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Solid Geometry




 
Related Topics: More Geometry Lessons , Geometry Games

What is solid geometry?
Solid geometry is concerned with three-dimensional shapes. Some examples of three-dimensional shapes are cubes, rectangular solids, prisms, cylinders, spheres, cones and pyramids. We will look at the volume formulas and surface area formulas of the solids. We will also discuss some nets of solids.

The following figures show some examples of shapes in solid geometry. Scroll down the page for more examples, explanations and worksheets for each shape.
solid geometry


Cubes

A cube is a three-dimensional figure with six matching square sides.

cube

The figure above shows a cube. The dotted lines indicate edges hidden from your view.

If s is the length of one of its sides, then the volume of the cube is s × s × s

Volume of the cube = s3

The area of each side of a cube is s2. Since a cube has six square-shape sides, its total surface area is 6 times s2.

Surface area of a cube = 6s2





Rectangular Prisms or Cuboids

A rectangular prism is also called a rectangular solid or a cuboid.

In a rectangular prism, the length, width and height may be of different lengths.

rectangular solid

The volume of the above rectangular prism would be the product of the length, width and height that is

Volume of rectangular prism = lwh

Total area of top and bottom surfaces is lw + lw = 2lw
Total area of front and back surfaces is lh + lh = 2lh
Total area of the two side surfaces is wh + wh = 2wh

Surface area of rectangular prism = 2lw + 2lh + 2wh = 2(lw + lh + wh)



Prisms

A prism is a solid that has two congruent parallel bases that are polygons. The polygons form the bases of the prism and the length of the edge joining the two bases is called the height.

triangle base pentagon base
Triangle-shaped base Pentagon-shaped base

The above diagrams show two prisms: one with a triangle-shaped base called a triangular prism and another with a pentagon-shaped base called a pentagonal prism.

A rectangular solid is a prism with a rectangle-shaped base and can be called a rectangular prism.

The volume of a prism is given by the product of the area of its base and its height.

Volume of prism = area of base × height

The surface area of a prism is equal to 2 times area of base plus perimeter of base times height.

Surface area of prism = 2 × area of base + perimeter of base × height




 

Cylinders

A cylinder is a solid with two congruent circles joined by a curved surface.

cylinder

In the above figure, the radius of the circular base is r and the height is h. The volume of the cylinder is the area of the base × height.

volume of cylinder

The net of a solid cylinder consists of 2 circles and one rectangle. The curved surface opens up to form a rectangle.

Surface area = 2 × area of circle + area of rectangle

Surface area of cylinder = 2πr2 + 2πrh = 2πr (r + h)



Spheres

A sphere is a solid with all its points the same distance from the center.

sphere

vol & surface area of sphere



Cones

A circular cone has a circular base, which is connected by a curved surface to its vertex. A cone is called a right circular cone, if the line from the vertex of the cone to the center of its base is perpendicular to the base.

cone

volume of cone

The net of a solid cone consists of a small circle and a sector of a larger circle. The arc of the sector has the same length as the circumference of the smaller circle.

Surface area of cone = Area of sector + area of circle

= πrs + πr2 = πr(r + s)


Pyramids

A pyramid is a solid with a polygon base and connected by triangular faces to its vertex. A pyramid is a regular pyramid if its base is a regular polygon and the triangular faces are all congruent isosceles triangles.

pyramid

volume of pyramid formula




Nets Of A Solid

An area of study closely related to solid geometry is nets of a solid. Imagine making cuts along some edges of a solid and opening it up to form a plane figure. The plane figure is called the net of the solid.

The following figures show the two possible nets for the cube.

cube net of cube net of cube


This video shows how to calculate the volume of prisms, cylinders, pyramids and cones.
Volumes of Prisms and Cylinders = Area of Base × Height
Volumes of Pyramids and Cones = 1/3 × Area of Base × Height
Examples to show how to calculate the volumes of prisms, cylinders, pyramids and cones.


 
This video shows how to calculate the surface area of a pentagonal prism with the base edge of 6 and height of 8.
This video shows how to calculate the surface area of cylinders, pyramids and cones.


You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.


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