# 7 Ways to Determine the Surface Area of ​​a Geometric Solid

Surface area is the total amount of space occupied by all surfaces of an object. It is the sum of the area of ​​all the object's surfaces. Finding the surface area of ​​a three-dimensional figure is relatively easy, as long as you know which formula to use. Each geometric figure has a specific formula; so, before starting, you need to identify the way you are working. Memorizing the formula for the surface area of ​​various objects can make calculations easier in the future. See in this article some of the most common geometric figures.

## Steps

### Method 1 of 7: Cube

#### Step 1. Identify the formula for the surface area of ​​a cube

A cube has six identical square sides. Since the height and width of a square are equal, the area of ​​this figure is the2, where "a" is the length of one side. Since there are six identical sides in a cube, to find the surface area, simply multiply the area of ​​one side by six. The formula for the surface area (AS) of a cube is AS = 6a2, where a is the length of one side.

• The unit of surface area will be the unit of length squared: cm2m2, km2, etc.

#### Step 2. Measure the length of one side

Each side or edge of a cube should, by definition, be equivalent to the length of the others, so you only need to measure one side. Using a ruler, measure the length of one side. Pay attention to the units used.

• Label this measurement as "a".
• Example: a = 2 cm.

#### Step 3. Square the "a" measurement

Square the measurement taken from the length of the edge. To do this, multiply the number by itself. If you are learning these formulas for the first time, if you write them down it can help you memorize them, like AS= 6*a*a.

• Note that this Step calculates the area of ​​one side of the cube.
• Example: a = 2 cm.
• The2 = 2 x 2 = 4 cm2

#### Step 4. Multiply this product by six

Remember that a cube has six identical sides. Now that you have the area on one side, you'll need to multiply it by six to total all six sides.

• This Step completes the calculation of the cube's surface area.
• Example: a2 = 4 cm2
• Surface area = 6 x a2 = 6 x 4 = 24 cm2

### Method 2 of 7: Rectangular Prism

#### Step 1. Identify the surface area formula of a rectangular prism

As with the cube, a rectangular prism has six sides; however, unlike him, the sides are not identical. In a rectangular prism, only opposite sides are identical. Therefore, to calculate its surface, it is necessary to consider the different lengths of its side. Therefore, its formula is as follows: AS = 2ab + 2bc + 2ac.

• In this formula, "a" is the prism's width, "b" is the height, and "c" is the length.
• By breaking this formula apart, it is possible to identify that it simply sums all the areas of each face of the object.
• The unit of surface area will be the unit of length squared: cm2m2, km2, etc.

#### Step 2. Measure the length, height and width of each side

These three measurements can vary, so measure them separately. Using a ruler, measure and record each measurement, using the same units for each.

• Measure the base length to find the prism length, and assign that value to "c".
• Example: c = 5 cm.
• Measure the base width to find the prism width, and assign this value to "a".
• Example: a = 2 cm.
• Measure the height of the side to find the height of the prism, and assign that value to "b".
• Example: b = 3 cm.

#### Step 3. Calculate the area of ​​one side of the prism and multiply it by two

Remember that there are six faces in a rectangular prism, but the opposite sides are identical. Multiply the length by the height, or c by a, to find the area of ​​a face. Take this measurement and multiply it by two because of the opposite side equivalent.

• Example: 2 x (a x c) = 2 x (2 x 5) = 2 x 10 = 20 cm2

#### Step 4. Calculate the area on the other side of the prism and multiply it by two

As with the first pair of faces, multiply the width by the height, or a by b, to find the area of ​​another face of the prism. Multiply this measurement by two because of the opposite side equivalent.

• Example: 2 x (a x b) = 2 x (2 x 3) = 2 x 6 = 12 cm2.

#### Step 5. Calculate the area of ​​the ends of the prism and multiply it by two

The two end faces will be the ends. Multiply the length by the width, or c by b, to find their area. Multiply these measurements by two because of the opposite side.

• Example: 2 x (b x c) = 2 x (3 x 5) = 2 x 15 = 30 cm2

#### Step 6. Add the three measurements together

Since the surface area is the value of the total area of ​​an object's faces, the final step is to add the individually calculated values. Add the measurements from all sides to find the total surface area.

• Example: Surface area = 2ab + 2bc + 2ac = 12 + 30 + 20 = 62 cm2.

### Method 3 of 7: Triangular Prism

#### Step 1. Identify the surface area formula of a triangular prism

A triangular prism has two identical triangular sides and three rectangular faces. To find the surface area, you have to calculate and add up the area of ​​all sides. The formula for the surface area of ​​a triangular prism is AS = 2a + ph, where a is the area of ​​the triangular base, p is the perimeter of the triangular base, and h is the height of the prism.

• In this formula, a is the area of ​​the triangle, that is, a= 12bh{displaystyle {frac {1}{2bh}}}

, onde b é a base do triângulo e h é a altura.

• O p é o perímetro do triângulo, que pode ser calculado pela soma dos três lados do triângulo.

#### Step 2. Calculate the area of ​​the triangular face and multiply it by two

The area of ​​a triangle is 12{displaystyle {frac {1}{2}}}

b*h, onde b é a base dele e h' é a altura. Como existem duas faces idênticas do triângulo, a fórmula é multiplicada por dois. Isso facilita o cálculo de ambas as faces, b*h.

• A base, b, equivale ao comprimento da base do triângulo.
• Exemplo: b = 4 cm.
• A altura, h, da base triangular equivale à distância da borda da base e do ponto mais alto.
• Exemplo: h = 3 cm.
• A área de um triângulo multiplicada por 2= 2(12{displaystyle {frac {1}{2}}}
• )b*h = b*h = 4*3 =12 cm.

#### Step 3. Measure each side of the triangle and the height of the prism

To finish calculating the surface area, you will need to know the measurement of the length of each side of the triangle and the height of the prism. Height is the distance between two triangular faces.

• Example: h = 5 cm.
• The three sides refer to the three sides of the base of the triangle.
• Example: s1 = 2 cm, s2 = 4 cm, s3 = 6 cm.

#### Step 4. Identify the perimeter of the triangle

The perimeter of a triangle can be calculated simply by adding the measure of all sides: s1 + s2 + s3.

### Example: p = s1 + s2 + s3 = 2 + 4 + 6 = 12 cm

#### Step 5. Multiply the perimeter of the base by the height of the prism

Remember that the prism height is the distance between two triangular bases. In other words, multiply p by h.

• Example: p x h = 12 x 5 = 60 cm2.

#### Step 6. Add the two measurements together

You will need to add the two measurements from the previous two steps together to calculate the surface area of ​​the triangular prism.

• Example: 2a + ph = 12 + 60 = 72 cm2.

### Method 4 of 7: Sphere

#### Step 1. Identify the formula for the surface area of ​​a sphere

The sphere has a curved surface. Therefore, to calculate your surface area, you will need to use the mathematical constant pi. The surface area of ​​a sphere can be calculated by the formula AS = 4π*r2.

• In this formula, r equals the radius of the sphere. Pi, or π, should be approximated to 3, 14.
• The unit of surface area will be the unit of length squared: cm2m2, km2, etc.

#### Step 2. Measure the radius of the sphere

The radius of the sphere is half the diameter value, or half the distance from one side of the sphere's center to the other.

### Example: r = 3 cm

#### Step 3. Square the radius

To do this, just multiply the number by itself. Multiply measure r by itself. Remember that the formula can be rewritten as AS = 4π*r*r.

• Example: r2 = r x r = 3 x 3 = 9 cm2

#### Step 4. Multiply the squared radius by the approximate constant pi

Pi is a constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number with many decimal numbers, often approximated to 3, 14. Multiply the square radius by π, or 3, 14, to find the area of ​​a circular section of the sphere.

• Example: π*r2 = 3.14 x 9 = 28.26 cm2

#### Step 5. Multiply this product by four

To complete the calculation, multiply the result by four. Find the surface area of ​​the sphere by multiplying the flat circular area by four.

• Example: 4π*r2 = 4 x 28, 26 = 113, 04 cm2.

### Method 5 of 7: Cylinder

#### Step 1. Identify the formula for the surface area of ​​a cylinder

A cylinder has two circular ends delimiting a rounded surface. The formula for finding the surface area of ​​a cylinder is AS = 2π*r2 + 2π*rh, where r equals the radius of the circular base and h equals the height of the cylinder. Round pi or π to 3, 14.

• The formula *2π*r2 represents the surface area of ​​the two circular ends, while 2πrh equals the surface area of ​​the column that connects them.
• The unit of surface area will be the unit of length squared: cm2m2, km2, etc.

#### Step 2. Measure the radius and height of the cylinder

The radius of a circle is half the diameter value, or half the distance from one side of the circle's center to the other. Height is the total distance of the cylinder from one end to the other. Using a ruler, measure and note these values.

• Example: r = 3 cm.
• Example: h = 5 cm.

#### Step 3. Calculate the area of ​​the base and multiply it by two

To find the area of ​​the base, just use the formula for the area of ​​the circle, or π*r2. To complete the calculation, square the radius and multiply it by pi. Multiply the result by two to find the second identical circle at the other end of the cylinder.

• Example: base area = π*r2 = 3.14 x 3 x 3 = 28.26 cm2
• Example: 2π*r2 = 2 x 28, 26 = 56, 52 cm2

#### Step 4. Calculate the surface area of ​​the cylinder using the formula 2π*rh

This is the formula for calculating the surface area of ​​a tube. The tube is the space between the two circular ends of the cylinder. Multiply the radius by two, by pi and by the height.

• Example: 2π*rh = 2 x 3, 14 x 3 x 5 = 94, 2 cm2

#### Step 5. Add the two measurements together

Add the surface area of ​​the two circles to the surface area of ​​the space between them to calculate the total surface area of ​​the cylinder. Note that when you add these values ​​together, you are using the original formula: AS =2π*r2 + 2π*rh.

• Example: 2π*r2 + 2π*rh = 56, 52 + 94, 2 = 150, 72 cm2

### Method 6 of 7: Quadrangular Pyramid

#### Step 1. Identify the formula for the surface area of ​​a quadrangular pyramid

A quadrangular pyramid has a square base and four triangular sides. Remember that the squared area is the length of one side squared. The area of ​​the triangle is 12sl{displaystyle {frac {1}{2sl}}}

1/2sl (lado do triângulo vezes o comprimento ou a altura). Como existem quatro triângulos, para encontrar a área da superfície total, é preciso multiplicar esse valor por quatro. Somar o valor de todas essas faces resulta na área da superfície da pirâmide quadrangular: AS = s2 + 2sl.

#### Step 2. Measure the slant height and side of the base

The slanted height, l, equals the height of the triangular sides. It is the distance between the base and the top of the pyramid measured on the flat side. The base side, s, is the length of one side of the square base. Since the base is a square, the measurement is the same on all sides. Use a ruler to take each measurement.

• Example: l = 3 cm.
• Example: s = 1 cm.

#### Step 3. Find the area of ​​the square base

The area of ​​the base square can be calculated by squaring one side, that is, multiplying s by itself.

• Example: s2 = s x s = 1 x 1 = 1 cm2

#### Step 4. Calculate the total area of ​​the four triangular faces

The second part of the equation involves the surface area of ​​the remaining four triangular sides. Using the formula 2ls, multiply s by 1 and by two. Doing this allows you to find the area on each side.

• Example: 2 x s x l = 2 x 1 x 3 = 6 cm2

#### Step 5. Add the measurement of the two areas

Add the total area of ​​the sides to the base area to calculate the total surface area.

• Example: s2 + 2sl = 1 + 6 = 7 cm2

### Method 7 of 7: Cone

#### Step 1. Identify the formula for the surface area of ​​a cone

A cone has a circular base and a rounded surface that ends at a point. To find the surface area, you will need to calculate the area of ​​the circular base and the surface of the cone, and add these two values ​​together. The formula for the surface area of ​​a cone is: AS = π*r2 + π*rl, where r is the radius of the circular base, l is the inclined height of the cone, and π is the mathematical constant pi (3, 14).

• The unit of surface area will be the unit of length squared: cm2m2, km2, etc.

#### Step 2. Measure the radius and height of the cone

The radius is the distance from the center of the circular base to the side of the base. Height is the distance from the center of the base to the highest point of the cone, measured by the center of the cone.

• Example: r = 2 cm.
• Example: h = 4 cm.

#### Step 3. Calculate the sloped height (l) of the cone

Since the sloped height is equivalent to the hypotenuse of the triangle, you must use the Pythagorean Theorem to calculate it. Use a rearranged form, l = √ (r2 + h2), where r is the radius and h is the height of the cone.

• Example: l = √ (r2 + h2) = √ (2 x 2 + 4 x 4) = √ (4 + 16) = √ (20) = 4.47 cm

#### Step 4. Find the area of ​​the circular base

The base area is calculated by the formula π*r2. After measuring the radius, square it (multiply it by itself) and multiply the product by pi.

• Example: π*r2 = 3.14 x 2 x 2 = 12.56 cm2.

#### Step 5. Calculate the surface area of ​​the top of the cone

Using the formula π*rl, where r is the radius of the circle and l is the previously calculated sloped height, you can find the surface area of ​​the top of the cone.

### Example: π*rl = 3.14 x 2 x 4.47 = 28.07 cm

#### Step 6. Add the two areas together to find the total surface area

Calculate the area of ​​the final surface of the cone by adding the area of ​​the circular base with the calculation from the previous step.

• Example: π*r2 + π*rl = 12, 56 + 28, 07 = 40, 63 cm2