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.

Cubes

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

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

Worksheets to calculate volume and surface area of cubes.	More examples about the volume of cubes.
More examples about the surface area of cubes.

Rectangular Solids or Cuboids

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

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

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

Volume of rectangular solid = 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 solid = 2lw + 2lh + 2wh = 2(lw + lh + wh)

Worksheets to calculate the volume and surface area of rectangular prisms.	More examples about the volume of cuboids.
More examples about the surface area of cuboids.

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-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

Worksheets to calculate volume of prisms and pyramids.	More examples about the volume of prisms.
More examples about the surface area of prisms.

Cylinders

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

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.

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)

Worksheets to calculate volume of cylinders.	Worksheets to calculate surface area of cylinders.
Worksheets to calculate volume and surface area of cylinders.	Worksheets to calculate surface area of cylinders and pipes.
More examples about the volume of cylinders.	More examples about the surface area of cylinders.

Spheres

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

Worksheets to calculate the volume of spheres.	Worksheets to calculate the surface area of spheres.
More examples about the volume of spheres.	More examples about the surface area of spheres.

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.

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

Worksheets to calculate the volume of cones.	More examples about the volume of cones.
More examples about the surface area of cones.

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.

Worksheets to calculate the volume of square pyramids.	Worksheets to calculate the volume of prisms and pyramids.
More examples about the volume of pyramids.	More examples about the surface area of pyramids.

Worksheets to calculate the surface area of different solids and composite solids.

Worksheet 1	Worksheet 2
Worksheet 3	Worksheet 4
Worksheet 5

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.