Saturday, June 17, 2023

Areas of Right Triangle Knowing the Hypotenuse and the Angle

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

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

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