Solid figures, unlike squares, rectangles, triangles, quadrilaterals or circles that have two dimensions, have three dimensions.
The most popular solid figures are prisms (triangular, quadrilateral, trapezoidal, pentagonal, hexagonal, heptagonal, octagonal, etc.), pyramids (rectangular, pentagonal, hexagonal, etc.), cylinders, cones and spheres.
Prisms
In this area everything you learned about polygons will come in handy.
What is a prism?
Imagine that you have two polygons, and they are in 3D space, parallel to each other. If you connect every side of first polygon to its parallel one of second polygon, you’ll get prism.
To be mathematically precise, prism is a polyhedron bounded by two congruent polygons which lie in parallel planes and parallelograms as sides.
Every prism has two bases (parallel polygons) and lateral faces (sides that connect specific sides). Prisms can be regular or right and irregular. Lateral sides in a right prism are rectangles and are perpendicular to both bases. Height of the prism is defined as the distance between two bases.
How would you know at first sight if the prism is regular or irregular? Imagine yourself cutting your prism like bread. If the section you got is regular polygon, your prism is regular.
Surface area of prisms
Area of any prism is the sum of areas of its lateral sides and bases- the sum of areas of all polygons that that prism is bound with. If you want to calculate it, first you’d find all separate areas and then just sum them.
Volume of prisms
Volume of prisms is equal to the product of the area of the base and height of that prism.
Prisms are named mostly by its base, which means that the prism whose base is a triangle is called triangular prism, whose base is a quadrilateral a quadrilateral prism and so on.
Triangular prism
Triangular prism has two parallel triangles as bases and parallelograms as lateral sides.
Regular triangular prism is a prism whose lateral sides are perpendicular to its bases, and it’s bases are equilateral triangles. If the prism is regular, its sides are rectangles.
Surface area is equal to the sum of three congruent rectangles and two congruent triangles, and those parts you know how to calculate.
Volume of the regular triangular prism is equal to the product of surface of one base and its height. Height of the regular triangular prism is the distance between two bases.
Example 1. Find the volume and surface area of regular triangular prism whose height is equal to 5 and its base is a equilateral triangle with one side length equal to 3.
$ A = 2 Base + 3 Lateral side$
First we’ll have to calculate the area of one base. The base is equilateral triangle with one side whose length is equal to 3. Using Pythagoras theorem we get to the altitude of the base – $ h = 2,598$.
Now the area of the base is equal to $ 3 \cdot \frac{2.598}{2} = 3.897$.
Lateral sides are rectangles whose one side is the height and other is one side of the base.
Which means that the area of one lateral side is equal to: $ 5 \cdot 3 = 15$.
Now we can get our whole area: $ A = 2 \cdot 3.87 + 3 \cdot 15 = 52.74$
We can also calculate the volume right away since we already calculated the area of the base:
$ V = 3.897 \cdot 5 = 19.485$
Since the condition for a prism to be regular, its lateral sides must be perpendicular to its bases, we can also introduce trigonometry. For example, if we are given the length of a diagonal of one lateral sides and height, we can use trigonometry to get to the base side.
If you’re having trouble seeing what you need to calculate for surface area you can easily spread your prism into 2D. Imagine you took it apart.
If you’re still unsure about what this is, try to construct it on paper, cut it out, and fold it where triangles end, if you did it correctly you should get your triangular prism.
From here you can clearly see that the surface area of your prism is equal to the sum of three rectangles and two equilateral triangles.
This is called the net of a triangle prism, and they can be drawn in more than one way.
This is also one acceptable form.
You can have many more, but you have to be careful, just imagine that you’re unfolding it and you’ll do it right.
Quadrilateral prisms
Quadrilateral prisms are prisms which have a quadrilateral. Regular prisms are divided by the sorts of quadrilateral which means that there are rectangular prisms, quadratic prisms, parallelepipeds and trapezoidal prism.
Formula for area of surface of all of these is the same, the sum of individual surfaces, and for the volume is height times area of the base.
Also every one of them has a specific net.
For trapezoidal prism:
This is the net of a trapezoidal prism whose bases have dimensions 2 and 8 (the length of the bases) and with height 5. If you are unsure what is what count the squares and conclude.
All nets are all made in a similar way, once you learn the pattern you’ll know it for every prism you can think of.
Here is the net for the hexagonal prism:
Note that you have to have as many rectangles as there are sides in your base polynomial.
Cylinders
Cylinder is a prism whose bases are circles, and are connected with a curved surface.
Cylinders can be oblique or right.
Right cylinder is a cylinder whose line that connects two centers of bases is perpendicular to those bases.
Oblique cylinder is a cylinder whose line that connects two centers of bases is not perpendicular to those bases.
What if we want to find the net of a cylinder? We know for sure that the two bases will be circles but what will our lateral side be?
It will also be a rectangle, but since we have only one ‘side’ here, it will be only one triangle. The question is which dimensions will that rectangle have?
We know one by default, it is our height. With second one we have to wrap up our circle to get enclosed area. That means that the second dimension of our rectangle will be equal to the circumference of the base.
That leads us to the following net:
Now it’s easy to see what will be the formula for our surface area, it will be area of two circles added to the surface of the triangle with a height h, and width 2 r π.
The volume of cylinder, oblique or right is always base times height.
Pyramids
Pyramids are polyhedrons bounded by one polynomial (base) with n sides, and n triangles. Those triangles are its lateral faces. Vertex of a pyramid is its top, a point where the free vertices of lateral sides merge. Height of a pyramid is a line that connects the base and vertex.
There are right and oblique pyramids. The height in right pyramids is perpendicular to the base, and its base is a regular polygon.
Pyramids are divided by the number of sides of its base.
Surface area of pyramids is equal to the sum of the area of its base and lateral sides.
Volume of prisms is equal to $\frac{1}{3}$ of the volume of the prism whose bases are equal to the base of that pyramid. This means that:
Volume of a pyramid = $\frac{1}{3}$ \cdot height.
From pyramids you can also make a net. Every pyramid will have a net in a shape of a star.
Rectangular pyramid
Pentagonal pyramid
Hexagonal pyramid
Net of a hexagonal pyramid
Cones
Cones are polyhedrons bounded with a circle and a circle sector.
The height of a cone is a line that is perpendicular to the base (circle) and goes through the vertex.
Volume of a cone is equal to the one third of the surface of the base times height.
Surface area is equal to the sum of the base (a circle) and it’s lateral side (circle sector).
Spheres
Spheres are a 3D generalization of a circle. Just imagine a circle constantly rotating in space.
Surface area:
$ S = 4R2\pi$
Volume:
$ V = \frac{4}{3} \pi R3$
Where R is the radius of a sphere.
Solid figures worksheets
Naming solid figures (162.2 KiB, 1,347 hits)
Surface area of solid figures (335.1 KiB, 1,602 hits)
Lateral area of solid figures (323.0 KiB, 1,297 hits)
Volume of solid figures (306.1 KiB, 1,204 hits)
Naming solids by its net (171.6 KiB, 1,377 hits)
Measure a net of solids (541.5 KiB, 1,986 hits)
Draw a solid from drawn net (401.4 KiB, 1,078 hits)
Draw a net from drawn solids (449.9 KiB, 1,174 hits)
Solids similarity (461.2 KiB, 1,246 hits)
