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Grade 8 · Mathematics · 35 minutes

Unit 3 · Geometry and bivariate evidence · Lesson 2 of 3

Right Triangles and Round Containers

Apply right-triangle reasoning and distinguish three round-solid volume formulas.

Mr. S Teaches · Original starter mini-lesson · MS8-MATH-05

Learning objective

Find an unknown right-triangle side and calculate cylinder, cone and sphere volumes from stated dimensions.

Before you begin

  • Squares and square roots
  • Circle area

Gather your materials

  • Paper and pencil
  • Ruler and graph paper

Language to know

hypotenuse
The side opposite the right angle.
volume
The space inside a three-dimensional object, measured in cubic units.
cone
A solid with one circular base and a single vertex.

Understand the concept

The Pythagorean relationship a² + b² = c² applies to a right triangle, with c the hypotenuse. It does not use any side as c interchangeably. A cylinder has volume πr²h; a cone with the same base and height has one-third that volume. A sphere has volume 4πr³/3. Radius, not diameter, belongs in these formulas.

Teach it step by step

  1. Sketch a right triangle with legs 6 and 8 cm; label the unknown hypotenuse c.
  2. Square and add the legs, then take the positive square root; check that c exceeds either leg.
  3. Draw a cylinder and cone each with r = 3 cm and h = 4 cm, and calculate their volumes.
  4. Calculate the volume of a sphere of radius 3 cm; compare without assuming that visual shape alone predicts volume.
  5. Require a diagram, substitution and cubic units for each independent measurement.

Make the thinking visible

Worked example or model response

c² = 36 + 64 = 100, so c = 10 cm. Cylinder: π(3²)(4) = 36π cm³. Cone: (1/3)π(3²)(4) = 12π cm³. Sphere: (4/3)π(3³) = 36π cm³. Equal volumes here do not imply identical shapes.

Guided practice

Find a missing leg when the hypotenuse is 13 m and the other leg is 5 m. Find a cylinder’s volume when diameter = 4 cm and height = 7 cm.

Independent practice

1. Find a right triangle’s hypotenuse when its legs are 9 and 12 cm. 2. Find a cone’s volume for r = 2 cm, h = 6 cm. 3. Find a sphere’s volume for r = 2 cm, leaving π in the answer.

Quick assessment

Why is √(13² + 5²) not the missing leg in the guided triangle?

Answer key & assessment criteria

Answers and teaching guidance

  1. Guided: Missing leg √(169−25) = 12 m. Cylinder radius 2 cm, volume 28π cm³.
  2. Independent 1: 15 cm. 2: 8π cm³. 3: 32π/3 cm³.
  3. Quick check: 13 is already the hypotenuse, so the known leg’s square must be subtracted from its square.

Look for: Look for an explanation of the choices, not only a final answer. Require units and a check against the context.

Make it work for your learner

More support

Label the right angle and hypotenuse first; supply formula cards and distinguish radius from diameter.

Extension challenge

Find the height of a cone whose radius is 3 cm and volume is 18π cm³.

Selected contributions · Official sources

Standards connections

Contributes to Common Core standard

8.G.B.7

Common Core · Grade 8

Apply the Pythagorean relationship to missing sides in two-dimensional right triangles.

Read the official standard ↗

This lesson uses two-dimensional examples; proof and three-dimensional applications need further instruction.

Source and mapping checked 2026-11-04. This mini-lesson contributes selected skills; it does not claim full attainment.

Contributes to Common Core standard

8.G.C.9

Common Core · Grade 8

Apply volume formulas for cylinders, cones and spheres.

Read the official standard ↗

This lesson contributes selected skills; it does not fully address the standard.

Source and mapping checked 2026-11-04. This mini-lesson contributes selected skills; it does not claim full attainment.