THE area on one solidgeometric it can be obtained by the sum of the areas of each one of the geometric figures that compose it. A tetrahedron, for example, is a pyramid of triangular base. This pyramid is formed by four triangles: one base and three side faces. Adding the areas of each of these triangles together, we get the area of the tetrahedron.
Regular tetrahedron on the right and its plane on the left
Below are the formulas used to calculate the area of some geometric solids and examples of how to use them.
cobblestone area
Consider a paving stone whose length measures "x", the width measures "y" and the height measures "z", as in the following figure:
The formula used to calculate your area é:
A = 2xy + 2yz + 2xz
This same formula applies to the cube area, which is a special case of paving stone. However, since all the edges of the cube are the same, this one formula Can be reduced. Thus, the area of an edge cube L is determined by:
A = 6L2
Example 1
what is the area of a blockrectangular with length and width equal to 10 cm and height equal to 5 cm?
As length = width = 10 cm, we will have x = 10 and y = 10. As height = 5 cm, we will have z = 5. Using the formula for the parallelepiped area, we will have:
A = 2xy + 2yz + 2xz
A = 2·10·10 + 2·10·5 + 2·10·5
A = 200 + 100 + 100
H = 400 cm2
Example 2
What is the area of a cube whose edge measures 10 cm?
A = 6L2
A = 6·102
A = 6·100
H = 600 cm2
Cylinder area
Given the cylinder of radius r and height h, illustrated by the figure below, a formula used to calculate your area é:
A = 2πr (r + h)
Example 3
Determine the area of a cylinder whose height measures 40 cm and the diameter measures 16 cm. Consider π = 3.
a damn circle is equal to half its diameter (16:2 = 8). Thus, the radius of the base of the cylinder is equal to 8 cm. Just replace these values in the formula:
A = 2πr (r + h)
A = 2·3·8(8 + 40)
A = 2·3·8·48
A = 6·384
H = 2304 cm2
cone area
The formula used to determine the cone area é:
A = πr (r + g)
The following figure shows that r is the radius of the cone and g is the measure of its generatrix.
Example 4
calculate the area on one cone whose diameter is 24 cm and whose height measures 16 cm. Consider π = 3.
To discover the measuregivesgeneratrix of the cone, use the following expression:
g2 = r2 + h2
Since the radius of the cone is equal to half its diameter, the measure of the radius is 24:2 = 12 cm. Replacing the values in the expression, we will have:
g2 = r2 + h2
g2 = 122 + 162
g2 = 144 + 256
g2 = 400
g = √400
g = 20 cm
Replacing the cone radius and generatrix measure in the formula in area, we will have:
A = πr (r + g)
A = 3·12(12 + 20)
A = 36·32
H = 1152 cm2
sphere area
The formula used to calculate the sphere area of radius r is:
A = 4πr2
Example 5
Calculate the area of the sphere in the following image. Consider π = 3.
Using the formulagivesarea gives ball, we will have:
A = 4πr2
A = 4·3·52
A = 12·25
H = 300 cm2
Pyramid area
You prisms and pyramids don't have a formulaspecific for calculating area, as the shape of its lateral faces and bases is very variable. However, it is always possible to calculate the area of a geometric solid by flattening it and adding the individual areas of each of its faces.
When these solids are straight, like the prismstraight and the pyramidstraight, it is possible to identify relations between the measures of its lateral faces.
See too:Calculating the area of a prism
Example 6
One pyramid straight with a square base has an apothema equal to 10 cm and a base edge equal to 5 cm. What is your area?
To solve this example, look at the image of the pyramid below:
A straight pyramid with a square base has all side faces congruent. So, just calculate the area of one of them, multiply the result by 4 and add this to the result obtained in the calculation of area of the base of the pyramid.
To calculate the area of one of these triangles, we need the measure of its height. This measure is equal to the apothema of the pyramid, therefore 10 cm. In the following formula, the apothema will be represented by the letter h. In addition, all bases of triangles are congruent, as they are all sides of a square and measure 5 cm.
Area of a side face:
A = bh
2
A = 5·10
2
A = 50
2
H = 25 cm2
Area of the four lateral faces:
A = 4·25
H = 100 cm2
Base area (which is equal to the area of a square):
A = 12
A = 52
H = 25 cm2
Total area of this pyramid:
A = 100 + 25 = 125 cm2
prism area
As stated, there is no specific formula for the prism area. We must calculate the area of each of its faces and add them up at the end.
Example 7
What is the prism area straight base square, knowing that the height of this solid is 10 cm and that the edge of its base measures 5 cm?
Solution:
Below, see an image of the prism in question to help build the solution:
The exercise informs that the baseofprism it's square. Furthermore, the two prism bases are congruent, that is, finding the area of one of these bases, just multiply this measurement by 2 to determine the area of the two prism bases.
THEB = 12
THEB = 52
THEB = 25 cm2
Also, as it has a square base, it's easy to see that it has fourfacessides, which are also congruent, since the solid is straight. So, finding the area of one of the lateral faces, just multiply this value by 4 to find the lateral area of the prism.
THEfl = b·h
THEfl = 5·10
THEfl = 50 cm2
THEthere = 4Afl
THEthere = 4·50
THEthere = 200 cm2
THE areatotalofprism é:
A = AB + Athere
A = 25 + 200
H = 225 cm2
By Luiz Paulo Silva
Degree in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/matematica/area-solidos-geometricos.htm
