Solid Geometry

Three Dimensions Solid Geometry is the geometry of three-dimensional space It is called three-dimensional, or 3D because there are three dimensions: width, depth and height.

Polyhedron A geometric object with flat faces and straight edges. each face is a polygon.

Polyhedron FACE: Polygon shaped sides of a polyhedron EDGE: Line segment formed by intersection of two faces VERTEX: Point where three or more edges meet

Base The surface that a solid object stands on or the bottom line of a shape such as a triangle or rectangle.

Polyhedron Just like a 2D polygon a Polyhedron can be regular

Polyhedron …or Irregular

Polyhedron A Polyhedron can also be semi-Regular

Polyhedron Just like a 2D polygon, a polyhedron can be convex

Polyhedron …or concave

Prism A solid object that has two identical bases and all flat sides. The shape of the bases give the prism it’s name "triangular prism“ It is a polyhedron.

Pyramid A solid object where: * The base is a polygon (a straight-sided shape) * The sides are triangles which meet at the top (the apex). It is a polyhedron.

Cylinder A cylinder is a solid object with: * two identical flat circular (or elliptical) ends * and one curved side.

Cone A solid (3-dimensional) object that has a circular base and one vertex

Sphere A solid (3-dimensional) object that has one curved side

Prisms & Pyramids Triangular Prism Rectangular Prism Cube Type Examples Properties Triangular Prism ● 5 faces 2 triangular bases 3 rectangular faces ● 9 edges ● 6 vertices Rectangular Prism 6 faces 2 rectangular bases 4 rectangular faces ● 12 edges ● 8 vertices Cube ● 6 faces 2 square bases 4 square faces Square Pyramid 1 square base 4 triangular faces ● 8 edges ● 5 vertices Triangular Pyramid ● 4 faces 1 triangular base 3 triangular faces ● 6 edges ● 4 vertices

Three Dimensional Figures with Curved Surfaces Type Example Properties Cylinder ● 2 circular bases ● 1 curved surface Cone ● 1 circular base ● 1 vertex Sphere

VOLUME

Prism V=Bh B: Area of the base h: height/length of the prism

Prism V=Bh V= (Area of Triangle) * h V= (½bh) * h V= (½*19*24) *47 V= 10,716 cm3

Prism V=Bh V= (Area of Rect.) * h V= (bh) * h V= (2 * 3) * 6 V=(6) * 6 V= 36 ft3

Cylinder V=Bh B: Area of the base h: height/length of the prism

Cylinder V=Bh V= (Area of Circle) * h V= (πr2) * h V= (π * 32) *10 V= 282.6 cm3

Cylinder V=Bh V= (Area of Circle) * h V= (πr2) * h V= (π * 52) *21 V= 1,648.5 ft3

Pyramid V=1/3Bh B: Area of the base h: height/length of the prism

Pyramid V=1/3Bh V= 1/3(Area of Tri.) * h V= 1/3 * (½bh) * h V= 20 cm3

Pyramid V=1/3Bh V= 1/3(Area of Sq.) * h V= 1/3 * (b * h) * h V= 250/3 units3

Cone V=1/3Bh B: Area of the base h: height/length of the prism

Cone V=1/3Bh V= 1/3(Area of Circle)* h V= 1/3(πr2) * h V= 11.78 in3

Cone V=1/3Bh V= 1/3(Area of Circle)* h V= 1/3(πr2) * h V= 1,473.71 cm3

Sphere V=4/3 πr3

Sphere V=4/3 πr3 V= 4/3 πr3 V= 4/3 * π * 143 V= 4/3 * π * 2744 V= 11,488.21 cm3

Sphere V=4/3 πr3 V= 4/3 πr3 V= 4/3 * π * 33 V= 4/3 * π * 27 V= 113.04 cm3

SURFACE AREA

Surface Area The sum of the area of the bases and lateral surfaces

Right Prism SA=2B+Ph B: Area of the base P: Perimeter of a base h: height/length of the prism

Right Prism SA=2B+Ph SA=2(Area of Rect.)+Ph SA= 2(bh) + Ph SA= 62 cm2

Right Prism SA=2B+Ph SA=2(Area of Tri.)+Ph SA= 2(1/2bh) + Ph (3+8+√73)7 SA=2(12) + (11+√73)5 SA= 24 + 55 + 5√73 SA= 79 + 5√73 m2 SA = 121.72 m2 SA=2B+Ph

Right Cylinder SA=2B+Ch B: Area of the base C: Circumference of a base h: height/length of the cylinder

Right Cylinder SA=2B+Ch SA=2(Area of Circ.)+Ch SA= 2(πr2) + (2πr)h SA= 5934.6 cm2

Right Cylinder SA=2B+Ch SA=2(Area of Circ.)+Ch SA= 2(πr2) + (2πr)h SA= 226.08 cm2

Right Pyramid SA=B + ½Ps B: Area of the base P: Perimeter of a base s: slant height of the lateral side

Right Pyramid (& Cone) What is “slant height”?

Right Pyramid SA=B + ½Ps SA=(Area of Rect.)+½Ps SA= (bh) + ½Ps SA= 423.89 units2

Right Pyramid SA=B + ½Ps SA=(Area of Rect.)+½Ps SA= (bh) + ½Ps SA= 602 in2 10 in

Right Cone SA=B + ½Cs B: Area of the base P: Circumference of a base s: slant height of the lateral side

Right Cone SA=B + ½Cs SA=(Area of Circ.)+½Cs SA= (πr2) + ½(2πr)s SA= 301.59 in2

Right Cone SA=B + ½Cs SA=(Area of Circ.)+½Cs SA= (πr2) + ½(2πr)s SA= 4.9 ft2

Sphere SA=4πr2 r: radius

Sphere SA=4πr2 SA= 4πr2 SA= 4*π*42 SA= 4*π*16 SA= 64π SA= 200.96 units2

Sphere SA=4πr2 SA= 4πr2 SA= 4*π*322 SA= 4*π*1024 SA= 4096π SA= 12,861.44 units2