GMAT Exam: A Simple Guide To Geometry - Perimeter, Area, Surface Area, and Volume

Perimeter
The perimeter of a simple, closed plane figure is the distance around it.
Perimeter is measured in units of length such as inches (in), feet (ft), miles (mi), centimeters (cm), or meters (m).

- The perimeter of a triangle is the sum of its three sides.
The perimeter, P, of a triangle that has sides of lengths 6 inches, 8 inches, and 10 inches is P = 6 in + 8 in + 10 in = 24 inches.

- The perimeter of a quadrilateral is the sum of its four sides.
The perimeter, P, of a trapezoid that has sides of lengths 4 meters, 8 meters, 5 meters, and 10 meters is P = 4 m + 8 m + 5 m + 10 m = 27 meters.

- P = 2a + 2b = 2(a + b), where a is the length of one of the congruent sides and b is the length of one of the congruent bases.
The perimeter of a parallelogram that has congruent sides of length 14 feet and congruent bases of length 23 feet is P = 2(14 ft + 23 ft) = 74 feet.

- P = 2l + 2w = 2(l + w), where l is its length and w is its width.
The perimeter of a rectangle that has length of 8 meters and width of 5 meters is P = 2(8 m + 5 m) = 26 meters.

- P = 4s, where s is the length of one of its sides.
The perimeter of a square whose sides are 12 centimeters is P = 4(12 cm) = 48 centimeters.

- circumference. The circumference of a circle is C = πd = 2πr, where d and r are the circle’s diameter and radius, respectively.
To the nearest inch, the circumference of a circle that has diameter 16 inches is C = π(16 in) ≈ (3.14)(16 in) ≈ 50 inches.

For the GMAT, use π ≈ 3.14. The symbol ≈ is read “is approximately equal to.”

Area
The area of a closed plane figure is the amount of surface enclosed by the boundary of the figure.
Area is measured in square units such as square inches (in2), square feet (ft2), square miles (mi2), square centimeters (cm2), and square meters (m2).
Regardless of the shape of the figure, the area units are always square units.

- b is the length of a base of the triangle, and h is the height for that base.
The area of a triangle that has a base of 8 inches and a height of 6 inches is .

Pick any convenient side of the triangle to serve as the base. The height for the base is the perpendicular distance from the opposite vertex to that base (or an extension of it).

- a is the length of a side.
The area of an equilateral triangle that has sides of length 20 centimeters is .

- b1 and b2 are the lengths of its bases and h is the perpendicular distance between the two bases.

The area of a trapezoid that has bases of 7 meters and 12 meters and a height of 4 meters is .

- A = bh, where b is a base of the parallelogram and h is the perpendicular distance between the two bases.
The area of a parallelogram that has a base of 15 feet and a height of 8 feet is A = (15 ft) × (8 ft) = 120 ft2.

- A = lw, where l is its length and w is its width.
The area of a rectangle that has length of 25 meters and width of 7 meters is A = (25 m) × (7 m) = 175 m2.

- A = s^2, where s is the length of one of its sides.
The area of a square whose sides are 12 centimeters is A = (12 cm)2 = 144 cm2.

- A = πr^2, where r is the circle’s radius.
To the nearest square inch, the area of a circle that has a radius of 8 inches is A = π(8 in)2 = π(64 in2) π (3.14)(64 in2) π 201 in2.

Surface Area
The surface area (SA) of a three-dimensional closed figure is the total area of its surface.

- The surface area of a prism or pyramid is the sum of the areas of all faces of the figure.
The surface area of the right rectangular prism (box) shown below is SA = 2(10 in)(8 in) + 2(10 in)(6 in) + 2(8 in)(6 in) =

160 in^2 + 120 in^2 + 96 in^2 = 376 in^2.

- SA = 2(πr^2) + (2πr)h, where r is the radius of one of the cylinder’s congruent bases and h is the perpendicular distance between the two bases.
The approximate surface area of a right cylinder in which the radius is 5 feet and the height is 14 feet is SA =

2[π(5 ft)^2] + [2π(5 ft)](14 ft) = 2[π(25 ft^2)] + [2π(5 ft)](14 ft) ≈ 2[(3.14)(25 ft2)] + [2(3.14)(5 ft)](14 ft) ≈ 597 ft^2.

- SA = 4πr2, where r is the radius of the sphere.
The approximate surface area of a sphere with radius of 7 feet is SA = 4πr2 =

4π(7 ft)^2 = 4π(49 ft^2) = 4(3.14)(49 ft2) ≈ 615 ft^2.

Volume
Volume is a measure of the space or capacity inside a three-dimensional closed figure.
Volume of geometric solids is measured in cubic units such as cubic inches (in3), cubic feet (ft3), cubic miles (mi3), cubic centimeters (cm3), and cubic meters (m3).

- V = Bh, where B is the area of one of the prism’s or cylinder’s congruent bases and h is the perpendicular distance between the two bases.
The volume of the right rectangular prism (box) shown below is V = Bh = [(l)(w)](h) =

[(10 in)(8 in)](6 in) = 480 in^3.

The approximate volume of a cylinder in which the radius, r, is 5 feet and the height, h, is 14 feet is V = Bh =

[(πr2)](h) = [π(5 ft)2](14 ft) π [(3.14)(25 ft2)](14 ft) = 1,099 ft^3.

- B is the area of the pyramid’s or cone’s base and h is the perpendicular distance from the apex to the base.

The volume of a square pyramid that has a 6-centimeter by 6-centimeter base and a height of 4 centimeters is .

The volume of a cone that has a radius, r, of 10 inches and a height, h, of 9 inches is .

- r is the radius of the sphere.
The approximate volume of sphere with radius of 7 feet is

.