Spheres, cylinders, prisms, right cones, right pyramids, and combinations of solids
The familiar cartesian plane with (x,y) coordinates can be extended to 3D space by introducing a z coordinate, making a 3D coordinate have the form (x,y,z). Each component (x,y and z) tells you how many unit "steps" to take from the origin in the direction of each axis to reach the described point.
By carefully imagining right triangles in 3D diagrams, we can find angles in space.
In the triangular prism below, we can find the length CB using Pythagoras. Then, since there is a right angle at C, we can write
A sphere is a perfectly round, three-dimensional geometric shape where every point on its surface is exactly the same distance (the radius) from a single central point. It's the three-dimensional analog of a circle. For example, a ball or globe is spherical in shape. The surface area A and volume V of a sphere are given by:
A cylinder is a three-dimensional geometric shape formed by two identical circular bases connected by a curved lateral surface. The segment connecting the centers of the circular bases is called the axis, which is perpendicular to each base in a right cylinder (the type usually studied).
The volume V of a cylinder with radius r is given by:
The curved surface of a cylinder (excluding the circular ends) is given by:
If we include the circular ends, each with area πr2, we get
A prism is a three-dimensional solid shape consisting of two parallel, congruent faces called bases, connected by rectangular lateral faces. Prisms are named according to the shape of their bases—for example, triangular prism, rectangular prism, or hexagonal prism.
The volume V of a prism is calculated by multiplying the area A of its base by its height h:
A right circular cone is a three-dimensional geometric shape whose apex (vertex) lies directly above the center of its circular base.
Key Parts:
Circular Base: Flat circle with radius r.
Apex (Vertex): The point directly above the center of the base.
Height (h): Perpendicular distance from apex to base center.
Slant Height (l): Distance along the cone's surface from apex to edge of base.
Formulas:
Volume:
Surface Area of curved surface:
Slant Height Relationship:
A right pyramid is a three-dimensional shape with a polygonal base and triangular lateral faces, in which the apex (vertex) is located directly above the center (centroid) of the base.
Key Parts:
Polygonal base: a flat polygon (triangle, square, pentagon, etc.)
Apex (vertex): the point positioned vertically above the base's centroid
Height (h): perpendicular distance from apex to base centroid
Slant height (l): distance along a lateral face from the apex perpendicular to an edge of the base
The volume of a right pyramid is given by
where A is the area of the base.
Nice work completing 3D volume and surface area, here's a quick recap of what we covered:
Exercises checked off
Spheres, cylinders, prisms, right cones, right pyramids, and combinations of solids
The familiar cartesian plane with (x,y) coordinates can be extended to 3D space by introducing a z coordinate, making a 3D coordinate have the form (x,y,z). Each component (x,y and z) tells you how many unit "steps" to take from the origin in the direction of each axis to reach the described point.
By carefully imagining right triangles in 3D diagrams, we can find angles in space.
In the triangular prism below, we can find the length CB using Pythagoras. Then, since there is a right angle at C, we can write
A sphere is a perfectly round, three-dimensional geometric shape where every point on its surface is exactly the same distance (the radius) from a single central point. It's the three-dimensional analog of a circle. For example, a ball or globe is spherical in shape. The surface area A and volume V of a sphere are given by:
A cylinder is a three-dimensional geometric shape formed by two identical circular bases connected by a curved lateral surface. The segment connecting the centers of the circular bases is called the axis, which is perpendicular to each base in a right cylinder (the type usually studied).
The volume V of a cylinder with radius r is given by:
The curved surface of a cylinder (excluding the circular ends) is given by:
If we include the circular ends, each with area πr2, we get
A prism is a three-dimensional solid shape consisting of two parallel, congruent faces called bases, connected by rectangular lateral faces. Prisms are named according to the shape of their bases—for example, triangular prism, rectangular prism, or hexagonal prism.
The volume V of a prism is calculated by multiplying the area A of its base by its height h:
A right circular cone is a three-dimensional geometric shape whose apex (vertex) lies directly above the center of its circular base.
Key Parts:
Circular Base: Flat circle with radius r.
Apex (Vertex): The point directly above the center of the base.
Height (h): Perpendicular distance from apex to base center.
Slant Height (l): Distance along the cone's surface from apex to edge of base.
Formulas:
Volume:
Surface Area of curved surface:
Slant Height Relationship:
A right pyramid is a three-dimensional shape with a polygonal base and triangular lateral faces, in which the apex (vertex) is located directly above the center (centroid) of the base.
Key Parts:
Polygonal base: a flat polygon (triangle, square, pentagon, etc.)
Apex (vertex): the point positioned vertically above the base's centroid
Height (h): perpendicular distance from apex to base centroid
Slant height (l): distance along a lateral face from the apex perpendicular to an edge of the base
The volume of a right pyramid is given by
where A is the area of the base.
Nice work completing 3D volume and surface area, here's a quick recap of what we covered:
Exercises checked off