In mathematics, the perimeter and area of various shapes are calculated using specific formulas. Here’s a summary of some common shapes and their respective formulas:
1. Rectangle
- Perimeter (P) = ( 2 \times (l + w) )
- Where (l) is the length and (w) is the width.
- Area (A) = ( l \times w )
2. Square
- Perimeter (P) = ( 4 \times s )
- Where (s) is the length of a side.
- Area (A) = ( s^2 )
3. Triangle
For a general triangle with base (b) and height (h):
- Area (A) = ( \frac{1}{2} \times b \times h )
For equilateral triangle (where all sides are equal):
- Perimeter (P) = ( 3 \times s )
- Where (s) is the length of one side.
- Area (A) = ( \frac{s^2 \sqrt{3}}{4} )
4. Circle
- Circumference (C) (equivalent to perimeter) = ( 2 \times \pi \times r )
- Where (r) is the radius.
- Area (A) = ( \pi \times r^2 )
5. Parallelogram
- Perimeter (P) = ( 2 \times (a + b) )
- Where (a) and (b) are the lengths of the adjacent sides.
- Area (A) = ( a \times h )
- Where (a) is the base and (h) is the height (perpendicular distance).
6. Trapezoid (Trapezium)
- Perimeter (P) = ( a + b + c + d )
- Where (a), (b), (c), and (d) are the lengths of the sides.
- Area (A) = ( \frac{1}{2} \times (a + b) \times h )
- Where (a) and (b) are the lengths of the parallel sides, and (h) is the height.
7. Rhombus
- Perimeter (P) = ( 4 \times a )
- Where (a) is the length of one side.
- Area (A) = ( \frac{1}{2} \times d_1 \times d_2 )
- Where (d_1) and (d_2) are the lengths of the diagonals.
8. Ellipse
- Perimeter (P) (approximation) = ( 2 \pi \sqrt{\frac{a^2 + b^2}{2}} )
- Where (a) is the length of the semi-major axis and (b) is the length of the semi-minor axis.
- Area (A) = ( \pi \times a \times b )
9. Sector of a Circle
- Arc Length (L) = ( \theta \times r )
- Where ( \theta ) is the central angle (in radians) and (r) is the radius.
- Area (A) = ( \frac{\theta}{2} \times r^2 )
- Where ( \theta ) is the central angle (in radians).
10. Regular Polygon (n sides)
- Perimeter (P) = ( n \times s )
- Where (n) is the number of sides, and (s) is the length of one side.
- Area (A) = ( \frac{n \times s^2}{4 \times \tan\left(\frac{\pi}{n}\right)} )
These formulas provide the basic way to calculate the perimeter (boundary length) and area (the interior space) of common geometric shapes. If you need more specific examples or clarification, feel free to ask!
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