Geometry defines a three-dimensional shape as a solid figure or object with three dimensions – length, width, and height. In contrast to two-dimensional shapes, three-dimensional shapes have thickness or depth. They are a part of three-dimensional geometry. As we see in our everyday life, solid geometry includes objects of three-dimensional shapes such as cylinders, cubes, cuboids, cones, spheres, etc. There are so many 3D shapes, and every 3D shape occupies some space based on its dimensions. By nature, three-dimensional shapes have an inside and an outside, separated by a surface. All physical objects, things you can touch, are three-dimensional. Examples of three-dimensional shapes that can be seen in real life include a Rubik’s cube, a globe, a gas cylinder, a cubical box, a cuboidal board, etc.
Real-life examples 3D shapes
In three dimensions, shapes are defined by vertices, faces, and edges. Faces are the flat surfaces of 3D shapes. An edge is a line segment connecting two faces. The vertex is a point where three edges meet. Additionally, they have depth, so they occupy some volume. There are some 3D shapes that have 2D portions, such as the bases and tops. One example is a cube, which has square faces on all sides. Shapes in 3D can be classified into several categories. Others are shaped like pyramids or prisms; some have curved surfaces.
In mathematics, we explore 3-dimensional objects in the concept of solids and apply them to real-world situations. Soccer balls, cubes, buckets, and books are a few examples of 3D shapes in real life.
3D shapes faces, edges, vertices
3D shapes and objects differ from 2D shapes and objects because they have three dimensions – length, breadth, and height. This means that objects have three dimensions, which translates into faces, edges, and vertices. Let us examine these three in detail.
- A face refers to any curved surface or single flat of a solid object
- 3D shapes can have multiple faces
- An edge is a line segment that joins one vertex (corner point) to another
- They serve as the junction of two faces
- A vertex is a point where two or more lines meet.
- It is a corner.
- Vertices are points of intersection of edges.
Types of 3D Shapes
The 3D shapes exist in many shapes that have different bases, volumes, and surface areas. They are:
A sphere has a round shape. The surface of this shape consists of points that are equidistant from its center. Earth appears to be a sphere, but it is not. Planet Earth is shaped like a spheroid. Spheroids resemble spheres, but their radius from center to surface is not constant at every point. A sphere has the following characteristics.
- It is symmetrical and shaped like a ball.
- Any spherical object has a radius, diameter, circumference, volume, and surface area.
- The points on the sphere are all equally distant from each other.
- It has one face, no edges, and no vertices.
- Since it does not have flat faces, it is not a polyhedron.
Cube and Cuboid
The cube and cuboid are 3D geometric shapes with the same number of faces, vertices, and edges. The difference between a cube and a cuboid is that a cube has all six faces as squares, while a cuboid has all six faces as rectangles. The volume and surface area of a cube and cuboid are different. For a cube, the length, width, and height are all the same, while for a cuboid they are all different.
Cylinders are 3D shapes with two circular faces, one at the top and one at the bottom, and a curved surface. Cylinders have a height and a radius. In a cylindrical shape, the height is the perpendicular distance between the top and bottom faces. The following are some important characteristics of a cylinder.
- It has one curved face.
- From the base to the top, the shape remains the same.
- A three-dimensional object, cylinders have two identical ends, which are either circular or oval.
- Right cylinders have both their circular bases on the same line. Oblique cylinders are cylinders with one base separated from another.
Cones are also 3D shapes that have a flat, circular base and a pointed tip at their top. The pointed end which is at the top of the cone is called ‘Apex’. Cones also have curved surfaces. As with cylinders, a cone can also be classified as a right circular cone and an oblique cone.
- Cones have a circular or oval base with an apex (vertex).
- Cones are rotated triangles.
- According to the alignment of the apex with the center of the base, a right cone or an oblique cone is formed.
- A right circular cone has an apex (or a pointed tip) that is perpendicular to its base. An oblique cone has its apex located anywhere other than the center of the base.
- Cones have a height and a radius. Cones also have slant height, which is the distance from the point at which the apex meets any point on the circumference of the circular base of the cone.
l= length, h=height, r= radius
Toruses are 3D shapes. In three-dimensional space, it is formed by rotating a smaller circle of radius (r) around a larger circle of radius (R).
- A torus is a ring that resembles a tire or doughnut.
- It has no edges or vertices.
r= radius of the smaller circle. R= radius of the larger circle.
A pyramid has a polygon base and an apex with straight edges and flat faces. It is a polyhedron. Depending on how the apex aligns with the center of the base, regular and oblique pyramids can be distinguished. The pyramid consists of:
- A triangular base, known as Tetrahedron
- A quadrilateral base, known as a square pyramid
- A pentagon base, known as a pentagonal pyramid
- A regular hexagon base, known as a hexagonal pyramid
a – apothem length of the pyramid, b – base length of the pyramid, h – height of the pyramid
Polygonal solids with flat parallelogram sides are prisms. Here are some of the characteristics of prisms:
- All along its length, it has the same cross-section.
- Different types of prisms include triangular prisms, square prisms, pentagonal prisms, hexagonal prisms, etc.
- The prisms can also be classified as regular prisms or oblique prisms.
Next, let us learn about 3-dimensional shapes with regular polyhedrons (platonic solids).
A polyhedron is a 3D shape with polygonal faces (triangle, square, hexagon) and straight edges and vertices. A polyhedron is defined as having:
- Edges that are straight.
- Faces are the flat sides.
- Corners are called vertices.
It is also known as a platonic solid. There are five regular polyhedrons. The faces of a regular polyhedron are all the same. Cubes are among the most basic and familiar polyhedrons. The cube is a regular polyhedron with six square faces, twelve edges, and eight vertices.
Below are some more examples of regular polyhedrons:
- Tetrahedron with four equilateral-triangular faces
- Cube with six square faces also known as a hexahedron
- Octahedron having eight equilateral-triangular faces
- Dodecahedron with twelve regular pentagon faces
- Icosahedron having twenty equilateral-triangular faces
Three-Dimensional Shapes with Curves
Polyhedrons do not include solid shapes with curved or round edges. Only straight sides can be found on polyhedrons.
A cylinder, cone, sphere, and torus are the four most common curves in geometry. Curves can be found in many of the objects around us.
Surface Area and Volume of 3D shapes
The two distinct measures used to define 3D shapes are:
- Surface Area
It is defined by the total area of the surface of the 3-D object. It is commonly denoted as “SA”. The surface area is measured in terms of square units. Below are the three different classifications of surface area. You will find them here:
- The area of all the curved regions is known as Curved Surface Area (CSA)
- The area of all the curved regions and all the flat surfaces excluding base areas is known as Lateral Surface Area (LSA)
- The area of all the surfaces including the base of a 3D object is known as the Total Surface Area (TSA)
The volume is defined by the total space occupied by the solid object or three-dimensional shape. The volume is commonly denoted as “V”. It is measured in terms of cubic units. All 3D shapes have different surface areas and volumes.
Examples of Surface Area Calculations
An area of a cube can be calculated by multiplying (length x width) by 6 since all six faces are identical.
Since the face of a cube is a square, you need to take one measurement – the length and width of a square are, by definition, the same.
One face of this cube is therefore 10 × 10 cm = 100cm2. Multiply by 6, the number of faces on a cube, and we find that the surface area of this cube is 600cm2.
Other Regular Polyhedrons
In the same way, the surface area of other regular polyhedrons (platonic solids) can be worked out by finding the area of one side and then multiplying that number by the total number of sides – see the Basic Polyhedrons diagram above.
If the area of one pentagon making up a dodecahedron is 22cm2 multiply this by the number of sides (12) to give the answer 264cm2.
The area of a standard pyramid with four equal triangular sides and a square base is calculated by first working out the area of the base (square) length × width.
Next, calculate the area of one side (triangle). Then measure the width along the base and the height of the triangle (also referred to as the slant length) from the central point on the base to the apex.
Divide your answer by two to get the surface area of one triangle, then multiply it by four to get the surface area of all four sides, or simply multiply one triangle’s surface area by two to get the surface area of all four sides.
Lastly, total up the base and side areas of the pyramid to find its total area.
To calculate the surface area of other types of pyramids, add together the base (known as the base area) and the sides (called the lateral area). You may need to measure the sides individually.
A geometric net is a two-dimensional representation of a three-dimensional object. Surface areas of three-dimensional objects can be calculated by means of nets. In the diagram below you can see how basic pyramids are built, if the pyramid is unfurled you are left with the net.
Right Pyramid Oblique Pyramid
To calculate the surface area of a prism:
A prism has two ends that are the same and flat parallelogram sides.
Multiply the area of one end by 2.
In the case of a regular prism (where all the sides are the same), multiply the area of one side by the number of sides.
If the prism has irregular sides (different lengths), calculate each side’s area.
The total area of the prism is equal to your two answers (ends x sides).
It is useful to think about the component parts of the shape when calculating the surface area of a cylinder. Imagine a tin of sweetcorn that has a top and a bottom, both of which are circles. You would get a rectangle if you cut the side along its length and flattened it. Therefore, you need to find the area of two circles and a rectangle.
First work out the area of one of the circles.
The area of a circle is π(pi) × radius2.
Assuming if the radius of 5cm, the area of one of the circles is 3.14 × 52 = 78.5cm2.
Multiply the answer by 2 since there are two circles 157cm2
The area of the side of the cylinder is the perimeter of the circle × the height of the cylinder.
Perimeter is equal to π x 2 × radius. Taking our example, 3.14 × 2 × 5 = 31.4
Measure the height of the cylinder – for this example, the height is 10 cm. The surface area of the side is 31.4 × 10 = 314cm2.
You can calculate the total surface area by adding the area of the circles and of the sides:
157 + 314 = 471cm2
The length of the ‘slant’, as well as the radius of the base, must be taken into account when calculating the surface area of a cone.
Calculating it is, however, relatively simple:
Let us say the radius is 5cm and the length of the slant is 10cm
The area of the circle at the base of the cone is, π(pi) × radius2.
In this example the sum is 3.14 × 52 = 3.14 × 25 = 78.5cm2
The area of the side, the sloping section, can be found using this formula:
π(pi) × radius × length of slant.
In our example the sum is 3.14 × 5 × 10 = 157cm2.
Finally, add the base area to the side area to get the total surface area of the cone.
78.5 + 157 = 235.5cm2
The surface area of a sphere is a relatively simple expansion of the formula for a circle’s area.
4 × π × radius2.
When determining the diameter of a sphere, it is often easiest to measure the distance across the sphere. You can determine the radius which is half of the diameter.
Let us consider the diameter of the tennis ball as 2.6 inches
The radius is therefore 1.3 inches.
For the formula we need the radius squared.
1.3 × 1.3 = 1.69.
The surface area of a tennis ball is, therefore:
4 × 3.14 × 1.69 = 21.2264 inches2.
In order to find the area of a torus, two radius values must be determined.
The large or major radius (R) is the diameter from the middle of the hole to the middle of the ring.
The small or minor radius (r) is calculated by measuring the ring from the center to the outside edge.
The diagram shows two views of an example torus and how to measure its radiuses (or radii).
R (Large Radius) = 20 cm
r (Small Radius) = 4 cm
Surface areas can be calculated using two methods (one for each radius). For each part, the calculation is the same.
The formula is: surface area = (2πR)(2πr)
To calculate the surface area of the example torus.
(2 × π × R) = (2 × 3.14 × 20) = 125.6
(2 × π × r) = (2 × 3.14 × 4) = 25.12
You can find the total surface area of the example torus by multiplying the two answers together.
125.6 × 25.12 = 3155.072cm2.
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