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 example
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.
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: 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.
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