Every existing object is composed of a flat shape or a geometric shape. These structures are formed according to the tools or things that humans need. These shapes can be calculated, such as area, length, width, and volume.
A spatial figure is a three-dimensional building that has space or volume and sides that limit it. The shape of the room itself is grouped into two, namely the curved side room and the flat side shape. Construct a curved side chamber consisting of a cone, sphere, and tube. Meanwhile, build flat side spaces in the form of cubes, pyramids, blocks, and prisms.
Build the familiar spaces in our lives in the form of tubes and blocks. The pyramid is a form that is rarely applied in everyday tools. Then, what kind of space are actually the five lakes? Next, you can listen to the explanation below.
The formula for the volume of a pyramid and the surface area of a pyramid
Limas is a three-dimensional geometric shape bounded by many square bases and has one vertex. Meanwhile, in the Big Indonesian Dictionary (KBBI), a pyramid is defined as a spatial object whose base is triangular (rectangle and so on) and its sides are triangular with the vertices that coincide.
Limas are grouped into several categories such as triangular pyramids, rectangular pyramids, pentagonal pyramids, and so on. A pyramid that has a square base is called a pyramid. While a pyramid with a circular base is called a cone. For example pyramid-shaped pyramids in Egypt with a square base.
The characteristics of the pyramid in detail as follows.
- Had 2n ribs
- Has many sides depending on the base, namely: one square-shaped side (can be a quadrangle, pentagon, etc.) in the form of a base, the other four sides are in the shape of a triangle standing upright and forming angles
- Has (n+1) facets
- Has (n+1) vertices
The following is the formula for calculating the volume and surface area of a pyramid.
- Limas Volume
V = 1/3 xpxlxt
- Surface Area of the Limas
L = base area + sheath area of the pyramid
Properties and Classification of Limas
On the Bobo.grid.id page, pyramids have several of these characteristics.
- Has a rectangular base
- Has 8 ribs
- It has five vertices including four base angles and one apex angle
- It has five sides, namely one side in the form of a rectangular base and the other four sides is called a triangular vertical plane
While pyramids can be grouped into several categories below.
1. Triangular pyramid
A triangular pyramid is a geometric shape that has a triangular base. Usually the triangles used are isosceles triangles, equilateral triangles, and other triangular shapes.
A triangular pyramid is a shape bounded by a polygonal base and a triangular plane whose base coincides with the sides of the polygonal plane. Meanwhile, the vertex coincides with a point that is outside the polygon.
The elements forming a triangular pyramid are detailed as follows.
- The corner point is formed from the meeting of 2 or more edges
- The rib is the line that is the intersection between the 2 sides of the pyramid
- The side plane is the plane that consists of the base plane and the straight side plane
- The base plane is the plane which is the base of a pyramid
- The vertical side plane is the plane that intersects the base plane
- The apex point is the point which is the joint point between the base blankets
- The height of the pyramid is the distance between the base plane and the vertex
- Has 4 corner points
- Has 4 sides
- Has 6 ribs
While the formula for the volume and surface area of a triangular pyramid is as follows.
- Triangular Plumbing Volume
V = ½ x La xt
Or
V = ½ x (1/2 x as ts) xt
Information:
V = volume
La = area of the base
as = base of the triangle
ts = height of the base triangle
t = pyramid height
- Surface Area of a Triangular Pyramid
L = La + L∆ I + L∆ II + L∆ III
Information:
L = surface area
La = area of the base
L∆ = area of the triangle
2. Quadrilateral pyramid
A rectangular pyramid is a pyramid with a rectangular base. It can be a square, rectangle, rhombus, parallelogram, kite and trapezoid. The characteristics of a rectangular pyramid are as follows.
- Sum of sides of a triangular pyramid = n + 1 = 4 + 1 = 5 sides
- Number of triangular pyramid edges = 2 × n = 2 × 4 = 8 edges
- The number of vertices of a triangular pyramid = = n + 1 = 4 + 1 = 5 vertices
- Has 5 sides (1 base side and 4 upright sides)
- The sides of the base are rectangular
- 4 The vertical sides are triangular
- Has 5 corner points
- Has 8 ribs
While the formula for volume and surface area of a rectangular pyramid is as follows.
- Quadrilateral Plumbing Volume
V = ½ x La xt
Information:
V = volume
La = area of the base
as = base of the triangle
ts = height of the base triangle
t = pyramid height
- Surface Area of a Quadrilateral
L = La + L∆ I + L∆ II + L∆ III + L∆ IV
Information:
L = surface area
La = area of the base
L∆ = area of the triangle
3. The pentagonal pyramid
A pentagonal pyramid is a type of pyramid that has a pentagonal base. The following are the observable characteristics of a pentagonal pyramid.
- Has six sides (one side of the base and five sides of the pyramid)
- The side of the base is a flat pentagon
- The upright side is a flat, triangular shape
- It has five diagonal areas that are triangular in shape
- Has 10 ribs
- Has 6 corner points
- Has 1 vertex
While the formula for the volume and surface area of a pentagonal pyramid is as follows.
- Volume of the pentagonal pyramid
V = 1/3 x La xt
Information:
V = volume
La = area of the base
as = base of the triangle
ts = height of the base triangle
t = pyramid height
- Surface Area of a pentagonal pyramid
L = La + L∆ I + L∆ II + L∆ III + L∆ IV + L∆ V
Information:
L = surface area
La = area of the base
L∆ = area of the triangle
4. Hexagonal pyramid
A hexagon pyramid is a type of pyramid that has a flat hexagonal base. The characteristics are as follows.
- Has 7 corner points
- Has 12 ribs
- Has 6 straight sides
- Has 1 side base
- It has a triangular shape
- The side of the base is in the shape of a polygon
- Has one peak
- The name of the pyramid depends on the shape of the base
While the formula for the volume and surface area of a hexagonal pyramid is as follows.
- Volume of the Hexagonal Plumbing
V = 1/3 x La xt
Information:
V = volume
La = area of the base
as = base of the triangle
ts = height of the base triangle
t = pyramid height
- Surface Area of a Hexagonal Pyramid
L = La + L∆ I + L∆ II + L∆ III + L∆ IV + L∆ V + L∆ VI
Information:
L = surface area
La = area of the base
L∆ = area of the triangle
Examples of Limas Questions
The following are examples of questions regarding pyramids which are summarized from various sources on the internet.
1. A square pyramid has a base length of 18 cm. Meanwhile, the length of the vertical side is 24 cm.
Determine the distance between the top of the pyramid and its base!
Discussion:
First, you have to draw the square pyramid.
The distance between the top of the pyramid and its base is expressed as TO.
In a square, the length of the diagonal is the product of the side length and √2.
That is, the length of side AC = 18√2 cm. Based on the picture above, the length of OC can be formulated as follows.
Furthermore, you can find TO using the Pythagorean theorem as follows.
So, the distance between the top of the pyramid and the base is 3√46 cm.
2. Look at the following equilateral triangle pyramid!
If the length of the edge of the pyramid is 12 cm, find the distance between the line CD and the plane ABC!
Discussion:
First, you have to draw the distance between line CD and plane ABC.
The distance between line CD and plane ABC is the same as the length of point D to point P.
Because the base of the pyramid is an equilateral triangle, the length of the DP can be formulated as follows.
So, the distance between line CD and plane ABC is 6√3 cm.
3. There is a pentagon prism with a base area of 60 cm2. If the height of the prism is 8 cm, what is the volume of the pentagonal prism? . . .
Discussion
V = La xt
V = 60 cm2 x 8 cm
V = 480 cm 3
4. A pentagonal pyramid has a volume of 116 liters. If the height of the pyramid is 12 cm, the area of the base of the pyramid is . . . .
Discussion
V = 1/3 x La xt
La = V/(1/3 xt)
La = (3 x V)/t
La = (3 x 116 liters)/12 dm
Because 1 liter = 1 dm3 then
La = 348 dm 3/12 dm
La = 29 dm 2
5. A rectangular pyramid has a square base with a side length of 6cm and a height of 5cm. If one side of the triangle has a height of 4 cm. Then calculate the surface area and volume of the rectangular pyramid.
Is known:
Base shape = square
Square Side (Rib Base) = 6 cm
t pyramid = 4 cm
t Δ1 = 5 cm
asked:
Limas Area (L)
Plumbing volume (V)
Completion:
To find the surface area, we must find the area of all the sides.
First, calculate the surface area of one side of the triangle
L Δ1 =½ × a Δ1 × t Δ1
L Δ1 =½ × 6cm × 5cm
L Δ1 =15cm2
Because the shape of the base is a square, then
a Δ1 = a Δ2 = a Δ3 = a Δ4 = 6cm, and
t Δ1 = t Δ2 = t Δ3 = t Δ4 = 4cm
so that
L Δ1 = L Δ2 = L Δ3 = L Δ4 = 15cm2
Then, calculate the surface area of the base
L base = square side × square side
L base = 6cm × 6cm = 36cm2
Next, we just need to add up all the surface areas
L = L base + L Δ1 + L Δ2 + L Δ3 + L Δ4
L = 36 cm 2 + 15 cm 2 + 15 cm 2 + 15 cm 2 + 15 cm 2
L = 96 cm 2
V = ⅓ × L base × h
V = ⅓ × 36 cm 2 × 4 cm
V = 48 cm 3
Various Build Space
The following are various geometric shapes, both from curved side geometric shapes and flat sided geometric shapes.
1. Cones
In the Big Indonesian Dictionary (KBBI), a cone is defined as an object (space) that has a round base and reaches up to one point. It becomes part of a three-dimensional geometric shape or building.
Tubes with cones have in common, that is, they both have circular bases. Meanwhile, the difference lies in the blanket, the conical blanket has the upright side of the cone. Meanwhile, the rectangular tube.
The more detailed characteristics of cones can be seen in the following presentation.
- Has two side planes
- Has one curved rib
- Has one corner point as a vertex
- The cone has no diagonals
Cones have volume and surface area. Here’s the formula for both.
- Cone Volume
V = 1/3 x π × r² × t
- Cone Surface Area
L = (π × r²) + (π × r × s)
2. Ball
The ball is a three-dimensional space figure that has boundaries in the form of curved sides. It has no ribs and nooks due to its round shape. However, the ball has a curved side plane as a volume or space limiter. For example basketball, globe, and so on.
The characteristics of the spherical space are as follows.
- It has no edges, vertices and diagonals
- Has only one side plane that forms an arch
- The distance from the wall to the core or center of the ball is called the radius
- Having one core point or center
The formulas for the volume and surface area of a ball are as follows.
- Ball Volume
V = 4/3 × π × r³
- Ball Surface Area
L = 4 × π × r²
3. Tube
The tube is a three-dimensional figure consisting of a circular lid and base of the same size and a rectangular body covering the upright side. For example musical instruments drums, canned milk, and so on.
The main characteristic of the tube is that it has 3 sides, namely the base and lid in the form of a circle and the blanket in the shape of a rectangle and has no corners.
Meanwhile, the formula for volume and surface area of a cylinder is as follows.
- Tube Volume
V = π × r² × t
- Tube Surface Area
L = (2 × base area) + (base circumference × height)
4. Cubes
A cube is a three-dimensional shape bounded by a rectangular field. It consists of 6 identical rectangular sides, 12 equal sides, and 8 vertices. Its shape is a square. For example dice, cardboard, and so on.
The characteristics are detailed as follows.
- Has 6 side surfaces
- Has 12 ribs
- Has 8 corner points
- The sides of the cube are square
- The lengths of the room diagonals are the same
- The cubes are the same length
- The diagonal plane of each cube is a rectangle
Meanwhile, the volume and surface area formulas are as follows.
- Cube Volumes
V = sxsxs
- Surface Area of a Cube
L = 6 x (sxs)
5. Blocks
A beam is a three-dimensional geometric figure bounded by 2 squares and 4 rectangles that are perpendicular to each other. The blocks have the same magnitude on opposite sides. For example cupboards, pencil boxes, aquariums, and so on.
The detailed characteristics of the beam can be seen in the presentation below.
- The sides of the beam have two pairs of rectangles
- The parallel ribs are the same length
- Each diagonal on the opposite side is the same length
- Each diagonal is a rectangle
Meanwhile, the formula for the volume and surface area of a block is as follows.
- Beam Volume
V = pxlxt
- Beam Surface Area
L = 2 x (pl + lt + pt)
6. Prism
A prism is a three-dimensional shape that is bounded by a base and a lid in the form of various squares and has the same size. As for the Big Indonesian Dictionary (KBBI), a prism is a polygonal plane that has a pair of parallel and congruent sides called the base and another side called the height.
In daily life, you can find prism-shaped items such as roofs, camping tents, and so on. To find out more about prism features, you can listen to the following details.
- Has (n+2) facets
- Has 2n vertices
- Having a congruent (same) base and roof plane
The formula for calculating the volume and surface area of a prism is as follows.
- Prism Volumes
V = base area x height
- Prism Surface Area
L = (2 x base area) + (base circumference x height)