Surface Area of 3D Objects

• Surface area, perimeter, volume, and area are fundamental geometric concepts

Surface Area of 3D Objects

• Surface area is the total area of all the surfaces of a 3D object
• It is measured in square units (e.g., ( \text{cm}^2 ), ( \text{m}^2 ), ( \text{in}^2 ))
• To calculate surface area, find the area of each face and then sum these areas
• Common 3D shapes include cubes, rectangular prisms, triangular prisms, and pyramids

Cube

• A cube has six identical square faces
• If the side length of a cube is ( s ), the area of one face is ( s^2 )
• The surface area ( A ) of a cube is ( 6s^2 )

Rectangular Prism

• A rectangular prism has six rectangular faces
• If the length, width, and height are ( l ), ( w ), and ( h ), respectively, the surface area ( A ) is given by ( 2(lw + lh + wh) )
• This formula accounts for the pairs of identical faces

Triangular Prism

• A triangular prism has two triangular faces and three rectangular faces
• If the base of the triangular face is ( b ), the height is ( h ), and the sides of the rectangle are ( l ), ( s_1 ), and ( s_2 ), the surface area ( A ) is ( bh + l(b + s_1 + s_2) )
• ( bh ) is the area of the two triangles ( l(b + s_1 + s_2) ) is the area of the three rectangles

Square Pyramid

• A square pyramid has one square base and four triangular faces
• If the side length of the square base is ( s ) and the slant height of the triangular face is ( l ), the surface area ( A ) is ( s^2 + 2sl )
• ( s^2 ) is the area of the base, and ( 2sl ) is the area of the four triangles

Perimeters of Unequal Shapes

• Perimeter is the total distance around the outside of a 2D shape
• It is measured in linear units (e.g., cm, m, in, ft)
• To find the perimeter, add the lengths of all the sides of the shape
• This applies to polygons with any number of sides, whether regular or irregular

Triangle

• For a triangle with side lengths ( a ), ( b ), and ( c ), the perimeter ( P ) is ( a + b + c )

Quadrilateral

• For a quadrilateral with side lengths ( a ), ( b ), ( c ), and ( d ), the perimeter ( P ) is ( a + b + c + d )

Irregular Polygon

• For any irregular polygon, sum the lengths of all its sides to find the perimeter

Volume of 3D Objects

• Volume is the amount of space an object occupies
• It is measured in cubic units (e.g., ( \text{cm}^3 ), ( \text{m}^3 ), ( \text{in}^3 ))
• Volume calculations depend on the shape of the object

Cube

• For a cube with side length ( s ), the volume ( V ) is ( s^3 )

Rectangular Prism

• For a rectangular prism with length ( l ), width ( w ), and height ( h ), the volume ( V ) is ( lwh )

Triangular Prism

• For a triangular prism, the volume ( V ) is the area of the triangular base times the length ( l ), i.e., ( V = \frac{1}{2} \times b \times h \times l ), where ( b ) is the base and ( h ) is the height of the triangular face

Square Pyramid

• For a square pyramid with base side ( s ) and height ( h ), the volume ( V ) is ( \frac{1}{3}s^2h )
• The pyramid's volume is one-third of the base area times the height

Area of a Circle and Other Shapes

• Area is the amount of space a 2D shape covers
• It is measured in square units (e.g., ( \text{cm}^2 ), ( \text{m}^2 ), ( \text{in}^2 ))

Circle

• For a circle with radius ( r ), the area ( A ) is ( \pi r^2 ), where ( \pi ) (pi) is approximately 3.14159

Square

• For a square with side length ( s ), the area ( A ) is ( s^2 )

Rectangle

• For a rectangle with length ( l ) and width ( w ), the area ( A ) is ( lw )

Triangle

• For a triangle with base ( b ) and height ( h ), the area ( A ) is ( \frac{1}{2}bh )

Parallelogram

• For a parallelogram with base ( b ) and height ( h ), the area ( A ) is ( bh )