I have 8 slices of pizza in a box, all 8 are triangle shaped, my perimeter is 43.98 Octagon with each edge 5.49 inches. So, I need to know area of circle and area of octagon.

1) The area of the circle pizza is approximately 161.64 in.²

2) The area of the octagon shaped pizza is approximately 145.52 square inches

Step-by-step explanation:

The 8 slices of pizza combine to form the composite octagon

The parameters of the pizza are;

The perimeter of the octagon = 43.98 inches

The length of each (outer) edge of the octagon = 5.49 inches

The outer edge of the octagon = The base of the triangle shaped pizza

Let θ represent the vertex angle of the triangle shaped pizza

Therefore, we have;

θ = (The angle at the center point of the pizza)/8 = 360°/8 = 45°

The vertex angle of the triangle shaped pizza, θ = 45°

We note that a triangle shaped pizza is isosceles shaped, therefore, the base angles are given as follows;

The base angles of a piece of pizza = (180° - (angle at vertex))/2

∴ The base angles of a piece of pizza = (180° - 45°)/2 = 67.5°

The length, 'l', of the other sides of the triangle shaped pizza is given by sine rule as follows;

l/(sin 67.5°) = 5.49/(sin(45°)

∴ l = (sin 67.5°) × 5.49/(sin(45°)

l = (sin 67.5°) × 5.49/(sin(45°) ≈ 7.173

The length of the other sides of the triangle shaped pizza, l ≈ 7.173 inches

1) The length of the other sides of the triangle shaped pizza, l = The radius of the circle, r

The area of the circle = π·l²

∴ The area of the circle ≈ π × (7.173 in.)² ≈ 161.64 in.²

2) The area of each triangle piece of pizza is given as follows;

The area of a triangle, A = 1/2×l·b·sin(C)

∴ A = 1/2× 5.49 × 7.173 × sin(67.5°) ≈ 18.19

The area of each triangle piece of pizza, A ≈ 18.19 inches²

The total area of 8 slices of octagon shaped pizza, TA ≈ 8 × A

∴ TA ≈ 8 × 18.19 inches² ≈ 145.52 inches²

The total area of 8 slices of octagon shaped pizza, TA ≈ 145.52 inches²