Here are the key formulas for both solid shapes and plane shapes:
Plane Shapes (2D)
-
Square:
- Area: \( A = s^2 \)
- Perimeter: \( P = 4s \)
- (where \( s \) is the length of a side)
-
Rectangle:
- Area: \( A = l \times w \)
- Perimeter: \( P = 2(l + w) \)
- (where \( l \) is the length and \( w \) is the width)
-
Triangle:
- Area: \( A = \frac{1}{2} b h \)
- Perimeter: \( P = a + b + c \)
- (where \( b \) is the base, \( h \) is the height, and \( a, b, c \) are the lengths of the sides)
-
Circle:
- Area: \( A = \pi r^2 \)
- Circumference: \( C = 2\pi r \)
- (where \( r \) is the radius)
-
Trapezoid:
- Area: \( A = \frac{1}{2} (b_1 + b_2) h \)
- Perimeter: \( P = a + b_1 + b_2 + c \)
- (where \( b_1 \) and \( b_2 \) are the lengths of the parallel sides, and \( h \) is the height)
Solid Shapes (3D)
-
Cube:
- Volume: \( V = s^3 \)
- Surface Area: \( SA = 6s^2 \)
- (where \( s \) is the length of a side)
-
Rectangular Prism:
- Volume: \( V = l \times w \times h \)
- Surface Area: \( SA = 2(lw + lh + wh) \)
- (where \( l \) is the length, \( w \) is the width, and \( h \) is the height)
-
Cylinder:
- Volume: \( V = \pi r^2 h \)
- Surface Area: \( SA = 2\pi r(h + r) \)
- (where \( r \) is the radius of the base and \( h \) is the height)
-
Sphere:
- Volume: \( V = \frac{4}{3} \pi r^3 \)
- Surface Area: \( SA = 4\pi r^2 \)
- (where \( r \) is the radius)
-
Cone:
- Volume: \( V = \frac{1}{3} \pi r^2 h \)
- Surface Area: \( SA = \pi r (r + l) \)
- (where \( r \) is the radius of the base, \( h \) is the height, and \( l \) is the slant height)
-
Pyramid:
- Volume: \( V = \frac{1}{3} b h \)
- Surface Area: \( SA = b + \frac{1}{2} P l \)
- (where \( b \) is the area of the base, \( h \) is the height, \( P \) is the perimeter of the base, and \( l \) is the slant height)
Summary
These formulas allow you to calculate the area, perimeter, volume, and surface area of various geometric shapes, both in two-dimensional and three-dimensional contexts.