**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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