supplementary conditions (such as aspect ratio) are required to solve the problem |
| regular hexagon | P = 6a (a is the side length) | S = (3√3/2)a² | 1. Find the side length a = P/6 through P 2. Substitute the area formula S = (3√3/2)(P/6)² |
3. Practical application cases
Case 1: Calculation of area of circular flower bed
It is known that the circumference of the circular flower bed is 20 meters, then the radius r = 20/(2×3.14) ≈ 3.18 meters, and the area S = 3.14×3.18² ≈ 31.8 square meters.
Case 2: Estimation of materials for square floor tiles
If the perimeter of the floor tile is 1.6 meters, the side length a = 1.6/4 = 0.4 meters, and the area of a single tile is S = 0.4² = 0.16 square meters.
4. Precautions
1.Graphic type needs to be clear: The calculation logic of different graphics is different, so you need to confirm the graphics category first.
2.Rectangle requires additional conditions: The area cannot be uniquely determined by knowing only the perimeter, and additional information (such as length-to-width ratio) is required.
3.unit consistency: Make sure perimeter and area are in the same units (e.g. meters and square meters).
Through the above analysis and structured data, I believe readers can understand the conversion relationship between perimeter and area more clearly and use it flexibly in practical applications.
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