Geometric Formulas Formula Bank

Introduction to Geometric Formulas

Geometric formulas are mathematical equations used to calculate the properties of various shapes and figures. These properties include area, perimeter, surface area, and volume. Mastering these formulas is essential for solving real-world problems in fields like engineering, architecture, design, and more.

Rectangle of Length \( b \) and Width \( a \)

1. Area:
\[
A = ab
\]
2. Perimeter:
\[
P = 2a + 2b
\]

Worked Example:
If \( a = 5 \) and \( b = 8 \), then:
\[
A = 5 \times 8 = 40
\quad
P = 2(5) + 2(8) = 10 + 16 = 26
\]

Parallelogram of Altitude \( h \) and Base \( b \)

1. Area:
\[
A = bh = ab \sin(\theta)
\]
2. Perimeter:
\[
P = 2a + 2b
\]

Worked Example:
If \( b = 6 \), \( h = 4 \), and \( \theta = 45^\circ \), then:
\[
A = 6 \times 4 = 24 \quad \text{or} \quad A = 6 \times 4 \times \sin(45^\circ) \approx 24
\]

Triangle of Altitude \( h \) and Base \( b \)

1. Area:
\[
A = \frac{1}{2}bh = \frac{1}{2}ab \sin(\theta)
\]
2. Perimeter:
\[
P = a + b + c
\]

Worked Example:
If \( b = 7 \), \( h = 5 \), then:
\[
A = \frac{1}{2} \times 7 \times 5 = 17.5
\quad \text{and} \quad P = 7 + 5 + 8 = 20
\]

Trapezoid of Altitude \( h \) and Parallel Sides \( a \) and \( b \)

1. Area:
\[
A = \frac{1}{2}h(a + b)
\]
2. Perimeter:
\[
P = a + b + h \left( \csc(\theta) + \csc(\phi) \right)
\]

Worked Example:
For \( a = 6 \), \( b = 4 \), \( h = 3 \):
\[
A = \frac{1}{2} \times 3 \times (6 + 4) = 15
\quad \text{and} \quad P = 6 + 4 + 3(\csc(\theta) + \csc(\phi))
\]

Circle of Radius \( r \)

1. Area:
\[
A = \pi r^2
\]
2. Perimeter (Circumference):
\[
C = 2\pi r
\]

Worked Example:
If \( r = 3 \):
\[
A = \pi (3)^2 = 28.27 \quad \text{and} \quad C = 2\pi(3) = 18.85
\]

Regular Polygon of \( n \) Sides Each of Length \( b \)

1. Area:
\[
A = \frac{1}{4} n b^2 \cot\left(\frac{\pi}{n}\right)
\]
2. Perimeter:
\[
P = n b
\]

Worked Example:
If \( n = 6 \) (a hexagon) and \( b = 4 \), then:
\[
A = \frac{1}{4} \times 6 \times 4^2 \times \cot\left(\frac{\pi}{6}\right) = 41.57
\quad \text{and} \quad P = 6 \times 4 = 24
\]

Sector of a Circle of Radius \( r \)

1. Area:
\[
A = \frac{1}{2} r^2 \theta \quad (\theta \text{ in radians})
\]
2. Arc Length:
\[
s = r \theta
\]

Worked Example:
For \( r = 6 \) and \( \theta = \frac{\pi}{4} \):
\[
A = \frac{1}{2} \times 6^2 \times \frac{\pi}{4} = 14.14 \quad \text{and} \quad s = 6 \times \frac{\pi}{4} = 4.71
\]

Radius of a Circle Inscribed in a Triangle of Sides \( a \), \( b \), \( c \)

1. Radius:
\[
r = \frac{\sqrt{(s – a)(s – b)(s – c)}}{s}
\]
where \( s = \frac{a + b + c}{2} \) is the semiperimeter.

Worked Example:
If \( a = 7 \), \( b = 8 \), and \( c = 5 \), then:
\[
s = \frac{7 + 8 + 5}{2} = 10 \quad \text{so} \quad r = \frac{\sqrt{(10 – 7)(10 – 8)(10 – 5)}}{10} = 1.26
\]

Radius of a Circle Circumscribing a Triangle of Sides \( a \), \( b \), \( c \)

1. Radius:
\[
R = \frac{abc}{4 \sqrt{s(s – a)(s – b)(s – c)}}
\]
where \( s \) is the semiperimeter:
\[
s = \frac{a + b + c}{2}
\]

Worked Example:
For \( a = 6 \), \( b = 8 \), and \( c = 10 \):
\[
s = \frac{6 + 8 + 10}{2} = 12
\]
\[
R = \frac{6 \times 8 \times 10}{4 \sqrt{12(12 – 6)(12 – 8)(12 – 10)}} = 5
\]

Radius of a Circle Inscribed in a Triangle of Sides \( a \), \( b \), \( c \)

1. Radius:
\[
r = \frac{\sqrt{(s – a)(s – b)(s – c)}}{s}
\]
where \( s = \frac{a + b + c}{2} \) is the semiperimeter.

Worked Example:
If \( a = 6 \), \( b = 7 \), and \( c = 8 \):
\[
s = \frac{6 + 7 + 8}{2} = 10.5 \quad \text{so} \quad r = \frac{\sqrt{(10.5 – 6)(10.5 – 7)(10.5 – 8)}}{10.5} = 1.87
\]

Radius of a Circle Circumscribing a Triangle of Sides \( a \), \( b \), \( c \)

1. Radius:
\[
R = \frac{abc}{4 \sqrt{s(s – a)(s – b)(s – c)}}
\]
where \( s \) is the semiperimeter:
\[
s = \frac{a + b + c}{2}
\]

Worked Example:
For \( a = 5 \), \( b = 12 \), and \( c = 13 \):
\[
s = \frac{5 + 12 + 13}{2} = 15
\]
\[
R = \frac{5 \times 12 \times 13}{4 \sqrt{15(15 – 5)(15 – 12)(15 – 13)}} = 6.5
\]

Regular Polygon of \( n \) Sides Inscribed in a Circle of Radius \( r \)

1. Area:
\[
A = \frac{1}{2} n r^2 \sin\left(\frac{2\pi}{n}\right)
\]
2. Perimeter:
\[
P = 2nr\sin\left(\frac{\pi}{n}\right)
\]

Worked Example:
If \( n = 6 \) (hexagon) and \( r = 4 \):
\[
A = \frac{1}{2} \times 6 \times 4^2 \times \sin\left(\frac{2\pi}{6}\right) = 41.57
\quad \text{and} \quad P = 2 \times 6 \times 4 \times \sin\left(\frac{\pi}{6}\right) = 24
\]

Segment of a Circle of Radius \( r \)

1. Area of Shaded Part:
\[
A = \frac{1}{2} r^2 (\theta – \sin(\theta))
\]

Worked Example:
If \( r = 10 \) and \( \theta = \frac{\pi}{3} \):
\[
A = \frac{1}{2} \times 10^2 \times \left(\frac{\pi}{3} – \sin\left(\frac{\pi}{3}\right)\right) = 35.44
\]

Regular Polygon of \(n\) Sides Inscribed in Circle of Radius \(r\)

1. Area:
\[
A = \frac{1}{2} n r^2 \sin\left(\frac{2\pi}{n}\right)
\]
2. Perimeter:
\[
P = 2nr \sin\left(\frac{\pi}{n}\right)
\]

Worked Example:
For a hexagon (\(n = 6\)) inscribed in a circle of radius \(r = 4\):
\[
A = \frac{1}{2} \times 6 \times 4^2 \times \sin\left(\frac{2\pi}{6}\right) \approx 41.57
\]
\[
P = 2 \times 6 \times 4 \times \sin\left(\frac{\pi}{6}\right) = 24
\]

Regular Polygon of \(n\) Sides Circumscribing a Circle of Radius \(r\)

1. Area:
\[
A = \frac{1}{2}nr^2 \cot\left(\frac{\pi}{n}\right)
\]
2. Perimeter:
\[
P = 2nr\tan\left(\frac{\pi}{n}\right)
\]

Worked Example:
For a regular pentagon (\(n = 5\)) circumscribing a circle of radius \(r = 3\):
\[
A = \frac{1}{2} \times 5 \times 3^2 \times \cot\left(\frac{\pi}{5}\right) \approx 43.01
\]
\[
P = 2 \times 5 \times 3 \times \tan\left(\frac{\pi}{5}\right) = 21.84
\]

Segment of Circle of Radius \(r\)

1. Area of the Shaded Part:
\[
A = \frac{1}{2}r^2(\theta – \sin(\theta))
\]

Worked Example:
For a circle with radius \(r = 5\) and central angle \(\theta = \frac{\pi}{4}\):
\[
A = \frac{1}{2} \times 5^2 \times \left(\frac{\pi}{4} – \sin\left(\frac{\pi}{4}\right)\right) \approx 8.18
\]

Ellipse of Semi-major Axis \(a\) and Semi-minor Axis \(b\)

1. Area:
\[
A = \pi ab
\]
2. Perimeter (Approximation):
\[
P \approx \pi \left( 3(a + b) – \sqrt{(3a + b)(a + 3b)} \right)
\]

Worked Example:
For an ellipse with semi-major axis \(a = 6\) and semi-minor axis \(b = 4\):
\[
A = \pi \times 6 \times 4 = 75.40
\]
\[
P \approx \pi \left(3(6 + 4) – \sqrt{(3 \times 6 + 4)(6 + 3 \times 4)}\right) \approx 31.42
\]

Right Circular Cone of Radius \(r\) and Height \(h\)

1. Volume:
\[
V = \frac{1}{3} \pi r^2 h
\]
2. Lateral Surface Area:
\[
A = \pi r \sqrt{r^2 + h^2}
\]

Worked Example:
For a cone with radius \(r = 3\) and height \(h = 5\):
\[
V = \frac{1}{3} \pi \times 3^2 \times 5 = 47.12
\]
\[
A = \pi \times 3 \times \sqrt{3^2 + 5^2} \approx 51.42
\]

Sphere of Radius \(r\)

1. Volume:
\[
V = \frac{4}{3} \pi r^3
\]
2. Surface Area:
\[
A = 4\pi r^2
\]

Worked Example:
For a sphere with radius \(r = 6\):
\[
V = \frac{4}{3} \pi \times 6^3 = 904.32
\]
\[
A = 4\pi \times 6^2 = 452.39
\]

Frustum of a Right Circular Cone with Radii \(r_1\) and \(r_2\) and Height \(h\)

1. Volume:
\[
V = \frac{1}{3} \pi h (r_1^2 + r_1r_2 + r_2^2)
\]
2. Lateral Surface Area:
\[
A = \pi (r_1 + r_2) \sqrt{(r_1 – r_2)^2 + h^2}
\]

Worked Example:
For a frustum with \(r_1 = 4\), \(r_2 = 2\), and \(h = 5\):
\[
V = \frac{1}{3} \pi \times 5 \times (4^2 + 4 \times 2 + 2^2) = 150.80
\]
\[
A = \pi \times (4 + 2) \times \sqrt{(4 – 2)^2 + 5^2} \approx 87.96
\]

Torus with Inner Radius \(a\) and Outer Radius \(b\)

1. Volume:
\[
V = \frac{\pi^2}{4} (b – a)(b + a)^2
\]
2. Surface Area:
\[
A = 2\pi^2 (b^2 – a^2)
\]

Worked Example:
For a torus with inner radius \(a = 3\) and outer radius \(b = 6\):
\[
V = \frac{\pi^2}{4} \times (6 – 3) \times (6 + 3)^2 \approx 1,579.14
\]
\[
A = 2\pi^2 \times (6^2 – 3^2) \approx 592.91
\]

Applications of Geometric Formulas

In Education

• Geometric formulas are fundamental in mathematics education, helping students develop spatial reasoning and problem-solving skills.

In Engineering and Architecture

• Engineers and architects use geometric formulas to design structures, calculate material requirements, and ensure stability and efficiency in their projects.

In Everyday Life

• From determining the amount of paint needed for a room to calculating the volume of a swimming pool, geometric formulas have practical everyday applications.

Tips for Mastering Geometric Calculations

– Practice Regularly: Consistent practice helps reinforce your understanding of different formulas.
– Understand the Concepts: Instead of memorizing formulas, comprehend the principles behind them.
– Use Visual Aids: Diagrams and drawings can help visualize problems and solutions.
– Apply Real-World Examples: Relating formulas to real-life situations enhances retention and understanding.
– Utilize Online Resources: Leverage educational websites, tutorials, and interactive tools for additional practice and explanations.

Frequently Asked Questions

Q1: What is the most important geometric formula to memorize?

A1: While all geometric formulas are important, the area and perimeter formulas for basic shapes like triangles, circles, rectangles, and squares are fundamental and widely used.

Q2: How can I easily remember geometric formulas?

A2: Understanding the derivation of formulas, practicing regularly, and applying them to real-world scenarios can significantly aid in memorization.

Q3: Are geometric formulas used in other subjects besides math?

A3: Yes, geometric formulas are integral in fields such as physics, engineering, architecture, computer graphics, and even art and design.

Q4: How do I choose which geometric formula to use in a problem?

A4: Identify the shape involved and the property you need to calculate (area, perimeter, volume). Match these with the corresponding formula for that shape.

Q5: Can geometric formulas be applied to irregular shapes?

A5: Yes, by decomposing irregular shapes into regular ones, calculating each part’s area or volume, and then summing them up.