GED Math: Perimeter, Area, Volume, and Surface Area
To find the volume of a combined solid, split it into the basic solids you know (prisms, cylinders, cones, pyramids, spheres), find each volume, and add them. Be careful with heights: when a cone sits on a cylinder, the total height is the sum of the two heights, and the cone's own height is only its top part. If an object has the same pieces repeated (two cones, four corner blocks), find one piece and multiply. If a piece is removed, subtract its volume. Keep π = 3.14 consistently and round only at the end.
1. Perimeter and Circumference
Perimeter is the distance around a flat figure. Add the lengths of all the sides, and measure it in ordinary linear units (inches, feet, meters). Formulas to memorize: square P = 4s; rectangle P = 2l + 2w; triangle P = a + b + c. For a circle, the distance around is the circumference: C = 2πr = πd, and on the GED you may use π ≈ 3.14. When a side is missing, use what you know about the shape (opposite sides of a rectangle are equal) or work backward from the perimeter with an equation.
Worked Example: A rectangle is 16 inches long and 9 inches wide. Find its perimeter.
P = 2(16) + 2(9) = 32 + 18 = 50 in.
1. A square measures 6 centimeters on one side. What is its perimeter?
Explanation: P = 4s = 4(6) = 24 cm.
2. A rectangle is 16 inches long and 9 inches wide. What is its perimeter?
Explanation: P = 2(16) + 2(9) = 50 in.
3. A triangle has sides of 7.8, 10, and 13. What is its perimeter?
Explanation: 7.8 + 10 + 13 = 30.8.
4. A rectangular yard is 40 ft by 25 ft. How many feet of fence are needed to go all the way around?
Explanation: P = 2(40) + 2(25) = 130 ft.
5. A rectangle has a perimeter of 50 and a length of 16. What is its width?
Explanation: 2(16) + 2w = 50, so 2w = 18 and w = 9.
6. A square has a perimeter of 28 feet. What is the length of one side?
Explanation: s = 28 ÷ 4 = 7 ft.
7. An isosceles triangle has two equal sides of 9 and a perimeter of 25. What is the third side?
Explanation: 9 + 9 + c = 25, so c = 7.
8. What is the circumference of a circle with radius 5? Use π = 3.14.
Explanation: C = 2πr = 2(3.14)(5) = 31.4.
9. A circular track has a diameter of 12 m. What is its circumference? Use π = 3.14.
Explanation: C = πd = 3.14(12) = 37.68 m.
2. Area of Plane Figures
Area is the amount of flat space inside a figure, measured in square units (in2, cm2, ft2). The key formulas: square A = s2; rectangle A = lw; parallelogram A = bh; triangle A = ½bh; trapezoid A = ½h(b1 + b2); circle A = πr2. The base is one side; the height is the perpendicular distance from the base to the opposite vertex or side, forming a right angle with the base. It is not the slanted side. To find a shaded region, find the area of the whole shape and subtract the area of the part that is not shaded.
Worked Example: Find the area of the parallelogram with base 15 and height 6.
A = bh = 15 × 6 = 90 square units. The slanted side (8) is not used.
1. What is the area of the parallelogram shown?
Explanation: A = base × height = 15 × 6 = 90. Ignore the slanted side 8.
2. What is the area of the triangle shown?
Explanation: A = ½(13)(6) = 39.
3. What is the area of the trapezoid shown?
Explanation: A = ½(6)(9 + 17) = 3(26) = 78.
4. The four sides of a rectangle measure 9 ft, 6 ft, 9 ft, and 6 ft. What is its area?
Explanation: Area = 9 × 6 = 54 square feet.
5. A square has a side of 9 inches. What is its area?
Explanation: A = 92 = 81 in2.
6. What is the area of a circle with radius 4? Use π = 3.14.
Explanation: A = πr2 = 3.14(16) = 50.24.
7. What is the area of a circle with a diameter of 10? Use π = 3.14.
Explanation: The radius is 5, so A = 3.14(25) = 78.5.
8. A yard is 20 ft by 15 ft. A patio 10 ft by 6 ft sits on one side, and the rest is grass. What is the area of the grass?
Explanation: Whole yard 20 × 15 = 300; patio 10 × 6 = 60; grass = 300 − 60 = 240 square feet.
9. The area of a triangle is 48 cm2 and its base is 12 cm. What is its height?
Explanation: 48 = ½(12)h = 6h, so h = 8 cm.
10. A rectangle has area 84 and width 7. What is its length?
Explanation: 84 ÷ 7 = 12.
3. Volume of Prisms, Cylinders, and Spheres
Volume (or capacity) is the space inside a three-dimensional object, measured in cubic units (in3, ft3, m3). For any object with two identical, parallel bases (a prism or a cylinder), volume = area of one base × height: V = Bh. Specific formulas: rectangular prism V = lwh; cube V = e3; cylinder V = πr2h; sphere V = 43πr3. Test tip: the cylinder formula is the circle-area formula (πr2) with an h on the end. Use the radius, which is half the diameter, and round only at the end.
Worked Example: A storage tank has a radius of 1.5 m and a height of 3 m. Find its volume to the nearest cubic meter.
V = πr2h = 3.14(1.52)(3) = 3.14(2.25)(3) = 21.195 ≈ 21 m3.
1. What is the volume of the box shown?
Explanation: V = lwh = 5 × 4 × 3 = 60.
2. A wooden crate measures 5 feet along each edge. What is its volume in cubic feet?
Explanation: V = e3 = 53 = 125.
3. A rectangular box has a volume of 80 cubic feet. Its length and width are each 4 ft. What is its height?
Explanation: 80 = 4 × 4 × h = 16h, so h = 5.
4. What is the volume of the tank shown, to the nearest cubic meter? (π = 3.14)
Explanation: V = 3.14(1.5)2(3) = 21.195 ≈ 21.
5. A pool is a cylinder with radius 6 ft and height 3 ft. What is its volume to the nearest cubic foot? (π = 3.14)
Explanation: V = 3.14(36)(3) = 339.12 ≈ 339.
6. The pool instructions say to add one scoop of stabilizer for every 20 cubic feet of water. About how many scoops does the pool need?
Explanation: 339 ÷ 20 ≈ 17 scoops.
7. A cylinder has a diameter of 6 and a length of 5. What is its volume to the nearest whole unit? (π = 3.14)
Explanation: The radius is 3: V = 3.14(9)(5) = 141.3 ≈ 141.
8. What is the volume of a sphere with radius 3, to the nearest whole unit? (π = 3.14)
Explanation: V = 43(3.14)(27) = 113.04 ≈ 113.
9. A cube has a volume of 64 cubic inches. What is the length of one edge?
Explanation: e3 = 64, so e = 4 (since 4 × 4 × 4 = 64).
4. Volume of Pyramids and Cones
A pyramid has a base (on the GED, a square base) and triangular faces that meet at one point (the apex). A pyramid holds exactly one-third of the volume of a prism with the same base and height: V = 13Bh, where B is the area of the base and h is the perpendicular height from the base up to the apex (not the slanted side). A cone is one-third of a cylinder with the same base and height: V = 13πr2h. Test tip: multiply the other factors first, then divide by 3, unless one of the numbers divides evenly by 3.
Worked Example: Find the volume of a square pyramid with base edge 5 cm and height 6 cm.
V = 13(52)(6) = 13(25)(6) = 50 cm3.
1. What is the volume of the pyramid shown?
Explanation: V = ⅓(5)(5)(6) = 50.
2. A square pyramid has a base edge of 12 and a height of 10. What is its volume?
Explanation: V = ⅓(144)(10) = 480.
3. A square pyramid and a square prism have the same base and the same height. The prism holds 300 cubic units. How much does the pyramid hold?
Explanation: A pyramid holds one-third of the prism: 300 ÷ 3 = 100.
4. What is the volume of the cone shown, to the nearest cubic inch? (π = 3.14)
Explanation: V = ⅓(3.14)(22)(9) = 37.68 ≈ 38.
5. A cone has a base radius of 6 inches and a height of 10 inches. What is its volume to the nearest cubic inch? (π = 3.14)
Explanation: V = ⅓(3.14)(36)(10) = 376.8 ≈ 377.
6. The height of a cone equals the radius of its base, and the height is 3 inches. What is the volume to the nearest cubic inch? (π = 3.14)
Explanation: With r = h = 3: V = ⅓(3.14)(9)(3) = 28.26 ≈ 28.
7. A cone and a cylinder have the same radius and the same height. Which statement is true?
Explanation: A cone holds one-third of what a cylinder with the same base and height holds.
8. A square pyramid has a volume of 96 cm3 and a height of 8 cm. What is its base area?
Explanation: 96 = ⅓B(8), so B = 96 × 3 ÷ 8 = 36 cm2.
5. Surface Area of Prisms and Pyramids
Surface area is the total area of the outside faces of a solid, measured in square units. Volume is the space inside; surface area is the covering outside. Cube: six identical squares, so SA = 6s2. Rectangular prism: SA = ph + 2B (perimeter of the base × height, plus two bases), which is the same as 2(lw + lh + wh). Square pyramid: SA = ½ps + B, where p is the perimeter of the base, s is the slant height (from the apex down the middle of a triangular face), and B is the base area. These formulas are on the GED formula sheet, but you should know the cube.
Worked Example: Find the surface area of a cube with 3-inch edges.
SA = 6s2 = 6(9) = 54 in2.
1. What is the surface area of a cube with edges of 3 inches?
Explanation: SA = 6(32) = 54 in2.
2. What is the surface area of the box shown (dimensions 4, 6, and 3)?
Explanation: SA = 2(lw + lh + wh) = 2(24 + 18 + 12) = 108.
3. A box is 5 m by 6 m by 7 m. What is its surface area? (Use ph + 2B.)
Explanation: p = 22 and h = 7 gives 154; 2B = 2(30) = 60; total 214 m2.
4. All the edges of a metal cube are equal. If its surface area is 150 square inches, what is the length of each edge?
Explanation: 6s2 = 150, so s2 = 25 and s = 5.
5. A square pyramid has a base edge of 6 and a slant height of 5. What is its surface area?
Explanation: The base perimeter is 24, so ½(24)(5) = 60; the base area is 36. SA = 60 + 36 = 96.
6. A pyramid has a slant height of 5 ft, a base perimeter of 14 ft, and a base area of 12 ft2. What is its approximate surface area?
Explanation: SA = ½(14)(5) + 12 = 35 + 12 = 47 ft2.
7. Every dimension of a cube with 3-cm edges is doubled. By how many square centimeters does its surface area increase?
Explanation: Original SA = 6(9) = 54. New edge 6, SA = 6(36) = 216. Increase = 216 − 54 = 162.
8. Figure A is a 5 by 5 by 5 cube. Figure B is a 10 by 3 by 3 box. Which statement is true?
Explanation: A: 6(25) = 150. B: 2(30 + 30 + 9) = 138. So A is greater.
6. Surface Area of Cylinders, Cones, and Spheres
These formulas all use π and the radius. Cylinder: SA = 2πrh + 2πr2 (the curved side, which unrolls to a rectangle of height h and width 2πr, plus two circular ends). Cone: SA = πrs + πr2, where s is the slant height (the side length, not the vertical height). Sphere: SA = 4πr2, four times the area of a circle with the same radius. Answers may be left in terms of π (for example, 36π) or computed with π ≈ 3.14. Test tip: if a problem gives the diameter, halve it before using these formulas.
Worked Example: Find the surface area of a cylinder with radius 5 in. and height 8 in. (π = 3.14).
SA = 2πrh + 2πr2 = 2(3.14)(5)(8) + 2(3.14)(25) = 251.2 + 157 = 408.2 in2.
1. What is the surface area of the cylinder shown (radius 5, height 8)? Use π = 3.14.
Explanation: SA = 251.2 + 157 = 408.2.
2. A cylinder has a radius of 7 and a height of 5. What is its surface area in terms of π?
Explanation: 2π(7)(5) + 2π(49) = 70π + 98π = 168π.
3. A cone has a radius of 3 and a slant height of 7. What is its surface area in terms of π?
Explanation: πrs + πr2 = π(3)(7) + π(9) = 21π + 9π = 30π.
4. A cone has radius 2 ft and slant height 5 ft. What is its surface area to the nearest tenth? (π = 3.14)
Explanation: SA = 3.14(2)(5) + 3.14(4) = 31.4 + 12.56 = 43.96.
5. What is the surface area of a sphere with a radius of 10 ft, to the nearest square foot? (π = 3.14)
Explanation: SA = 4(3.14)(100) = 1,256.
6. The diameter of a sphere is 8. What is its surface area in terms of π?
Explanation: The radius is 4: 4π(16) = 64π.
7. In a cylinder the diameter of the base equals the height. If the diameter is 6 inches, what is the surface area to the nearest square inch? (π = 3.14)
Explanation: r = 3, h = 6: 2π(3)(6) + 2π(9) = 36π + 18π = 54π = 169.56 ≈ 170.
8. A sphere has radius 3. What is its surface area to the nearest whole unit? (π = 3.14)
Explanation: 4(3.14)(9) = 113.04 ≈ 113.
7. Combined Figures: Perimeter and Area
A combined figure is made of two or more simple shapes. For perimeter, add the lengths of every outside side; first find any missing lengths by using what you know (opposite sides of a rectangle are equal, and lengths can be added or subtracted). Do not count inside lines. For area, break the figure into parts (rectangles, triangles, half-circles), find each area, then add them; or take a big rectangle and subtract a missing piece. There is usually more than one way to split a figure, and every way gives the same answer. Test tip: for a half-circle, use half of the circumference and half of the area.
Worked Example: Find the area of the house-shaped figure: an 8 by 8 square topped by a triangle with base 8 and height 3.
Square: 8 × 8 = 64. Triangle: ½(8)(3) = 12. Total = 76 square units.
1. In the L-shaped room shown, what is x (the left side)?
Explanation: The left side equals the right side plus the notch drop: x = 9 + 5 = 14.
2. In the same L-shaped room, what is y (the step length)?
Explanation: The top is 20 and the bottom is 12, so y = 20 − 12 = 8.
3. What is the perimeter of the L-shaped room?
Explanation: 20 + 9 + 8 + 5 + 12 + 14 = 68.
4. What is the area of the L-shaped room?
Explanation: Rectangle 20 × 14 = 280 minus the missing corner 8 × 5 = 40 gives 240 (or 20 × 9 + 12 × 5 = 180 + 60).
5. What is the area of the house-shaped figure (8 by 8 square with a triangle of height 3 on top)?
Explanation: 64 + ½(8)(3) = 76.
6. A rectangle 12 long and 10 tall has a half circle (diameter 10) on its left end. What is the total area? (π = 3.14)
Explanation: Rectangle 120 plus half circle ½(3.14)(25) = 39.25, total 159.25.
7. For the rectangle with the half circle (12 by 10, diameter 10 at the left), what is the outside perimeter? (π = 3.14)
Explanation: 12 + 12 + 10 + half the circumference (3.14 × 5 = 15.7) = 49.7.
8. A rectangle is 14 ft by 9 ft. A 4 ft by 3 ft rectangular piece is cut from one corner. What is the area that remains?
Explanation: 14 × 9 = 126; 126 − 12 = 114 ft2.
8. Combined Solids: Volume
To find the volume of a combined solid, split it into the basic solids you know (prisms, cylinders, cones, pyramids, spheres), find each volume, and add them. Be careful with heights: when a cone sits on a cylinder, the total height is the sum of the two heights, and the cone's own height is only its top part. If an object has the same pieces repeated (two cones, four corner blocks), find one piece and multiply. If a piece is removed, subtract its volume. Keep π = 3.14 consistently and round only at the end.
Worked Example: A silo is a cylinder (radius 2 m, height 5 m) with a cone on top (height 3 m). Find its volume.
Cylinder: 3.14(4)(5) = 62.8. Cone: 13(3.14)(4)(3) = 12.56. Total = 75.36 ≈ 75 m3.
1. What is the volume of the silo shown, to the nearest cubic meter? (π = 3.14)
Explanation: 62.8 + 12.56 = 75.36 ≈ 75.
2. A square pyramid (height 8) sits on top of a square prism (base 7 by 7, height 3). What is the total volume to the nearest cubic unit?
Explanation: Prism: 49 × 3 = 147. Pyramid: ⅓(49)(8) = 130.67. Total 277.67 ≈ 278.
3. A concrete slab is 10 ft by 8 ft and 2 ft thick. Four blocks, each 2 ft by 2 ft by 1.5 ft, sit on it. What is the total volume in cubic feet?
Explanation: Slab: 10 × 8 × 2 = 160. Each block: 6, so four blocks = 24. Total = 184.
4. A candy package is a cylinder with a cone on each end. The cylinder has radius 3 in. and length 12 in.; each cone is 4 in. long. What is the capacity to the nearest cubic inch? (π = 3.14)
Explanation: Cylinder: 3.14(9)(12) = 339.12. Each cone: ⅓(3.14)(9)(4) = 37.68, so two = 75.36. Total 414.48 ≈ 414.
5. A square slab 11 by 11 and 2 thick has a cylinder (radius 5, height 2) on top. What is the total volume to the nearest cubic unit? (π = 3.14)
Explanation: Slab: 121 × 2 = 242. Cylinder: 3.14(25)(2) = 157. Total 399.
6. A solid has a hemisphere (half sphere) of radius 3 on a cylinder with radius 3 and height 5. What is its volume to the nearest cubic unit? (π = 3.14)
Explanation: Cylinder: 3.14(9)(5) = 141.3. Hemisphere: ½(113.04) = 56.52. Total 197.82 ≈ 198.
Common Mistakes to Avoid
- Mixing up perimeter (distance around, in linear units) and area (space inside, in square units).
- Using the slanted side as the height of a parallelogram, triangle, or trapezoid. The height is perpendicular to the base.
- Forgetting the 1/2 in the area of a triangle or trapezoid, or the 1/3 in the volume of a pyramid or cone.
- Using the diameter instead of the radius in circle, cylinder, cone, and sphere formulas.
- Squaring the wrong thing: πr2 means r is squared first, not πr.
- Mixing up the vertical height of a cone or pyramid with its slant height.
- Reporting volume in square units or area in cubic units.
- Rounding in the middle of a problem instead of at the end.
Frequently Asked Questions
What is the difference between perimeter, area, and volume?
Perimeter is the distance around a flat figure (linear units), area is the space inside a flat figure (square units), and volume is the space inside a solid (cubic units).
What are the area formulas I should memorize?
Square s2, rectangle lw, parallelogram bh, triangle ½bh, trapezoid ½h(b1 + b2), and circle πr2.
How do I find the volume of a cylinder?
Multiply the area of the circular base by the height: V = πr2h.
Why is the volume of a cone one-third of a cylinder?
A cone holds exactly one-third of a cylinder with the same base and height, so V = ⅓πr2h.
What is surface area?
The total area of all the outside faces of a solid, measured in square units.
How do I find the area or volume of a combined figure?
Break it into simple shapes, find each area or volume, and add (or subtract a removed piece).
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.