Areas of Right Triangle Knowing the Hypotenuse and the Angle
General Right Triangle
Let:
D = the length of a hypotenuse
A = the length of side A opposite of angle α°
B = the length of side B opposite of angle β°
Let's assume that we only know of hypotenuse D and angle α°. Find the area:
Area = 1/2 × A × B
Determined by trigonometric ratios:
A = H × sin α° and B = H × cos α°
Then:
Area = 1/2 × A × B
Area = 1/2 × H × sin α° × H × cos α°
Area = 1/2 × H^2 × sin α° × cos α°
Let's assume that we only know the angle β° instead:
Area = 1/2 × H^2 × sin α° × cos α°
Since α° + β° = 90°,
Area = 1/2 × H^2 × sin (90° - β°) × cos (90° - β°)
With the trigonometric identities:
sin(90° - θ°) = cos θ°, and cos(90° - θ°) = sin θ°
Area = 1/2 × H^2 × cos β° × sin β°
In a remarkable conclusion:
Area = 1/2 × H^2 × sin α° × cos α° = 1/2 × H^2 × cos β° × sin β°
Let's look at specific right triangles.
Area of 30°-60°-90° Triangles
Assume that α = 60° and β = 30°. Then:
Area = 1/2 × H^2 × sin 60° × cos 60°
Area = 1/2 × H^2 × √3/2 × 1/2
Area = (H^2 × √3) / 8
Similarly,
Area = 1/2 × H^2 × sin 30° × cos 30°
Area = 1/2 × H^2 × 1/2 × √3/2
Area = (H^2 × √3) / 8
Area of 45°-45°-90° Triangles
On a 45-45-90 triangle, the measures A and B are equal. Then:
Area = 1/2 × H^2 × sin 45° × cos 45°
Area = 1/2 × H^2 × √2 / 2 × √2 / 2
Area = H^2 / 4
Area of 75°-15°-90° Triangles
Assume that α = 75° and β = 15°. Then:
Area = 1/2 × H^2 × sin 75° × cos 75°
Area = 1/2 × H^2 × (√6 + √2)/4 × (√6 - √2)/4
Area = 1/32 × H^2 × (√6 + √2) × (√6 - √2)
Area = 1/32 × H^2 × (6 - √6 × √2 + √6 × √2 - 2)
Area = 1/32 × H^2 × 4
Area = H^2 / 8
Similarly,
Area = 1/2 × H^2 × sin 15° × cos 15°
Area = 1/2 × H^2 × (√6 - √2)/4 × (√6 + √2)/4
Area = 1/32 × H^2 × (6 + √6 × √2 - √6 × √2 - 2)
Area = 1/32 × H^2 × 4
Area = H^2 / 8
Summary
Area of a Right Triangles knowing only the Hypotenuse and One (does not matter which one as it turns out) Angle:
Area = 1/2 × H^2 × sin θ × cos θ
Area of 30°-60°-90° Triangles: (H^2 × √3) / 8
Area of 45°-45°-90° Triangles: H^2 / 4
Area of 75°-15°-90° Triangles: H^2 / 8
Eddie
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