Study a list of of more than 15 basic triangles formulas. Key formulas related to: Equilateral, Isosceles, Right triangles. Useful to solve a math and geometry problems involving triangles.
Triangle Formulas
Equilateral, Isosceles, Right Triangles
Perimeter of a Triangle: The perimeter (P) of a triangle is the sum of its three sides:
P = a + b + c where a, b, and c are the lengths of the triangle's three sides.
Semiperimeter of a Triangle: The semi perimeter (s) of a triangle is half of its perimeter:
s = (a + b + c) / 2
Area of a Triangle : The area (A) of a triangle can be calculated using different Formulas:
Heron's Formula: A = √(s * (s - a) * (s - b) * (s - c)) Where:
- A is the area of the triangle.
- s is the semi-perimeter (s = (a + b + c) / 2).
- a, b, and c are the lengths of the triangle's sides.
Base and Height Formula: A = (base × height) / 2 Where:
- A is the area of the triangle.
- "base" is the length of the base of the triangle.
- "height" is the height of the triangle perpendicular to the base.
Sine Rule Formula: A = (0.5 × a × b × sin(γ)) Where:
- A is the area of the triangle.
- a and b are the lengths of two sides of the triangle.
- γ is the angle between the two sides a and b.
Area of a Triangle Using Side Lengths: A = 0.25 × √((a + b + c) × (-a + b + c) × (a - b + c) × (a + b - c)) Where:
- A is the area of the triangle.
- a, b, and c are the lengths of the triangle's sides.
Area of a Triangle Using Angles: A = (a² × sin(β) × sin(γ)) / (2 × sin(β + γ)) Where:
- A is the area of the triangle.
- a is the length of one side of the triangle.
- β and γ are the angles opposite to side a.
Pythagorean Theorem: For a right triangle, the Pythagorean theorem relates the lengths of the two shorter sides (legs, a and b) and the length of the longest side (hypotenuse, c):
a² + b² = c²
Trigonometric Ratios: In a right triangle, you can use trigonometric ratios to relate the angles to the sides. The most common ratios are:
- Sine (sin): sin(θ) = opposite side / hypotenuse
- Cosine (cos): cos(θ) = adjacent side / hypotenuse
- Tangent (tan): tan(θ) = opposite side / adjacent side
Law of Sines: For any triangle (not just right triangles), the law of sines relates the lengths of sides to the sines of their opposite angles:
(a / sin(A)) = (b / sin(B)) = (c / sin(C))
where a, b, and c are the side lengths, and A, B, and C are the opposite angles.
Law of Cosines: The law of cosines relates the lengths of sides and angles in any triangle:
c² = a² + b² - 2ab * cos(C)
a² = b² + c² - 2bc * cos(A)
b² = a² + c² - 2ac * cos(B)
Altitude (Height) of a Triangle: The altitude (h) of a triangle can be calculated using the formula:
h = (2 * Area) / base
where Area is the area of the triangle, and the base is the length of the side to which the altitude is drawn.
Area of an Equilateral Triangle: For an equilateral triangle (all sides are of equal length, and all angles are 60 degrees), the area can be calculated using:
A = (s²√3) / 4,
where s is the length of each side.
Area of an Isosceles Triangle: For an isosceles triangle (two sides of equal length), you can find the area using:
A = (b * h) / 2
where b is the base and h is the height (altitude) drawn to the base.
Area of a Right Triangle: For a right triangle, you can calculate the area using the formula:
A = (1/2) * base * height
where the base and height are the two sides that form the right angle.
Angle Bisector Theorem: In a triangle, the angle bisector theorem states that the ratio of the lengths of the two segments created by an angle bisector is proportional to the ratio of the lengths of the two opposite sides. Specifically:
(BD / CD) = (AB / AC),
where BD is the bisector, CD is the other segment created by the bisector, and AB and AC are the two sides of the triangle.
Median of a Triangle: The median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. The medians of a triangle are concurrent, and their point of intersection is called the centroid of the triangle. The length of a median can be calculated using the formula:
Length of Median = (1/2) * √(2b² + 2c² - a²),
where a, b, and c are the side lengths.
Inradius and Circumradius: The inradius (r) is the radius of the incircle (the largest circle that can fit inside the triangle), and the circumradius (R) is the radius of the circumcircle (the circle that passes through all three vertices of the triangle). These radii are related to the triangle's sides and area:
- Inradius (r) = Area / Semiperimeter (s)
- Circumradius (R) = (abc) / (4 * Area), where a, b, and c are the side lengths.
Law of Tangents: The law of tangents is another trigonometric formula used to relate the sides and angles of a triangle:
(a - b) / (a + b) = tan[(A - B) / 2] / tan[(A + B) / 2]
where a and b are side lengths, and A and B are the opposite angles.
Triangle Calculation Examples
Calculation examples for each of the 15 triangle formulas using Imperial units of measurement:
Perimeter of a Triangle: Example: In a triangle with side lengths of 5 inches, 6 inches, and 7 inches, calculate the perimeter. P = 5 in + 6 in + 7 in = 18 inches
Semiperimeter of a Triangle: Using the same triangle as in the previous example: s = (5 in + 6 in + 7 in) / 2 = 9 inches
Area of a Triangle (Heron's Formula): Continuing with the same triangle: s = 9 inches (as calculated above) A = √(9 in * (9 in - 5 in) * (9 in - 6 in) * (9 in - 7 in)) A ≈ √(9 in * 4 in * 3 in * 2 in) ≈ 6 square inches
Pythagorean Theorem: Example: In a right triangle with legs of 3 feet and 4 feet, calculate the length of the hypotenuse. c² = 3 ft² + 4 ft² = 9 ft² + 16 ft² = 25 ft² c = √25 ft² = 5 feet
Trigonometric Ratios: Example: In a right triangle with angle θ = 30 degrees, the opposite side is 2 yards, and the hypotenuse is 4 yards, calculate the adjacent side. Using the sine ratio: sin(30°) = 2 yd / 4 yd Solve for the adjacent side: adjacent side = 4 yd * sin(30°) = 2 yd
Law of Sines: Example: In a triangle with angles A = 40 degrees, B = 70 degrees, and a side length of 5 feet (opposite angle A), calculate side b. (5 ft / sin(40°)) = (b / sin(70°)) Solve for b: b ≈ (5 ft * sin(70°)) / sin(40°) ≈ 7.2 feet
Law of Cosines: Example: In a triangle with sides a = 5 yards, b = 6 yards, and angle C = 45 degrees, calculate side c. c² = 5 yd² + 6 yd² - 2 * 5 yd * 6 yd * cos(45°) c² = 25 yd² + 36 yd² - 60 yd² * (1 / √2) c ≈ √(61 yd²) ≈ 7.8 yards
Altitude (Height) of a Triangle: Example: Given a triangle with a base of 8 inches and an area of 24 square inches, calculate the height (altitude). A = (base * height) / 2 24 in² = (8 in * height) / 2 height = (24 in² * 2) / 8 in = 6 inches
Area of an Equilateral Triangle: Example: For an equilateral triangle with each side measuring 6 feet, calculate the area. A = (6 ft * 6 ft * √3) / 4 ≈ 15.6 square feet
Area of an Isosceles Triangle: Example: Given an isosceles triangle with a base of 10 inches and a height of 8 inches, calculate the area. A = (10 in * 8 in) / 2 = 40 square inches
Area of a Right Triangle: Example: In a right triangle with a base of 5 feet and a height of 12 feet, calculate the area. A = (1/2) * 5 ft * 12 ft = 30 square feet
Angle Bisector Theorem: Example: In a triangle with sides AB = 6 inches, AC = 9 inches, and the angle bisector BD, calculate the length of CD. (BD / CD) = (AB / AC) (6 in / CD) = (6 in / 9 in) CD = (6 in * 9 in) / 6 in = 9 inches
Median of a Triangle: Example: In a triangle with sides a = 5 feet, b = 6 feet, and c = 8 feet, calculate the length of the median from vertex A. Length of Median = (1/2) * √(2b² + 2c² - a²) Length of Median ≈ (1/2) * √(2(6 ft)² + 2(8 ft)² - (5 ft)²) ≈ 7.35 feet
Inradius and Circumradius: Example: In a triangle with sides a = 7 inches, b = 24 inches, and c = 25 inches, calculate the inradius (r) and circumradius (R). Calculate the semiperimeter: s = (7 in + 24 in + 25 in) / 2 = 28 inches
Calculate the area using Heron's formula: A ≈ √(28 in * (28 in - 7 in) * (28 in - 24 in) * (28 in - 25 in)) ≈ 84 square inches
Now, calculate the inradius and circumradius:
- Inradius (r) = 84 in² / 28 in = 3 inches
- Circumradius (R) = (7 in * 24 in * 25 in) / (4 * 84 in²) ≈ 7.5 inches
Law of Tangents: Example: In a triangle with sides a = 5 inches, b = 6 inches, and angle C = 60 degrees, calculate side c using the law of tangents. (-1/11) = tan[(60° - B) / 2] / tan[(60° + B) / 2]
You would need to solve for angle B and then use it to calculate side c using the law of sines or cosines. This involves trigonometric calculations beyond a simple formula substitution.