Question
Give all formulas needed in dealing with as well their individual properties
pyramids
Cuboid
Cube
Cone
Sphere
Hemisphere
Cone
pyramids
Cuboid
Cube
Cone
Sphere
Hemisphere
Cone
Answers
GPT-4o mini
Here are the main formulas and properties associated with pyramids, cuboids, cubes, cones, spheres, and hemispheres:
### 1. **Pyramid**
**Formulas:**
- **Volume (V)**:
\
\[ V = \frac{1}{3} B h \]
where \( B \) is the area of the base and \( h \) is the height of the pyramid.
- **Surface Area (SA)**:
\
\[ SA = B + \frac{1}{2} P l \]
where \( B \) is the area of the base, \( P \) is the perimeter of the base, and \( l \) is the slant height.
**Properties:**
- A pyramid has a polygonal base and triangular faces converging at a point (the apex).
- The height is perpendicular to the base.
### 2. **Cuboid (Rectangular Prism)**
**Formulas:**
- **Volume (V)**:
\
\[ V = l \times w \times h \]
where \( l \), \( w \), and \( h \) are the length, width, and height respectively.
- **Surface Area (SA)**:
\
\[ SA = 2(lw + lh + wh) \]
**Properties:**
- All angles are right angles.
- Opposite faces are equal and parallel.
### 3. **Cube**
**Formulas:**
- **Volume (V)**:
\
\[ V = a^3 \]
where \( a \) is the length of a side.
- **Surface Area (SA)**:
\
\[ SA = 6a^2 \]
**Properties:**
- All faces are squares, and all sides are equal in length.
- All angles are right angles.
### 4. **Cone**
**Formulas:**
- **Volume (V)**:
\
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height.
- **Surface Area (SA)**:
\
\[ SA = \pi r (r + l) \]
where \( l \) is the slant height.
**Properties:**
- Has a circular base and a single vertex (the apex).
- The height is the perpendicular distance from the base to the apex.
### 5. **Sphere**
**Formulas:**
- **Volume (V)**:
\
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius.
- **Surface Area (SA)**:
\
\[ SA = 4 \pi r^2 \]
**Properties:**
- Every point on the surface is equidistant from the center.
- No edges or vertices.
### 6. **Hemisphere**
**Formulas:**
- **Volume (V)**:
\
\[ V = \frac{2}{3} \pi r^3 \]
- **Surface Area (SA)**:
\
\[ SA = 3 \pi r^2 \]
(This includes the curved surface area plus the base area of the circular face.)
**Properties:**
- Half of a sphere.
- Contains a circular flat base and a curved surface.
### Summary of Variables:
- \( V \) = Volume
- \( SA \) = Surface Area
- \( l \) = Length
- \( w \) = Width
- \( h \) = Height
- \( r \) = Radius
- \( a \) = Side length of the cube
- \( B \) = Area of the base of a pyramid
- \( P \) = Perimeter of the base of a pyramid
- \( l \) = Slant height (for pyramids and cones)
These formulas and properties provide a comprehensive basis for understanding and calculating the volume and surface area of these geometric shapes.
### 1. **Pyramid**
**Formulas:**
- **Volume (V)**:
\
\[ V = \frac{1}{3} B h \]
where \( B \) is the area of the base and \( h \) is the height of the pyramid.
- **Surface Area (SA)**:
\
\[ SA = B + \frac{1}{2} P l \]
where \( B \) is the area of the base, \( P \) is the perimeter of the base, and \( l \) is the slant height.
**Properties:**
- A pyramid has a polygonal base and triangular faces converging at a point (the apex).
- The height is perpendicular to the base.
### 2. **Cuboid (Rectangular Prism)**
**Formulas:**
- **Volume (V)**:
\
\[ V = l \times w \times h \]
where \( l \), \( w \), and \( h \) are the length, width, and height respectively.
- **Surface Area (SA)**:
\
\[ SA = 2(lw + lh + wh) \]
**Properties:**
- All angles are right angles.
- Opposite faces are equal and parallel.
### 3. **Cube**
**Formulas:**
- **Volume (V)**:
\
\[ V = a^3 \]
where \( a \) is the length of a side.
- **Surface Area (SA)**:
\
\[ SA = 6a^2 \]
**Properties:**
- All faces are squares, and all sides are equal in length.
- All angles are right angles.
### 4. **Cone**
**Formulas:**
- **Volume (V)**:
\
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height.
- **Surface Area (SA)**:
\
\[ SA = \pi r (r + l) \]
where \( l \) is the slant height.
**Properties:**
- Has a circular base and a single vertex (the apex).
- The height is the perpendicular distance from the base to the apex.
### 5. **Sphere**
**Formulas:**
- **Volume (V)**:
\
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius.
- **Surface Area (SA)**:
\
\[ SA = 4 \pi r^2 \]
**Properties:**
- Every point on the surface is equidistant from the center.
- No edges or vertices.
### 6. **Hemisphere**
**Formulas:**
- **Volume (V)**:
\
\[ V = \frac{2}{3} \pi r^3 \]
- **Surface Area (SA)**:
\
\[ SA = 3 \pi r^2 \]
(This includes the curved surface area plus the base area of the circular face.)
**Properties:**
- Half of a sphere.
- Contains a circular flat base and a curved surface.
### Summary of Variables:
- \( V \) = Volume
- \( SA \) = Surface Area
- \( l \) = Length
- \( w \) = Width
- \( h \) = Height
- \( r \) = Radius
- \( a \) = Side length of the cube
- \( B \) = Area of the base of a pyramid
- \( P \) = Perimeter of the base of a pyramid
- \( l \) = Slant height (for pyramids and cones)
These formulas and properties provide a comprehensive basis for understanding and calculating the volume and surface area of these geometric shapes.