Area Formulas Explained: Triangles to Circles

Start with the rectangle

Area counts how many unit squares fit inside a shape. A rectangular room measuring 3.6 m by 2.7 m has an area of 3.6 × 2.7 = 9.72 square meters. Its perimeter is 2 × (3.6 + 2.7) = 12.6 m, and by the Pythagorean theorem its diagonal is the square root of (3.6² + 2.7²) = 4.5 m. Area and perimeter measure different things: two rectangles can share a perimeter and still have very different areas.

In feet, a 12 ft by 15 ft bedroom covers 180 square feet. Since one foot is exactly 0.3048 m, one square foot is 0.09290304 square meters, so the room is about 16.72 square meters. Notice that you convert area with the square of the length factor, not the length factor itself.

A parallelogram can be turned into a rectangle by cutting a triangle off one end and sliding it to the other, so its area is base × height. The height here is the perpendicular distance between the two parallel sides, not the length of the slanted side. A parallelogram with a 7 cm base, a 4 cm height and 5 cm slanted sides has an area of 28 square centimeters. Using the slanted side would give 35, which is wrong.

The diagonals of a rhombus cross at right angles, so a rhombus fills exactly half of the rectangle drawn around its diagonals. Its area is p × q ÷ 2, where p and q are the diagonals. With diagonals of 6 cm and 8 cm, the area is 24 square centimeters, and each side is the square root of (3² + 4²) = 5 cm. Because a rhombus is also a parallelogram, side × height works too.

• Rectangle: length × width
• Parallelogram: base × perpendicular height
• Rhombus: diagonal × diagonal ÷ 2

Triangles and trapezoids: where the half comes from

Place a copy of any triangle upside down next to the original and the two form a parallelogram. That is why a triangle's area is base × height ÷ 2. A triangle with an 8 cm base and a 5 cm height covers 8 × 5 ÷ 2 = 20 square centimeters. In an obtuse triangle the height can fall outside the shape; measure it to the extended base line and the formula still holds.

If you know the three sides but not the height, use Heron's formula. Let s be half the perimeter, (a + b + c) ÷ 2. The area is the square root of s(s − a)(s − b)(s − c). For sides 13, 14 and 15, s = 21 and the area is the square root of 21 × 8 × 7 × 6, which is the square root of 7,056, or exactly 84. The three lengths must satisfy the triangle inequality: each pair has to add up to more than the third side. Sides of 3, 4 and 8 cannot form a triangle at all.

A trapezoid works the same way. Join two copies, one flipped, and you get a parallelogram whose base is the sum of the two parallel sides. The area is (a + b) × h ÷ 2. With parallel sides of 4 cm and 10 cm and a height of 6 cm, the area is (4 + 10) × 6 ÷ 2 = 42 square centimeters. The average of the two parallel sides is the length of the midline, so you can also think of it as midline × height.

• Triangle: base × height ÷ 2
• Heron's formula: square root of s(s − a)(s − b)(s − c), with s = (a + b + c) ÷ 2
• Trapezoid: (a + b) × height ÷ 2

Circles, sectors and ellipses

A circle's area is πr². Slice a circle into thin wedges and lay them alternately point-up and point-down: the result looks more and more like a rectangle whose height is the radius r and whose width is half the circumference, πr. Multiply and you get πr². A circle with a 10 cm radius covers about 314.16 square centimeters. Using 3.14 for π gives 314, close enough for a rough estimate but not for precise work.

A sector is a slice of a circle, so its area is the full circle's area times the fraction of 360 degrees it covers. For a 12 cm radius and a 60 degree angle, that is 144π × 60 ÷ 360 = 24π, about 75.40 square centimeters. The arc length is 2π × 12 × 60 ÷ 360 = 4π, about 12.57 cm, so the sector's perimeter, arc plus two radii, is about 36.57 cm. If the angle is in radians, the area formula simplifies to r²θ ÷ 2.

An ellipse is a circle stretched in one direction, and its area is πab, where a and b are the semi-major and semi-minor axes, that is, half the longest and half the shortest width. With a = 5 cm and b = 3 cm, the area is 15π, about 47.12 square centimeters. Plugging in the full widths instead of the halves makes the answer four times too large. Unlike the area, an ellipse's perimeter has no simple exact formula and is usually computed with an approximation.

Here is a practical comparison. One 30 cm pizza (radius 15 cm) has about 706.86 square centimeters of pizza. Two 21 cm pizzas (radius 10.5 cm) have about 346.36 each, or 692.72 together. The pair has the larger total diameter, yet the single pizza is slightly bigger.

• Circle: πr²
• Sector: πr² × angle ÷ 360 (or r²θ ÷ 2 in radians)
• Ellipse: πab, using the semi-axes

Regular polygons: split them into triangles

Draw lines from the center of a regular polygon with n sides to each corner and you get n identical isosceles triangles. Each has a base equal to the side length and a height equal to the apothem, the distance from the center to the middle of a side. Adding them up gives area = perimeter × apothem ÷ 2. In terms of the side length s alone, the area is n × s² ÷ (4 × tan(180° ÷ n)).

A regular hexagon with 4 cm sides has an area of 6 × 16 ÷ (4 × tan 30°), about 41.57 square centimeters. You can check it another way: a hexagon is six equilateral triangles, each with area √3 ÷ 4 × 16, about 6.93, and six of those give the same 41.57. A regular octagon with 2 cm sides covers about 19.31 square centimeters.

There is a neat link back to the circle. Inscribe regular polygons in a circle of radius 1: the hexagon has an area of about 2.598, the 12-gon exactly 3, and the 100-gon about 3.1395. As the number of sides grows, the area closes in on π, about 3.1416.

Units and the most common mistakes

Area units are length units squared. Since 1 m = 100 cm, one square meter is 100 × 100 = 10,000 square centimeters. The 3.6 m by 2.7 m room measured in centimeters is 360 × 270 = 97,200 square centimeters, which is 9.72 square meters after dividing by 10,000. Dividing by 100 instead, and ending up with 972, is a very common slip.

Scaling also works on the square. Double every length of a shape and the area grows four times; triple it and the area grows nine times. A pizza with twice the diameter has four times the area, which is why size upgrades often look generous on the menu.

• Convert all lengths to the same unit before multiplying
• 1 square meter = 10,000 square centimeters; 1 square foot = 0.09290304 square meters
• Use the perpendicular height, not a slanted side
• Use radii for circles and ellipses, not diameters
• If lengths scale by k, area scales by k²