Shaded Region Area Calculator

A shaded region area problem asks for the leftover space after one shape is cut out of (or fit inside) another — the classic “circle inside a square” diagram from geometry class, and several close relatives. This calculator handles five named configurations at once: a circle inscribed in a square, a square inscribed in a circle, a circle inscribed in a rectangle, a stadium shape (a rectangle capped with two semicircle ends, like a running track), and the incircle of a triangle.

Pick the configuration matching your problem, enter the measurements, and the calculator returns the outer shape’s area, the inner shape’s area, the shaded area, and what percentage of the outer shape the shaded region covers.

What Is a “Shaded Region” Problem?

Shaded Area = Outer Shape Area − Inner Shape Area

Most shaded-region problems work the same way: one shape sits inside another, and the “shaded” part is whatever’s left of the outer shape once the inner shape is subtracted out. This calculator also reports that shaded area as a percentage of the outer shape, so you can see at a glance how much of the total space the shaded region actually covers.

Circle Inscribed in a Square

Shaded Area = Side² − π(Side/2)²

Where the circle’s diameter exactly equals the square’s side, so the circle’s radius is half the side length.

Worked Example

A 10-ft square with an inscribed circle: Square Area = 10² = 100 square feet, Circle Area = π × 5² ≈ 78.54 square feet, Shaded Area = 100 − 78.54 ≈ 21.46 square feet — about 21.46% of the square.

Square Inscribed in a Circle

Shaded Area = πr² − 2r²

Where the square’s four corners all touch the circle, making the circle’s diameter equal to the square’s diagonal. This means the square’s area works out to 2r² (side length = r√2).

Worked Example

A circle with a 5-ft radius, with a square inscribed inside it: Circle Area = π × 5² ≈ 78.54 square feet, Square Area = 2 × 5² = 50 square feet, Shaded Area = 78.54 − 50 ≈ 28.54 square feet — about 36.34% of the circle.

Circle Inscribed in a Rectangle

Shaded Area = (Length × Width) − π(d/2)², where d is the rectangle’s shorter side

The largest circle that fits inside a rectangle is limited by the shorter of the two sides, so the circle’s diameter equals whichever of length or width is smaller.

Worked Example

A 10-ft by 6-ft rectangle with an inscribed circle: Rectangle Area = 10 × 6 = 60 square feet. The circle’s diameter is limited by the 6-ft width, so Circle Area = π × 3² ≈ 28.27 square feet, Shaded Area = 60 − 28.27 ≈ 31.73 square feet — about 52.88% of the rectangle.

Stadium Shape (Rectangle + Semicircle Ends)

Total Area = (Length × Width) + π(Width/2)²

Unlike the other modes, this one is additive rather than subtractive — it’s the shape of a running track or a rounded-end swimming pool: a rectangle with a semicircle capping each end, where the two semicircle ends together make one full circle whose diameter equals the rectangle’s width.

Worked Example: A Running Track

A straight section 20 ft long and 8 ft wide, with semicircle ends: Rectangular Part = 20 × 8 = 160 square feet, Semicircle Ends (together, one full circle) = π × 4² ≈ 50.27 square feet, Total Area = 160 + 50.27 ≈ 210.27 square feet.

Incircle of a Triangle

Incircle Radius = Triangle Area ÷ Semi-perimeter, then Incircle Area = πr²

The incircle is the largest circle that fits entirely inside a triangle, tangent to all three sides. Its radius comes from the triangle’s own area and semi-perimeter (computed via Heron’s formula from the three side lengths), not from a separate measurement.

Worked Example

A triangle with sides 5 ft, 6 ft, and 7 ft: Triangle Area (Heron’s formula) ≈ 14.70 square feet, Semi-perimeter = (5 + 6 + 7) ÷ 2 = 9 ft, Incircle Radius = 14.70 ÷ 9 ≈ 1.63 ft, Incircle Area = π × 1.63² ≈ 8.38 square feet, Shaded Area (triangle minus incircle) ≈ 14.70 − 8.38 ≈ 6.32 square feet — about 43.00% of the triangle.

Frequently Asked Questions

How do I find the area of a circle inscribed in a square?

Subtract the circle’s area from the square’s area: Shaded Area = Side² − π(Side/2)². Since the inscribed circle’s diameter equals the square’s side, its radius is half the side length.

What’s the shaded area between a square and its inscribed circle?

It’s the square’s area minus the circle’s area — for a 10-ft square, that’s 100 − 78.54 ≈ 21.46 square feet, or about 21.46% of the square.

How do I find the incircle radius of a triangle?

Divide the triangle’s area (found using Heron’s formula from the three side lengths) by its semi-perimeter: Incircle Radius = Area ÷ Semi-perimeter. A triangle with sides 5, 6, and 7 ft has an incircle radius of about 1.63 ft.

Is the stadium shape mode a subtraction like the other modes?

No — the Stadium mode is additive, not subtractive. It adds a rectangle’s area to the area of its two semicircle end-caps (which together form one full circle), giving the shape’s total area rather than a shaded leftover region.