Area of a Triangle with Sine
Aligned to G-SRT.D.9 — Common Core State Standards for Mathematics.
What this lesson teaches
For a triangle with two sides a and b and the included angle C, the area equals (1/2)*a*b*sin(C). This comes from the standard area (1/2)*base*height: dropping an altitude from one vertex makes the height equal to a*sin(C).
Worked examples
Example 1
Derive and use A = (1/2)ab sin C. Take side b as the base. The altitude h from the opposite vertex satisfies sin C = h/a, so h = a*sin C. Then area = (1/2)*base*height = (1/2)*b*(a sin C) = (1/2)ab sin C. For a = 8, b = 10, C = 30°: A = (1/2)(8)(10)(0.5) = 20 square units.
Example 2
A triangular plot has two sides of 40 m and 55 m meeting at an angle of 110°. Find its area. Use A = (1/2)ab sin C with a = 40, b = 55, C = 110°. Since sin 110° is about 0.940, A = (1/2)(40)(55)(0.940) = 1,100 × 0.940 which is about 1,034 square meters — no altitude ever needed measuring.
Example 3
Common mistake: using an angle that is not between the two sides. The formula requires the INCLUDED angle. Given sides 8 and 10 and an angle of 35° that touches only the side of length 8, plugging in (1/2)(8)(10) sin 35° gives a number, but not this triangle's area. Find the included angle first — with the triangle's angle sum or the Law of Sines — then apply the formula.
Practice questions
- Find the area of a triangle with sides 6 and 9 and included angle 90°. Confirm it matches (1/2)*base*height.
- A triangle has a = 12, b = 5, and included angle 30°. Find its area.
- Explain where the factor sin C comes from in the area formula (1/2)ab sin C.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Área de un triángulo con seno
Para un triángulo con dos lados a y b y el ángulo incluido C, el área es (1/2)*a*b*sen(C). Esto proviene del área estándar (1/2)*base*altura: al trazar una altura desde un vértice, la altura resulta a*sen(C).
Ejemplo 1: Deduce y usa A = (1/2)ab sen C. Toma el lado b como base. La altura h desde el vértice opuesto cumple sen C = h/a, así que h = a*sen C. Entonces área = (1/2)*base*altura = (1/2)*b*(a sen C) = (1/2)ab sen C. Para a = 8, b = 10, C = 30°: A = (1/2)(8)(10)(0.5) = 20 unidades cuadradas.
Ejemplo 2: Un terreno triangular tiene dos lados de 40 m y 55 m que se encuentran en un ángulo de 110°. Halla su área. Usa A = (1/2)ab sen C con a = 40, b = 55, C = 110°. Como sen 110° es aproximadamente 0.940, A = (1/2)(40)(55)(0.940) = 1,100 × 0.940, unos 1,034 metros cuadrados — sin necesidad de medir ninguna altura.
Ejemplo 3: Error común: usar un ángulo que no está entre los dos lados. La fórmula exige el ángulo COMPRENDIDO. Con lados de 8 y 10 y un ángulo de 35° que toca solo al lado de 8, sustituir en (1/2)(8)(10) sen 35° da un número, pero no el área de ese triángulo. Halla primero el ángulo comprendido — con la suma de ángulos del triángulo o con la ley de senos — y luego aplica la fórmula.
More Similarity, Right Triangles & Trigonometry lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required