How to Apply Sine Rule, Cosine Rule and Tangent Rule in Trigonometry

Sine Rule

In trigonometry, the sine law, law of sines, sine rule, or sine formula is an rational equation that relates to the lengths of the sides of a triangle (any shape or kind) to the sines of its angles.

According to the sine rule,

a / sin(A) = b / sin(B) = c / sin(C)

where a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the opposite angles.

Application of Sine Rule

  • Sine rule can be used to find the unknown side or length of a triangle.
  • Sine rule can also be used to find the unknown angle of a triangle.

To find the unknown side or length of a triangle, three essential parameters must be provided and these parameters are:

  • The angle opposite to the unknown side or length of the triangle.
  • The length of a side of the triangle (b).
  • The angle opposite to the length of side (b) of the triangle.

Let’s take for example we have a triangle and we want to determine the unknown side of this triangle, when the opposite angle is given, another length of the side of the triangle and its opposite angle is also given. Like the image below.

From the image above, one can see that;
The angle opposite to the unknown length of the side of the triangle is 80°. (A)
The length of the side of the triangle given is 7. (b)
The angle opposite to the length of the side of the triangle given is 60° (B)
The unknown length of the side of the triangle is x. (a)

Applying the sine rule to find the value of x:

a / sin(A) = b / sin(B)

x / sin(80°) = 7 / sin(60°)
x / 0.9848 = 7 / 0.8660

Applying cross multiplication

x (0.8660) = 7 (0.9848)
x (0.8660) = 6.8936

Dividing both sides by 0.8660

x = 6.8936 / 0.8660
x = 7.96

Therefore, the unknown length of the side of the triangle, x is 7.96.

Continue reading How to Apply Sine Rule, Cosine Rule and Tangent Rule in Trigonometry

How to Calculate and Solve for the Width, Height, Length and Area of a Rectangular Prism | The Calculator Encyclopedia

The image above is a rectangular prism.

To compute the area of a rectangular prism requires three essential parameters which are the length, width and height of the rectangular prism.

A = 2wl + 2hl + 2hw

Where;
A = Area of the rectangular prism
l = Length of the rectangular prism
w = Width of the rectangular prism
h = Height of the rectangular prism

Let’s solve an example;
Find the area of a rectangular prism with a width of 10 cm, a height of 17 cm and a length of 14 cm.

This implies that;
l = Length of the rectangular prism = 14 cm
w = Width of the rectangular prism = 10 cm
h = Height of the rectangular prism = 17 cm

A = 2wl + 2hl + 2hw
A = 2 x 10 x 14 + 2 x 17 x 14 + 2 x 17 x 10
A = 280 + 476 + 340
A = 1096

Therefore, the area of the rectangular prism is 1096 cm2.

Calculating the Length of a Rectangular Prism using the Area, Width and Height of the Rectangular Prism. 

l = A – 2hw / 2 (w + h)

Where;
l = Length of the rectangular prism
A = Area of the rectangular prism
w = Width of the rectangular prism
h = Height of the rectangular prism

Let’s solve an example;
Find the length of a rectangular prism with a width of 9 cm, a height of 12 cm and a Area of 250 cm2.

This implies that;
w = Width of the rectangular prism = 9 cm
h = Height of the rectangular prism = 12 cm
A = Area of the rectangular prism = 250 cm2

l = A – 2hw / 2 (w + h)
l = 250 – 2 x 12 x 9 / 2 (9 + 12)
l = 250 – 216 / 2 (21)
l = 34 / 42
l = 0.81

Therefore, the length of the rectangular prism is 0.81 cm.

Continue reading How to Calculate and Solve for the Width, Height, Length and Area of a Rectangular Prism | The Calculator Encyclopedia

How to Calculate and Solve for the Width, Height, Length and Volume of a Rectangular Prism | Nickzom Calculator

The image above is a rectangular prism.

To compute the volume of a rectangular prism requires three essential parameters which are the length, width and height of the rectangular prism.

V = lwh

Where;
V = Volume of the rectangular prism
l = Length of the rectangular prism
w = Width of the rectangular prism
h = Height of the rectangular prism

Let’s solve an example;
Find the volume of a rectangular prism with a length of 18 cm, a width of 12 cm and a height of 21 cm.

This implies that;
l = Length of the rectangular prism = 18 cm
w = Width of the rectangular prism = 12 cm
h = Height of the rectangular prism = 21 cm

V = lwh
V = 18 x 12 x 21
V = 4536

Therefore, the volume of the rectangular prism is 4536 cm3.

Calculating the Length of a Rectangular Prism using the Volume, Width and Height of the Rectangular Prism. 

l = V / wh

Where;
l = Length of the rectangular prism
V = Volume of the rectangular prism
w = Width of the rectangular prism
h = Height of the rectangular prism

Let’s solve an example;
Find the length of a rectangular prism with a width of 19 cm, a height of 24 cm and a Volume of 250 cm3.

This implies that;
w = Width of the rectangular prism = 19 cm
h = Height of the rectangular prism = 24 cm
V = Volume of the rectangular prism = 250 cm3

l = V / wh
l = 250 / 19 x 24
l = 250 / 456
l = 0.55

Therefore, the length of the rectangular prism is 0.55 cm.

Continue reading How to Calculate and Solve for the Width, Height, Length and Volume of a Rectangular Prism | Nickzom Calculator

How to Calculate and Solve for the Angle, Radius and Length of an Arc of a Circle | The Calculator Encyclopedia

The image above represents the length of an arc of a circle.

To compute the length of an arc of a circle, two essential parameters are needed and this parameters are radius of the circle (r) and angle of the circle (α).

The formula for calculating the length of an arc of a circle:

L = απr / 180

Where;
L = Length of an arc of the circle
α = Angle of the circle
r = radius of the circle

Let’s solve an example:
Find the length of an arc of a circle when the angle of the circle is 90° and the radius of the circle is 20 cm.

This implies that;
α = Angle of the circle = 90°
r = Radius of the circle = 20 cm

L = απr / 180
L = 90 x 3.142 x 20 / 180
L = 5655.6 / 180
L = 31.42

Therefore, the length of an arc of the circle is 31.42 cm.

Calculating the Angle of a Circle using the Radius of the Circle and Length of an Arc of the Circle.

α = 180L / πr

Where;
L = Length of an arc of the circle
r = Radius of the circle

Let’s solve an example;
Find the angle of the circle when the length of an arc of the circle is 60° and a radius of 140 cm.

This implies that;
L = Length of an arc of the circle = 60°
r = Radius of the circle = 140 cm

α = 180L / πr
α = 180 x 60 / 3.142 x 140
α = 10800 / 439.88
a = 24.55

Therefore, the angle of the circle is 24.55°.

Continue reading How to Calculate and Solve for the Angle, Radius and Length of an Arc of a Circle | The Calculator Encyclopedia

How to Calculate and Solve for the Length and Area of a Pentagon | Nickzom Calculator

The image above is a pentagon.

To compute the area of a pentagon, one essential parameter is needed and this parameter is length of side (a).

The formula for calculating the area of a pentagon:

A = a2(5(5 + 2√5) / 4

Where;
A = Area of the pentagon
a = length of side

Let’s solve an example;
Find the area of the pentagon when the length of side is 30 cm.

This implies that;
a = length of side = 30 cm

A = a2(5(5 + 2√5) / 4
A = 302(5(5 + 4.47) / 4
A = 900√(5(9.47) / 4
A = 900√47.36 / 4
A = 900√11.84
A = 900 x 3.44
A = 3096

Therefore, the area of the pentagon is 3096 cm2.

Calculating the length of side using the area of the pentagon.

a = √(4A / √5(5 + 2√5))

Where;
a = length of side
A = Area of the pentagon

Let’s solve an example;
Find the length of side with an area of 150 cm2.

This implies that;
A = Area of the pentagon = 150 cm2

a = √(4A / √5(5 + 2√5))
a = √(4 x 150 / √5(5 + 1.148))
a = √(600 / √5(6.148))
a = √(600 / √30.74)
a = √(600 / 5.54)
a = √108.30
a = 10.41

Therefore, the length of side is 10.41 cm.

Continue reading How to Calculate and Solve for the Length and Area of a Pentagon | Nickzom Calculator

How to Calculate the Volume, Length, Width and Height of a Cuboid | Nickzom Calculator

The image above represents a cuboid.
To compute the volume of a cuboid requires three essential parameters which are the length, width and height of the cuboid.

The formula for computing the volume of a cuboid is:

V = lwh

Where:
V = Volume of the Cuboid
l = Length of the Cuboid
w = Width of the Cuboid
h = Height of the Cuboid

Let’s solve an example
Find the volume of a cuboid with a length of 6 cm, width of 2 cm and a height of 10 cm.

This implies that:
l = length of the cuboid = 6
w = width of the cuboid = 2
h= = height of the cuboid = 10

V = lwh
V = 6 x 2 x 10
V = 120

Therefore, the volume of the cuboid is 120 cm3.

Calculating the Length of a cuboid when Volume, Width and Height are Given

The formula is l = V / (w)(h)

Where;
V = Volume of the Cuboid
l = Length of the Cuboid
w = Width of the Cuboid
h = Height of the Cuboid

Let’s solve an example:
Find the length of a cuboid with a volume of 440 cm3 , a width of 5 cm and a height of 11 cm.

This implies that;
V = Volume of the cuboid = 440 cm3
w = width of the cuboid = 5 cm
h  = height of the cuboid = 11 cm

l =  V / (w)(h)
l = 440 / (5)(11)
l = 440 / 55
l = 8 cm

Therefore, the length of the cuboid is 8 cm.

Calculating the Width of a cuboid when Volume, Length and Height are Given

The formula is w = V / (l)(h)

Where;
V = Volume of the Cuboid
l = Length of the Cuboid
w = Width of the Cuboid
h = Height of the Cuboid

Let’s solve an example:
Find the width of a cuboid with a volume of  180 cm3 , a length of 6 cm and a height of 10 cm

This implies that;
V = Volume of the cuboid = 180 cm3
l = length of the cuboid = 6 cm
h  = height of the cuboid = 10 cm

w =  V / (l)(h)
w = 180 / (6)(10)
w = 180 / 60
w = 3 cm

Therefore, the width of the cuboid is 3 cm.

Calculating the Height of a cuboid when Volume, Length and Width are Given

The formula is h = V / (l)(w)

Where;
V = Volume of the Cuboid
l = Length of the Cuboid
w = Width of the Cuboid
h = Height of the Cuboid

Let’s solve an example:
Find the height of a cuboid with a volume of 195 cm3 , a length of 5 cm and a width of 3 cm

This implies that;
V = Volume of the cuboid = 195 cm3
l = length of the cuboid = 5 cm
w  = width of the cuboid = 3 cm

h =  V / (l)(w)
h = 195 / (5)(3)
h = 195 / 15
h = 13 cm

Therefore, the height of the cuboid is 13 cm.

Continue reading How to Calculate the Volume, Length, Width and Height of a Cuboid | Nickzom Calculator

How to Calculate and Solve for the Length and Perimeter of a Rhombus | Nickzom Calculator

The image above is a rhombus.

To compute the area of a rhombus, one essential parameter is needed and this parameter is length of side (a).

The formula for calculating the perimeter of a rhombus:

P = 4a

Where;
P = Perimeter of the rhombus
a = length of side

Let’s solve an example;
Find the perimeter of a rhombus when the length of side is 30 cm.

This implies that;
a = Length of side = 30 cm

P = 4a
P = 4 x 30
P = 120

Therefore, the perimeter of the rhombus is 120 cm.

Calculating the Length of Side using the Perimeter of the Rhombus.

a = P / 4

Where;
a = Length of Side
P = Perimeter of the rhombus

Let’s solve an example;
Given that the perimeter of the rhombus is 180 cm. Find the length of side?

a = P / 4

This implies that;
P = perimeter of the rhombus = 180 cm

a = P / 4
a = 180 / 4
a = 45

Therefore, the length of side is 45 cm.

Continue reading How to Calculate and Solve for the Length and Perimeter of a Rhombus | Nickzom Calculator

How to Calculate and Solve for the Area, Length, Width and Height of a Cuboid | Nickzom Calculator

The image above represents a cuboid.
To compute the area of a cuboid requires three essential parameters which are the length, width and height of the cuboid.

The formula for computing the area of a cuboid is:

A = 2(lw + lh + wh)

Where:
A = Area of the Cuboid
l = Length of the Cuboid
w = Width of the Cuboid
h = Height of the Cuboid

Let’s solve an example
Find the area of a cuboid with a length of 5 cm, width of 3 cm and a height of 9 cm.

This implies that:
l = length of the cuboid = 5
w = width of the cuboid = 3
h= = height of the cuboid = 9

A = 2(lw + lh + wh)
A = 2(5 x 3 + 5 x 9 + 3 x 9)
A = 2(87)
A = 174

Therefore, the area of the cuboid is 174 cm2.

Calculating the Length of a cuboid when Area, Width and Height are Given

The formula is l = A – 2(w)(h) / 2(w) + 2(h)

Where;
A = Area of the Cuboid
l = Length of the Cuboid
w = Width of the Cuboid
h = Height of the Cuboid

Let’s solve an example:
Find the length of a cuboid with an area of 140 cm2 , a width of 4 cm and a height of 12 cm

This implies that;
A = Area of the cuboid = 140 cm2
w = width of the cuboid = 4 cm
h  = height of the cuboid = 12 cm

l =  A – 2(w)(h) / 2(w) + 2(h)
l = 140 – 2(4)(12) / 2(4) + 2(12)
l = 140 – 96 / 8 + 24
l = 44 / 32
l = 1.375 cm

Therefore, the length of the cuboid is 1.375 cm.

Calculating the Width of a cuboid when Area, Length and Height are Given

The formula is w = A – 2(l)(h) / 2(l) + 2(h)

Where;
A = Area of the Cuboid
l = Length of the Cuboid
w = Width of the Cuboid
h = Height of the Cuboid

Let’s solve an example:
Find the width of a cuboid with an area of 200 cm2 , a length of 5 cm and a height of 12 cm

This implies that;
A = Area of the cuboid = 200 cm2
l = length of the cuboid = 5 cm
h  = height of the cuboid = 12 cm

w =  A – 2(l)(h) / 2(l) + 2(h)
w = 200 – 2(5)(12) / 2(5) + 2(12)
w = 200 – 120 / 10 + 24
w = 80 / 34
w = 2.353 cm

Therefore, the width of the cuboid is 2.353 cm.

Calculating the Height of a cuboid when Area, Length and Width are Given

The formula is h = A – 2(l)(w) / 2(l) + 2(w)

Where;
A = Area of the Cuboid
l = Length of the Cuboid
w = Width of the Cuboid
h = Height of the Cuboid

Let’s solve an example:
Find the height of a cuboid with an area of 300 cm2 , a length of 6 cm and a width of 2 cm

This implies that;
A = Area of the cuboid = 300 cm2
l = length of the cuboid = 6 cm
w  = width of the cuboid = 2 cm

h =  A – 2(l)(w) / 2(l) + 2(w)
h = 300 – 2(6)(2) / 2(6) + 2(2)
h = 300 – 24 / 12 + 4
h = 276 / 16
h = 17.25 cm

Therefore, the height of the cuboid is 17.25 cm.

Continue reading How to Calculate and Solve for the Area, Length, Width and Height of a Cuboid | Nickzom Calculator

How to Calculate and Solve for the Length of the Diagonal and Area of a Rhombus | The Calculator Encyclopedia

The above image is a rhombus.

To compute the area of a rhombus, two essential parameters are needed and this parameters are length of the diagonal (p) and length of the diagonal (q).

The formula for calculating the area of a rhombus:

A = pq ⁄ 2

Where;
A = Area of the rhombus
p = length of the diagonal
q = length of the diagonal

Let’s solve an example;
Find the area of a rhombus when the length of the diagonal (p) is 10 cm and the length of the diagonal (q) is 18 cm.

This implies that;
p = length of the diagonal = 10 cm
q = length of the diagonal = 18 cm

A = pq2
A = 10 x 18 / 2
A = 180 / 2
A = 90

Therefore, the area of the rhombus is 90 cm2.

Calculating the length of the diagonal (p) using Area of the Rhombus and length of the diagonal (q).

p = 2A / q

Where;
A = Area of the rhombus
q = length of the diagonal

Let’s solve an example;
Given that the length of the diagonal (q) is 20 cm with an area of 60 cm2. Find the length of the diagonal (p)?

This implies that;
A = Area of the rhombus = 60 cm2
q = length of the diagonal = 20 cm


p = 2A / q
p = 2 x 60 / 20
p = 120 / 20
p = 6


Therefore, the length of the diagonal is 6 cm.

Continue reading How to Calculate and Solve for the Length of the Diagonal and Area of a Rhombus | The Calculator Encyclopedia

How to Calculate and Solve for the Area and Length of a Parallelogram | The Calculator Encyclopedia

The image above represents a parallelogram.
To compute the area of a parallelogram requires two essential parameters which are the base and height of the parallelogram.

The formula for computing the area of a parallelogram is:

A = b x h

Where:
A = Area of a parallelogram
b = base of the parallelogram
h = height of the parallelogram

Let’s solve an example
Find the area of a parallelogram with a base of 12 cm and a height of 4 cm.

This implies that:
b = base of the parallelogram = 12
h = height of the parallelogram = 4

A = b x h
A = 12 x 4
A = 48

Therefore, the area of the parallelogram is 48 cm2.

Calculating the base of a parallelogram when Area and Height is Given

The formula is b = A / h

Where;
A = Area of a parallelogram
b = base of the parallelogram
h = height of the parallelogram

Let’s solve an example:
Find the base of a parallelogram with an area of 40 cm² and a height of 8 cm

This implies that;
A = Area of the parallelogram = 40 cm²
h = height of the parallelogram = 8 cm

b = A / h
b = 40 / 8
b = 5

Therefore, the base of the parallelogram is 5 cm.

Calculating the height of a parallelogram when Area and Base is Given

The formula is h = A / b

Where;
A = Area of a parallelogram
b = base of the parallelogram
h = height of the parallelogram

Let’s solve an example:
Find the height of a parallelogram with an area of 60 cm² and a base of 6 cm

This implies that;
A = Area of the parallelogram = 60 cm²
b = base of the parallelogram = 6 cm

h = A / b
h = 60 / 6
h = 10

Therefore, the height of the parallelogram is 10 cm.

Continue reading How to Calculate and Solve for the Area and Length of a Parallelogram | The Calculator Encyclopedia