G.8(D)·Geometry·Measurement & Three-Dimensional FiguresReadiness

Surface Area and Volume

Calculate surface area and volume for three-dimensional figures in context.

Students calculate surface area and volume of common solids, distinguish lateral from total surface area, and choose the correct radius, perpendicular height, or slant height for each context.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Geometry released practice55 minutes

Learn the concept

Surface area covers the outside

Surface area is the total area of all outside surfaces. It is measured in square units, such as square centimeters. Painting, wrapping, and covering problems use surface area.

Key formulas

For a sphere, $SA=4\pi r^2$ . For a closed cylinder, $SA=2\pi r^2+2\pi rh$ . For a prism, add the areas of all faces, or use $SA=2B+Ph$ , where B is the base area and P is the base perimeter.

Read the context

Convert diameter to radius. Leave out surfaces that are not covered, such as an open top. If several identical objects are involved, multiply one object's surface area by the number of objects.

Volume of prisms and cylinders

Volume measures space inside in cubic units. For any prism, $V=Bh$ , where B is the area of a base and h is the perpendicular distance between bases. A rectangular prism has $V=lwh$ . For a cylinder, $B=\pi r^2$ , so $V=\pi r^2h$ . Convert diameter to radius before substituting.

Volume of pyramids, cones, and spheres

A pyramid has $V=\frac{1}{3}Bh$ , and a cone has $V=\frac{1}{3}\pi r^2h$ . Each has one-third the volume of a prism or cylinder with the same base area and perpendicular height. A sphere has $V=\frac{4}{3}\pi r^3$ . A cone's capacity uses its perpendicular height, not the sloping side length.

Lateral area of a square pyramid

A regular square pyramid has four triangular lateral faces. Their combined area is $LA=\frac{1}{2}P\ell$ , where P is the base perimeter and $\ell$ is the slant height measured along a face from the apex to the midpoint of a base edge. For base side s, $P=4s$ . Lateral area excludes the base; total surface area is $\frac{1}{2}P\ell+s^2$ . The vertical height is not the triangular face's height.

Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Geometry EOC · 2013 · Item 28 · 1 point

An inverted cone. The radius of the circular top is 4.5 centimeters, the dashed perpendicular height from the center of the top to the tip is 6 centimeters, and the slant height along the side is 7.5 centimeters. — Conical paper cupOriginal visual from the released STAAR test.

Archived released STAAR Geometry EOC · 2013 · Item 37 · 1 point

A square pyramid with a dashed vertical height from the apex to the center of the base, a right-angle mark at the base, and a dashed segment from the center of the base to the midpoint of a base edge, showing the slant height along a face. — Square pyramidOriginal visual from the released STAAR test.

Worked example

Three closed cylinders, each with radius 2 inches and height 5 inches, are painted. What total area is painted?

One cylinder: $2\pi(2)^2+2\pi(2)(5)=8\pi+20\pi=28\pi$ .
Three cylinders: $3(28\pi)=84\pi$ .
Approximate: $84\pi\approx 263.9$ square inches.

About 263.9 square inches

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Geometry: Composite Figures Volume and Surface Area

UT IPSI · Kid2Kid / Texas Education Agency

Find surface area and volume for three-dimensional and composite figures in context.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. Find the surface area of a sphere with a diameter of 10 centimeters.
Show answer and reasoning
$100\pi\approx 314.2$ square centimeters
The radius is 5, so $SA=4\pi(5)^2=100\pi$ .
2. Find the surface area of a cube with an edge length of 4 inches.
Show answer and reasoning
96 square inches
Each of the six faces has an area of 16 square inches, and 6(16) = 96.
3. A conical container has radius 3 cm and perpendicular height 10 cm. Find its exact capacity.
Show answer and reasoning
$30\pi$ cubic centimeters
$V=\frac{1}{3}\pi(3)^2(10)=30\pi$ . Use cubic units because capacity is volume.
4. A regular square pyramid has base side 12 m and slant height 10 m. Find its lateral surface area.
Show answer and reasoning
240 square meters
The base perimeter is $4(12)=48$ m, so $LA=\frac{1}{2}(48)(10)=240$ square meters. Do not add the base.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 6 · 1 point

Five spheres are being painted for a display at a store. If the diameter of each sphere is 7 centimeters, which value is closest to the total surface area that will be painted?

770 cm²
154 cm²
192 cm²
440 cm²

Show STAAR answer and explanation

770 cm²

Each radius is 3.5 cm. One sphere has area $4\pi r^2=4\pi(3.5)^2=49\pi$ , so five spheres have $245\pi\approx770$ cm².

154 cm²: $49\pi\approx154$ cm² is the surface area of only one sphere, not all five.

192 cm²: This does not apply $S=4\pi r^2$ with radius 3.5 and then multiply by five.

440 cm²: This underestimates $245\pi$ . Use the radius, not the diameter, in $4\pi r^2$ and multiply by five.

Ask the tutor about this question

Archived released STAAR Geometry EOC · 2013 · Item 28 · 1 point

A conical paper cup is shown in the diagram. Which value is closest to the maximum volume of water this cup can hold?

159 cm³
32 cm³
127 cm³
40 cm³

Show STAAR answer and explanation

127 cm³

The radius is 4.5 cm and perpendicular height is 6 cm. Thus $V=\frac13\pi(4.5)^2(6)=40.5\pi\approx127.2$ cm³.

159 cm³: Using slant height 7.5 instead of perpendicular height 6 gives about 159 cm³.

32 cm³: Halving the given radius incorrectly gives about 32 cm³.

40 cm³: This approximates 40.5 but omits the factor of $\pi$ .

Ask the tutor about this question

Archived released STAAR Geometry EOC · 2013 · Item 37 · 1 point

The main entrance to the Louvre art museum is shaped like a pyramid. The pyramid is 71 feet tall and has a slant height of approximately 91 feet. Each side of the square base measures 115 feet. Which of the following is closest to the lateral surface area of the pyramid?

20,930 ft²
16,330 ft²
10,465 ft²
34,155 ft²

Show STAAR answer and explanation

20,930 ft²

The lateral area is $\frac12Pl=\frac12(4\cdot115)(91)=20{,}930$ ft². Lateral area excludes the square base.

16,330 ft²: This uses perpendicular height 71 rather than slant height 91.

10,465 ft²: This counts only two of the four triangular faces.

34,155 ft²: This adds the base area $115^2$ and gives total surface area, not lateral area.

Ask the tutor about this question

Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Apply the sphere surface-area formula $SA=4\pi r^2$ .
Find surface areas of prisms and cylinders by adding the areas of their faces.
Decide which surfaces count in a context and scale the result for several objects.
Find volumes of prisms, cylinders, pyramids, cones, and spheres, and lateral surface area of regular square pyramids.

Instructional sequence

1
Engage
5 min
Ask whether painting a ball requires surface area or volume and why.
2
Direct instruction
10 min
Connect nets to surface-area formulas and introduce the sphere formula with radius.
3
Guided practice
12 min
Find surface areas for spheres, cylinders, and prisms, converting diameter to radius first.
4
Context problems
12 min
Solve problems with several identical objects or open surfaces, deciding what counts.
5
Volume and pyramid faces
10 min
Compare volume formulas, distinguish perpendicular height from slant height, and find a square pyramid's lateral surface area.
6
Closure
6 min
Complete the exit ticket with square units.

Misconceptions to address

““Use the diameter as r in the formula.””

Teacher move: Divide the diameter by 2 first. Using the diameter makes the surface area four times too large.

““Painting a figure uses its volume.””

Teacher move: Paint covers the outside, so it is measured in square units. Volume measures space inside in cubic units.

““Height means the same segment in every formula.””

Teacher move: Volume uses perpendicular height. A regular pyramid's lateral-area formula uses slant height along a triangular face.

Materials

•Formula chart
•Nets of prisms and cylinders
•Balls or cans for measuring
•Calculator

Teaching tips

Circle the radius on every diagram before substituting.
Ask students to write the units, such as square centimeters, at each step.
For multiple objects, find one surface area first and then multiply.

Exit ticket · 3–5 min

Find the total surface area of 4 balls, each with a radius of 3 inches.

Show expected answer

One ball has $4\pi(3)^2=36\pi$ square inches. Four balls have $144\pi\approx 452.4$ square inches.

Sources & alignment

TEA 2013 Geometry released test TEA 2013 Geometry answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Surface area covers the outside

Key formulas

Read the context

Volume of prisms and cylinders

Volume of pyramids, cones, and spheres

Lateral area of a square pyramid

Study released STAAR visuals

Archived released STAAR Geometry EOC · 2013 · Item 28 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 37 · 1 point

Worked example

Watch the lesson

Geometry: Composite Figures Volume and Surface Area

Try it yourself

1. Find the surface area of a sphere with a diameter of 10 centimeters.

2. Find the surface area of a cube with an edge length of 4 inches.

3. A conical container has radius 3 cm and perpendicular height 10 cm. Find its exact capacity.

4. A regular square pyramid has base side 12 m and slant height 10 m. Find its lateral surface area.

Released STAAR practice

Archived released STAAR Geometry EOC · 2013 · Item 6 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 28 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 37 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Guided practice

Context problems

Volume and pyramid faces

Closure

Misconceptions to address