Isosceles Triangles Calculator

How the Calculator Works:

The Isosceles Triangle Calculator allows you to compute various properties of an isosceles triangle, given the length of its two equal sides (side a) and its base (side b). It calculates the following:

• Side a (equal sides): The two equal sides of the triangle.
• Base (b): The base or the unequal side of the triangle.
• Side c: The third side, which is equal to side a in an isosceles triangle.
• Perimeter (P): The total distance around the triangle.
• Semi-perimeter (s): Half of the perimeter, used in calculating the area.
• Area (K): The area inside the triangle, calculated using Heron’s formula.
• Heights (ha, hb, hc): The perpendicular heights from the vertices of the triangle to the opposite sides.

Formulas:

1. Perimeter (P):

   $$P = 2a + b$$

   The perimeter is the sum of all sides of the triangle.

2. Semi-perimeter (s):

   $$s = \frac{P}{2}$$

   The semi-perimeter is half the perimeter, which is used in calculating the area of the triangle.

3. Area (K) – Heron’s Formula:

   $$K = \sqrt{s(s – a)(s – a)(s – b)}$$

   This formula calculates the area of the isosceles triangle based on its semi-perimeter and side lengths.

4. Height (ha, hb, hc):

   • The height from the base to the apex: $$h_a = \frac{2K}{a}$$
   • The height from side b to the opposite vertex: $$h_b = \frac{2K}{b}$$
   • Since sides a and c are equal, the height from the other equal side to the opposite vertex is the same as $h_a$: $$h_c = h_a$$

Example Calculation

Let’s calculate the properties of an isosceles triangle where:

• Side a = 5 meters
• Base b = 4 meters

Step-by-step Calculation:

1. Perimeter (P):

   $$P = 2 \times 5 + 4 = 10 + 4 = 14 \text{ meters}$$

2. Semi-perimeter (s):

   $$s = \frac{14}{2} = 7 \text{ meters}$$

3. Area (K):

   $$K = \sqrt{7 \times (7 – 5) \times (7 – 5) \times (7 – 4)} = \sqrt{7 \times 2 \times 2 \times 3} = \sqrt{84} \approx 9.16515 \text{ m}^2$$

4. Height from base (ha):

   $$h_a = \frac{2 \times 9.16515}{5} \approx 3.66606 \text{ meters}$$

5. Height from side b (hb):

   $$h_b = \frac{2 \times 9.16515}{4} \approx 4.58258 \text{ meters}$$

6. Since side a = side c, height hc = ha:

   $$h_c = h_a = 3.66606 \text{ meters}$$

Final Results:

• $a = 5.00000 \, \text{m}$
  
• $b = 4.00000 \, \text{m}$
  
• $c = 5.00000 \, \text{m}$
  
• $P = 14.00000 \, \text{m}$
  
• $s = 7.00000 \, \text{m}$
  
• $K = 9.16515 \, \text{m}^2$
  
• $h_a = 3.66606 \, \text{m}$
  
• $h_b = 4.58258 \, \text{m}$
  
• $h_c = 3.66606 \, \text{m}$