Answer:
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
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